ramp-energy-matches-source-work - Electronic Circuits XIII & XIV: Power Supplies and Switching Energy Step by Step

A ramp-energy-matches-source-work scan checks each linear current ramp so the source work W = V * I_avg * delta-t equals the stored-energy increase delta-E row by row. The arithmetic is exact only for the stated voltages, currents, times, work, energy changes, and rows in the ideal lossless inductor and lossless switching model. Real inductors add winding resistance, core saturation, core loss, switch loss, current-sense/probe calibration limits, finite precision, and measurement uncertainty.

A linear ramp must close its source-work and energy ledgers. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Ramp work uses the average ramp current

The work check cites the ramp endpoints and the stored-energy delta. The average current is not a free label.

W=V\left({I_i+I_f\over 2}\right)\Delta t

Each ramp row closes work against energy increase

Every row is built from a matching ramp and energy endpoint, so source work equals the energy increase.

\begin{array}{c|c|c|c|c}V&I_{\text{avg}}&\Delta t&W&\Delta E\\2\ \text{V}&\tfrac{1}{2}\ \text{A}&1\ \text{s}&1\ \text{J}&1\ \text{J}\\4\ \text{V}&1\ \text{A}&1\ \text{s}&4\ \text{J}&4\ \text{J}\\6\ \text{V}&\tfrac{3}{2}\ \text{A}&1\ \text{s}&9\ \text{J}&9\ \text{J}\\\end{array}

The ramp source work matches the energy increase

The average current is derived from the ramp endpoints, then source work is checked against the stored-energy increase.

W=V I_{\text{avg}}\Delta t=6\ \text{V}\cdot\tfrac{3}{2}\ \text{A}\cdot1\ \text{s}=9\ \text{J}