ramp-energy-matches-source-work - Electronic Circuits XIV: Inductors and Switching Energy Step by Step

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}