volt-second-balance-three-row-scan - Electronic Circuits XIV: Inductors and Switching Energy Step by Step

Balanced and unbalanced switching intervals are separated by exact sums. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Balanced volt-seconds make the ripple return

Each row uses one positive interval and one negative interval on the same inductance.

\sum V_L\Delta t=0\quad\Rightarrow\quad \sum\Delta I=0\ \text{A}

The balance gate separates zero and nonzero sums

Two rows cancel exactly. The third row keeps the same off voltage but too little off time, so it fails.

\begin{array}{c|c|c|c}V_{\text{on}}t_{\text{on}}&V_{\text{off}}t_{\text{off}}&\sum Vt&\text{gate}\\9\ \text{V}\cdot\text{s}&-9\ \text{V}\cdot\text{s}&0\ \text{V}\cdot\text{s}&\text{pass}\\6\ \text{V}\cdot\text{s}&-6\ \text{V}\cdot\text{s}&0\ \text{V}\cdot\text{s}&\text{pass}\\9\ \text{V}\cdot\text{s}&-6\ \text{V}\cdot\text{s}&3\ \text{V}\cdot\text{s}&\text{fail}\\\end{array}

The accepted balance row cites both interval ramps

The balance object is downstream of the on-ramp and off-ramp checks, so the pass bit follows the two signed areas.

9\ \text{V}\cdot\text{s}+\left(-9\ \text{V}\cdot\text{s}\right)=0\ \text{V}\cdot\text{s}