volt-second-balance-three-row-scan - Electronic Circuits XIII & XIV: Power Supplies and Switching Energy Step by Step

Inductor volt-second balance says the signed on and off voltage-time areas must cancel if the ripple returns to its starting current each cycle. The rows use exact ideal switching intervals; real converters include winding resistance, switch drops, dead time, core loss, ripple measurement limits, and timing uncertainty.

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}