opposed-dot-polarity-subtracts-voltage - Electronic Circuits XV & XVI: Transformers and Switched-Capacitor Conversion Step by Step

An opposed-dot-polarity-subtracts-voltage scan applies the dot sign rule to two cited winding voltages, checking that aiding windings add V_out = V_a + V_b while opposing windings subtract V_out = V_a - V_b row by row. The arithmetic is exact only for the stated winding voltages, dot polarities, output voltages, and rows in the ideal shared-flux, perfectly coupled transformer model. Real transformers add leakage flux, winding resistance, core loss, saturation, dot-marking/wiring error, probe/calibration limits, finite precision, and measurement uncertainty.

Dot convention is a source-bound sign rule in this finite model. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Dot polarity is a sign rule on cited winding voltages

The polarity row must cite two distinct winding sources before it can add or subtract them.

V_{\text{out}}=V_a\pm V_b

Aiding adds and opposing subtracts

Equal opposing windings cancel. Equal aiding windings add. Unequal opposing windings leave the difference.

\begin{array}{c|c|c|c}V_a&V_b&\text{mode}&V_{\text{out}}\\4\ \text{V}&4\ \text{V}&\text{opposing}&0\ \text{V}\\4\ \text{V}&4\ \text{V}&\text{aiding}&8\ \text{V}\\6\ \text{V}&4\ \text{V}&\text{opposing}&2\ \text{V}\\\end{array}

Opposed dot polarity subtracts equal winding voltages

The polarity row cites two distinct winding sources; it cannot cite the same voltage source twice.

4\ \text{V}+(-4\ \text{V})=0\ \text{V}\quad\text{but aiding gives}\quad8\ \text{V}