Delta–Wye (Y) Conversion

The delta-wye transformation emerged from two parallel traditions. In 1899 Arthur Edwin Kennelly (MIT) published the resistor-network version as a technique for simplifying bridge and ladder circuits. In three-phase power, the same topology (triangle vs star) described the two ways to connect generator or transformer windings — and converting between the two became essential for fault analysis when a Δ-connected source feeds a Y-connected load (or vice versa).

Graph-theoretically, Δ-Y is a local transformation that preserves the two-port behavior of a circuit at its three terminals. It's one of a small set of "network rewrites" that let complex graphs be reduced to simpler canonical forms. Paired with series/parallel reduction, most resistor networks collapse to a single resistance between any two terminals — a fact Gustav Kirchhoff proved in his 1847 paper that founded spectral graph theory.

Modern three-phase engineering still uses Δ-Y conversion constantly. Unbalanced-load analysis (one phase drawing more current than the others) is much easier with all elements in Y form — each phase becomes an independent circuit with a shared neutral. Δ-connected transformer windings, which carry phase currents rather than line currents, require conversion to analyze primary vs secondary quantities consistently.

Pick conversion direction, enter three resistor values (either the three delta legs Rab, Rbc, Rca, or the three star resistors R1, R2, R3). Delta-to-Wye: each Y resistor equals the product of its two adjacent Δ legs divided by the sum of all three. Wye-to-Delta: each Δ leg equals the sum of all pairwise Y products divided by the Y leg opposite to it.

A key special case: if all three Δ resistors are equal (R_Δ), the equivalent star has R_Y = R_Δ/3. Conversely, three equal Y resistors give R_Δ = 3·R_Y. This is why Δ-connected motor windings have one-third the per-winding impedance of Y-connected windings for the same terminal characteristics. Everything runs client-side; no values leave your browser.