Delta-Wye (Δ-Y) Conversion Calculator
Convert a delta (Δ, triangle) network of three resistors to its equivalent wye (Y, star) network or back, with the Kennelly formulas and a terminal-resistance check.
How it works
Δ → Y: Ra = Rab·Rca ÷ (Rab + Rbc + Rca), Rb = Rab·Rbc ÷ Σ, Rc = Rbc·Rca ÷ Σ · Y → Δ: Rab = (Ra·Rb + Rb·Rc + Rc·Ra) ÷ Rc, Rbc = (…) ÷ Ra, Rca = (…) ÷ Rb · balanced: R_Y = R_Δ ÷ 3
Some resistor networks — bridges, three-phase loads — have no series or parallel pairs to combine. The delta-wye (star-triangle) transformation, published by A. E. Kennelly in 1899, replaces three resistors connected in a triangle between nodes A, B and C with three resistors meeting at a centre point (or vice versa) without changing any resistance seen between the terminals. Each wye resistor equals the product of the two delta resistors touching its node divided by the sum of all three. The same formulas hold for impedances in AC circuits and for balanced three-phase loads.
Worked example
A delta with Rab = 10 Ω, Rbc = 20 Ω and Rca = 30 Ω (sum 60 Ω) becomes a wye with Ra = 10 × 30 ÷ 60 = 5 Ω, Rb = 10 × 20 ÷ 60 = 3.333 Ω and Rc = 20 × 30 ÷ 60 = 10 Ω. Check between A and B: delta gives 10 ∥ 50 = 8.333 Ω, wye gives 5 + 3.333 = 8.333 Ω.
Sources: A. E. Kennelly, The equivalence of triangles and three-pointed stars in conducting networks, Electrical World and Engineer 34 (1899) · Y-Δ transform — Wikipedia
Frequently asked questions
When do I need a delta-wye conversion?
When a circuit contains a triangle (or star) of resistors that cannot be reduced by series and parallel rules — the classic example is an unbalanced Wheatstone bridge. Converting one triangle into a star usually leaves only series-parallel combinations.
What is the shortcut for equal resistors?
For a balanced network, R_Y = R_Δ ÷ 3 and R_Δ = 3 × R_Y. Three 30 Ω resistors in delta behave like three 10 Ω resistors in star — the same rule used for balanced three-phase loads.
Is the star-delta transformation the same as star-delta motor starting?
They share the geometry but not the purpose. A star-delta starter reconnects the motor windings from star to delta to cut starting current to one third; the transformation here is a circuit-analysis identity that gives equivalent resistances.
Does it work with capacitors or inductors?
Yes, with complex impedances in place of resistances (Z = R + jX). For pure capacitors use 1/C in place of R, because capacitive impedance is inversely proportional to capacitance.
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This calculator is for educational purposes. Double-check important results before acting on them.