Why convert delta to wye
Some resistor networks cannot be simplified with series and parallel rules alone, such as an unbalanced bridge. The delta-wye (Δ-Y, also called pi-tee or star-mesh) transformation, published by Arthur Kennelly in 1899, replaces three resistors forming a triangle with three resistors forming a star, or the reverse, without changing what the rest of the circuit sees between nodes A, B and C.
Delta to wye
With the delta resistors Rab, Rbc and Rca and their sum S:
- Ra = Rab × Rca ÷ S
- Rb = Rab × Rbc ÷ S
- Rc = Rbc × Rca ÷ S
Each star resistor equals the product of the two delta resistors touching its node, divided by the sum of all three.
Wye to delta
With P = Ra × Rb + Rb × Rc + Rc × Ra:
- Rab = P ÷ Rc
- Rbc = P ÷ Ra
- Rca = P ÷ Rb
Each delta resistor equals the sum of the pairwise products divided by the star resistor on the opposite node.
Example
A delta of Rab = 30 Ω, Rbc = 60 Ω and Rca = 90 Ω (sum 180 Ω):
- Ra = 30 × 90 ÷ 180 = 15 Ω;
- Rb = 30 × 60 ÷ 180 = 10 Ω;
- Rc = 60 × 90 ÷ 180 = 30 Ω.
Converting back: P = 15 × 10 + 10 × 30 + 30 × 15 = 900, so Rab = 900 ÷ 30 = 30 Ω, Rbc = 900 ÷ 15 = 60 Ω and Rca = 900 ÷ 10 = 90 Ω.
Balanced networks
When all three resistors are equal, the rule is simple: R_Y = R_Δ ÷ 3 and R_Δ = 3 × R_Y. This is what you use for balanced three-phase loads.
Uses
- Solving bridge circuits, such as an unbalanced Wheatstone bridge, by hand.
- Three-phase power: converting a delta-connected motor or heater into its per-phase star equivalent.
- Star-delta starting of induction motors: starting in star reduces the phase voltage by √3 and the starting current and torque to one third.
- Filter and attenuator design (pi and tee networks).
The same formulas work with complex impedances for AC circuits, but this calculator handles real resistances only.