Wye ↔ Delta Calculator

What is a Wye-Delta Transformation?

The Wye-Delta (also written Y-Δ or star-triangle) transformation is a mathematical technique used to convert a three-terminal resistor network between two equivalent topologies: the Wye (Y) configuration and the Delta (Δ) configuration. The two networks are electrically equivalent when their terminal impedances are equal.

This transformation is widely used in circuit analysis, three-phase AC power systems, and filter design to simplify networks that cannot be reduced with simple series or parallel combinations.

Delta to Wye Formulas

Given a Delta network with resistors Ra (between nodes 1-2), Rb (between nodes 2-3), and Rc (between nodes 1-3), the equivalent Wye resistors are:

R1 = (Ra × Rb) ÷ (Ra + Rb + Rc)

R2 = (Rb × Rc) ÷ (Ra + Rb + Rc)

R3 = (Ra × Rc) ÷ (Ra + Rb + Rc)

Wye to Delta Formulas

Given a Wye network with resistors R1, R2, R3 (each connected to the common centre node), the equivalent Delta resistors are:

Ra = (R1R2 + R2R3 + R1R3) ÷ R3

Rb = (R1R2 + R2R3 + R1R3) ÷ R1

Rc = (R1R2 + R2R3 + R1R3) ÷ R2

Balanced Networks

When all three resistors are equal (a balanced network), the conversion is simple: a balanced Delta with resistance R converts to a balanced Wye with resistance R ÷ 3, and vice versa. For example, three 30 Ω resistors in a Delta are equivalent to three 10 Ω resistors in a Wye.

Where is this Used?

Wye-Delta transformations appear in three-phase power distribution (star vs. triangle motor winding connections), bridge circuit analysis (Wheatstone bridge reduction), passive filter design, and simplification of multi-loop resistive networks where Kirchhoff's laws alone become unwieldy.