Delta (Δ) ⇄ Wye (Y) Conversion Calculator

Understanding the delta–wye conversion calculator — concept and principle

Some resistor networks cannot be reduced with series and parallel rules alone, such as a bridge with no balance. The delta–wye (Δ–Y, also called Π–T) transformation replaces three resistors connected in a triangle with three connected in a star, or the reverse, without changing what the rest of the circuit sees at the three terminals. It is used in network analysis, three-phase power calculations and filter design.

How it is calculated

Δ → Y, with ΣR = Rab + Rbc + Rca:

• Ra = Rab · Rca / ΣR
• Rb = Rab · Rbc / ΣR
• Rc = Rbc · Rca / ΣR

Y → Δ, with P = Ra·Rb + Rb·Rc + Rc·Ra:

• Rab = P / Rc
• Rbc = P / Ra
• Rca = P / Rb

In the delta, Rab connects terminals a and b, and so on. In the wye, Ra connects terminal a to the centre point. Enter all values in ohms in the fields shown for the chosen direction.

The principle

Two networks are equivalent if every pair of terminals shows the same resistance when the third terminal is left open. Between a and b, the wye gives Ra + Rb, while the delta gives Rab in parallel with (Rbc + Rca). Writing this for all three pairs and solving gives the formulas above. A simple memory rule: each wye resistor is the product of the two delta resistors touching its terminal, divided by the sum of all three. Each delta resistor is the sum of the pairwise wye products divided by the opposite wye resistor. Arthur Kennelly published the transformation in 1899.

Worked example

1. Inputs: Rab = 30 Ω, Rbc = 60 Ω, Rca = 90 Ω, converting Δ → Y.
2. ΣR = 30 + 60 + 90 = 180 Ω.
3. Ra = 30 × 90 / 180 = 2,700 / 180 = 15 Ω.
4. Rb = 30 × 60 / 180 = 1,800 / 180 = 10 Ω.
5. Rc = 60 × 90 / 180 = 5,400 / 180 = 30 Ω.
6. Check between a and b: wye 15 + 10 = 25 Ω; delta 30 ∥ (60 + 90) = 30 × 150 / 180 = 25 Ω.

How to read the result

The three converted values replace the original three exactly at the terminals, so the rest of the circuit is unaffected. For a balanced network, the rule simplifies: R_Y = R_Δ / 3 and R_Δ = 3 · R_Y. This factor of 3 is the same one that appears in balanced three-phase loads.

Common mistakes

• Pairing the wrong resistors: each wye arm uses the two delta resistors that touch its terminal.
• Dividing by the wrong resistor in Y → Δ; use the one opposite the delta branch.
• Expecting internal currents to be the same; only the behaviour at the terminals is preserved.

Frequently asked questions

When do I need a delta–wye transformation?

When a network has a triangle or star that blocks series–parallel reduction, such as an unbalanced bridge.

Does it work for capacitors and inductors?

Yes, with complex impedances at one frequency.

Why is it also called Π–T?

A delta drawn with one side flat looks like the letter Π, and a wye looks like a T.

Sources: A. E. Kennelly, "The equivalence of triangles and three-pointed stars in conducting networks", Electrical World and Engineer 34 (1899) · C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, McGraw-Hill