What is the delta to wye conversion formula?

For a delta network with resistors Rab, Rbc and Rca, the equivalent wye resistances are: Ra = (Rab x Rca) / (Rab + Rbc + Rca), Rb = (Rab x Rbc) / (Rab + Rbc + Rca), and Rc = (Rbc x Rca) / (Rab + Rbc + Rca). Each wye resistor equals the product of the two adjacent delta resistors divided by the sum of all three delta resistors. If all three delta resistors are equal to R, each wye resistor equals R divided by 3.

Why convert between delta and wye configurations?

Delta (Δ) and wye (Y, also called star) are two ways of connecting three resistors, transformers, or three-phase loads between three terminals. Some circuits are difficult to simplify with standard series-parallel rules because the resistors are neither purely in series nor purely in parallel, such as bridge networks. Converting a delta section to its wye equivalent (or vice versa) lets you combine the transformed resistor with the rest of the circuit using ordinary series and parallel rules. It is also used in three-phase power systems to convert between delta-connected and star-connected loads or transformer windings.

Delta to Wye Conversion Calculator

Convert a delta (Δ) connected three-resistor network into its equivalent wye (Y, star) network, or convert a wye network into its equivalent delta. This uses the standard Y-Δ transform used in circuit analysis, bridge networks, and three-phase power systems.

Enter your three known resistances, choose the direction of conversion, and the equivalent values update instantly.

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Verified formula Standard Y-Δ (star-delta) transform used in electrical engineering and circuit theory.

1. Conversion Direction

Convert

In a delta network, resistors Rab, Rbc and Rca connect directly between each pair of the three terminals (a, b, c). In a wye network, resistors Ra, Rb and Rc each connect from one terminal to a common centre point.

2. Delta (Δ) Resistances

Rab (between terminal a and b)

Rbc (between terminal b and c)

Rca (between terminal c and a)

2. Wye (Y) Resistances

Ra (terminal a to centre)

Rb (terminal b to centre)

Rc (terminal c to centre)

Equivalent Wye (Y) Resistances

Ra
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Ohms

Rb
-
Ohms

Rc
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Ohms

Working

Input network-

Sum used in formula-

Output network-

Formula Applied

- - -

Result: Enter your resistor values above.

What Is the Delta-Wye (Y-Δ) Transform?

Delta (Δ) and wye (Y, also called star) are the two basic ways to connect three resistors, three-phase transformer windings, or three-phase loads across three terminals. In a delta configuration, a resistor connects directly between each pair of terminals, forming a triangle. In a wye configuration, a resistor runs from each terminal to a shared common (neutral) point, forming a star shape. The delta-wye transform, also called the Y-Δ transform or star-delta transform, lets you replace one configuration with an electrically equivalent version of the other, without changing the behaviour of the circuit as seen from the three terminals.

This is useful because some resistor networks, particularly bridge circuits, cannot be simplified using ordinary series and parallel combination rules alone. Converting one delta or wye section into its equivalent form often turns an awkward network into one that can be reduced with simple series-parallel steps.

Delta to Wye Formula

Given delta resistances Rab, Rbc and Rca, the equivalent wye resistances are:

Wye Resistor	Formula
Ra	(Rab × Rca) / (Rab + Rbc + Rca)
Rb	(Rab × Rbc) / (Rab + Rbc + Rca)
Rc	(Rbc × Rca) / (Rab + Rbc + Rca)

Each wye resistor is the product of the two delta resistors touching that terminal, divided by the sum of all three delta resistors. If all three delta resistors are equal to R, each wye resistor works out to R / 3.

Wye to Delta Formula

Given wye resistances Ra, Rb and Rc, the equivalent delta resistances are:

Delta Resistor	Formula
Rab	(RaRb + RbRc + RcRa) / Rc
Rbc	(RaRb + RbRc + RcRa) / Ra
Rca	(RaRb + RbRc + RcRa) / Rb

You first calculate the sum of the three pairwise products of the wye resistors, then divide that sum by the wye resistor "opposite" the delta resistor you are solving for. If all three wye resistors are equal to R, each delta resistor works out to 3R.

Where This Is Used

Circuit analysis: simplifying bridge networks and other non-series-parallel resistor arrangements.
Three-phase power: converting between delta-connected and star (wye) connected loads, motor windings, and transformer windings.
Power systems studies: fault analysis and load-flow calculations often require converting parts of a network between delta and wye form.
Filter and attenuator design: resistive pad and attenuator networks are sometimes specified in one form and need converting to the other.

Worked Example

Take a delta network with Rab = 30 Ω, Rbc = 60 Ω and Rca = 90 Ω. The sum is 30 + 60 + 90 = 180 Ω. The equivalent wye resistances are: Ra = (30 × 90) / 180 = 15 Ω, Rb = (30 × 60) / 180 = 10 Ω, and Rc = (60 × 90) / 180 = 30 Ω.

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Sources: Standard Y-Δ (star-delta) transform equations as used in circuit theory and power systems engineering (Boylestad, Introductory Circuit Analysis; IEEE three-phase transformer connection references).

This calculator applies the standard delta-wye transform formula to the resistor values you enter. It assumes ideal, purely resistive elements. For AC circuits with reactive components, the same transform applies using complex impedances instead of plain resistances.