Delta to Wye Conversion Calculator
This calculator converts a delta (Δ) connected three-resistor network into its electrically equivalent wye (Y, or star) network, and can also convert the other way, from wye back to delta. Electrical work often runs into resistor arrangements, such as bridge circuits, three-phase transformer windings or motor loads, that cannot be reduced with ordinary series and parallel rules because the resistors are neither purely in series nor purely in parallel. Swapping a delta section for its wye equivalent, or vice versa, lets you carry on simplifying the rest of the circuit without changing how the network behaves at its three terminals. Choose your conversion direction, then enter either the three delta resistances (Rab, Rbc and Rca) or the three wye resistances (Ra, Rb and Rc) in ohms. The calculator instantly applies the standard Y-Δ transform formulas and returns the three equivalent resistor values, along with a working panel showing the sum used in the calculation, the formula applied to each output, and a plain-English verdict summarising the conversion. It suits circuit analysis, three-phase power system studies, and filter or attenuator design where a network needs converting between the two forms. The tool assumes ideal, purely resistive components; for AC circuits with reactive elements, the same transform applies using complex impedances instead, so treat results here as indicative for resistive networks only.
1. Conversion Direction
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
Working
Formula Applied
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.