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Delta-to-Wye Transformation

Sometimes resistors are wired in a triangle, called delta, and sometimes in a star shape, called wye. The two arrangements look different but can behave in exactly the same way. This transformation lets you swap one for the other so the circuit becomes easier to solve. Each wye resistor equals the product of the two delta resistors touching the same node, divided by the sum of all three.

Written by the Vorzep Team

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Delta-Wye (Δ-Y) Transformation Calculator

Convert a delta resistor network to its equivalent wye (star) network, or back again. See both networks drawn with your values, and verify the equivalence from every pair of terminals.

Ra = (Rab·Rca) / (Rab+Rbc+Rca)Rab = (RaRb+RbRc+RcRa) / Rc
Direction
Resistor unit
Delta network (input)
Ra-
Rb-
Rc-
SUM-

Amber = the network you entered. Teal = the equivalent network. Terminals a, b and c are the same in both.

Terminal-pair verification

PairDelta viewWye viewDifference

Network facts

The two transformations

Label the terminals a, b, c. Rab sits between a and b, Rbc between b and c, Rca between c and a. In the wye, Ra, Rb, Rc run from each terminal to the centre point. Both networks are equivalent when every pair of terminals sees the same resistance.

Delta to Wye: product of adjacent / sum

Ra = Rab·Rca / (Rab+Rbc+Rca)
Rb = Rab·Rbc / (Rab+Rbc+Rca)
Rc = Rbc·Rca / (Rab+Rbc+Rca)

Wye to Delta: sum of pair products / opposite

S = RaRb + RbRc + RcRa
Rab = S / Rc
Rbc = S / Ra
Rca = S / Rb

Worked example

Delta: Rab = 60, Rbc = 30, Rca = 90 ohm
Sum = 60 + 30 + 90 = 180
Ra = 60·90 / 180 = 30 ohm | Rb = 60·30 / 180 = 10 ohm | Rc = 30·90 / 180 = 15 ohm
Check a-b: 60 || (30+90) = 40 ohm, and Ra+Rb = 40 ohm

Balanced case: if all three delta resistors equal R, every wye resistor equals R/3 (and in reverse, delta = 3 × wye).

FAQ

When a network has neither series nor parallel resistors you can simplify directly, such as a bridge circuit. Replacing a delta with a wye (or the reverse) often creates series/parallel pairs.

Delta to Wye: each wye resistor is the product of the two delta resistors touching its terminal, divided by the sum of all three. Wye to Delta: each delta resistor is the sum of pair products divided by the wye resistor opposite it.

Yes. The same formulas hold with complex impedances Z in place of R, as long as the circuit is linear and in steady state.

Variables

Symbol Quantity Unit
Ra Wye (star) resistance connected to terminal a Ω
Rab Delta resistance between terminals a and b Ω
Rbc Delta resistance between terminals b and c Ω
Rca Delta resistance between terminals c and a Ω

What it means

Delta‑to‑wye (Δ‑Y) transformation is used to simplify a three‑resistor network from a delta (triangle) configuration to an equivalent wye (star) configuration. For a delta with resistors R_ab (between a and b), R_bc, and R_ca, the corresponding wye resistors are: R_a = (R_ab * R_ca) / (R_ab + R_bc + R_ca), R_b = (R_ab * R_bc) / (R_ab + R_bc + R_ca), R_c = (R_bc * R_ca) / (R_ab + R_bc + R_ca). This transformation allows the application of series‑parallel reduction to networks that would otherwise be irreducible. It is widely used in power systems analysis, bridge circuits, and three‑phase networks. The inverse (wye‑to‑delta) also exists. Example: Δ resistors R_ab=10Ω, R_bc=20Ω, R_ca=30Ω. Sum = 60Ω. Then R_a = (10*30)/60 = 5Ω, R_b = (10*20)/60 = 3.33Ω, R_c = (20*30)/60 = 10Ω. The wye equivalent is 5Ω, 3.33Ω, 10Ω.

Worked example

Delta‑to‑Wye Transformation – Practical Example

Real‑World
Scenario: A delta network has Rab = 10 Ω, Rbc = 15 Ω, Rca = 20 Ω. Convert it to a wye (star) network.
ParameterValue
Rab10 Ω
Rbc15 Ω
Rca20 Ω
FormulaRa = (Rab·Rca) / (Rab+Rbc+Rca)
1Sum of delta: 10+15+20 = 45 Ω
2Compute Ra = (10·20)/45 = 4.44 Ω
3Compute Rb = (10·15)/45 = 3.33 Ω
4Compute Rc = (15·20)/45 = 6.67 Ω
5Final wye values:Ra≈4.44Ω, Rb≈3.33Ω, Rc≈6.67Ω
Final Design Wye: 4.44Ω, 3.33Ω, 6.67Ω ✓ Equivalent
Why: The transformation preserves the terminal behavior – useful for simplifying bridge circuits.

Common mistakes

  • Mapping: R_a is the wye resistor connected to node a – it is the product of the two delta resistors adjacent to node a divided by the sum of all three delta resistors.
  • Symmetry: If the delta is balanced (all resistors equal R), the wye resistors are R/3.
  • Wye‑to‑Delta: The reverse transformation exists: R_ab = R_a + R_b + (R_a·R_b)/R_c, etc.
  • Units: All resistances in ohms – the result is in ohms.
  • Useful for: Simplifying bridge circuits and star‑delta networks.

Applications

Delta‑to‑wye transformation converts a three‑resistor delta (Δ) network into an equivalent wye (Y) network, and vice versa. This transformation simplifies the analysis of three‑phase networks, bridge circuits, and complex resistor configurations. Electrical engineers use it to simplify circuits where resistors are connected in a delta shape, making series‑parallel reductions easier. In power systems, it helps in analysing transformer connections and fault currents. By using this transformation, professionals can reduce complex networks to simpler equivalent circuits, enabling easier calculation of currents and voltages. This technique is essential for solving unbalanced three‑phase circuits and for understanding network topology.

  • Simplification of three‑phase resistor networks
  • Analysis of bridge circuits (e.g., Wheatstone bridge)
  • Transformer and motor winding connections
  • Circuit reduction for fault current calculations
  • Educational tool for network transformations

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  • Published July 10, 2026
  • Last updated September 12, 2026