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Delta-to-Wye Transformation

Converts a delta-connected resistor network into an equivalent wye (star) network.

Written by the Vorzep Team

Circuit AnalysisNetwork Transformation

Delta-Wye Transformation Calculator Δ-Y Resistor Networks

Ra = (Rab·Rca) / (Rab+Rbc+Rca)
Δ (delta)  ↔  Y (wye)  ·  Equivalent resistor networks
⟹ Convert ⟷ Δ → Y  |  Y → Δ
Ω
Ω
Ω
Ω
Ω
Ω
Please fix the errors above.
Convert:
Presets:
Converted Values —
—
Rab: — Rbc: — Rca: — Ra: — Rb: — Rc: —
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Equivalent Resistance —
Low (< 50 Ω) Medium (50–200 Ω) High (> 200 Ω)
Δ→Y: Ra = (Rab·Rca)/Σ, Rb = (Rab·Rbc)/Σ, Rc = (Rbc·Rca)/Σ  ·  Σ = Rab+Rbc+Rca

Variables

SymbolQuantityUnit
RaWye (star) resistance connected to terminal aΩ
RabDelta resistance between terminals a and bΩ
RbcDelta resistance between terminals b and cΩ
RcaDelta resistance between terminals c and aΩ

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