sparkytools

Third year Three-phase · tool

Delta ↔ wye impedance conversion

Enter three impedances in ohms. Delta to wye gives you the three star arms that behave the same between the corners; wye to delta goes the other way. Balanced or not.

A B C Zab Zca Zbc delta A B C Za Zb Zc wye
Convert
  • Za—
  • Zb—
  • Zc—
Show the working

    All three values must be the same kind (resistance, or reactance of one sign). Equivalent only as seen from the three corners.

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    How it works

    • Za = Zab × Zca ÷ ΣΔeach wye arm: the two delta sides touching its corner, over the sum
    • Zab = ΣP ÷ Zceach delta side: sum of pair products (ZaZb + ZbZc + ZcZa), over the opposite arm
    • Balanced: ZY = ZΔ ÷ 3and ZΔ = 3 × ZY

    Worked: 10, 20 and 30 Ω in delta

    1. Sum of the delta: 10 + 20 + 30 = 60 Ω.
    2. Za (corner A touches Zab and Zca) = 10 × 30 ÷ 60 = 5 Ω.
    3. Zb (touches Zab and Zbc) = 10 × 20 ÷ 60 = 3.33 Ω.
    4. Zc (touches Zbc and Zca) = 20 × 30 ÷ 60 = 10 Ω.
    5. Check between A and B: in the delta it is 10 Ω in parallel with (20 + 30) = 8.33 Ω; in the wye it is Za + Zb = 5 + 3.33 = 8.33 Ω. Same.

    Questions people ask

    What is a delta-wye (or pi-tee) conversion for?

    It lets you swap three impedances joined in a triangle for three joined in a star (or back) without changing anything measured between the three corners. You use it to untangle a network that is neither series nor parallel — a bridge circuit, a three-phase load you want to treat per phase, or a fault-current study that mixes delta and wye pieces.

    Why is a balanced wye one-third of the delta?

    When all three are equal, each wye arm works out to Z × Z ÷ 3Z = Z ÷ 3. That is why a balanced delta load can be treated as a wye of one-third the impedance for per-phase arithmetic, using line-to-neutral voltage.

    Can I use it for reactances or full impedances?

    Yes, as long as all three are the same kind — all resistances, all inductive reactances, or all capacitive reactances. For a mix of resistance and reactance the same formulas apply with complex numbers, which this tool does not do.

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