Star-Delta Conversion Formulas for Resistors
Star-delta (wye-delta) conversion formulas for resistors or impedances in both directions, equal-resistor shortcut, with a worked example and a table.
The formulas
Label the delta resistors R12, R23 and R31, and the star resistors R1, R2 and R3, where R1 is the one at node 1, between R12 and R31.
| Conversion | Formula |
|---|---|
| Delta to star | R1 = R12 × R31 / (R12 + R23 + R31) |
| R2 = R12 × R23 / (R12 + R23 + R31) | |
| R3 = R23 × R31 / (R12 + R23 + R31) | |
| Star to delta | R12 = (R1×R2 + R2×R3 + R3×R1) / R3 |
| R23 = (R1×R2 + R2×R3 + R3×R1) / R1 | |
| R31 = (R1×R2 + R2×R3 + R3×R1) / R2 | |
| Equal resistors | RY = RΔ / 3, and RΔ = 3 × RY |
The same relations hold for complex impedances. The two networks are equivalent only as seen from the three terminals. The conversion is useful for bridges and for three-phase loads that are neither simple series nor parallel.
Worked example: delta 30, 60 and 90 Ω
R12 = 30 Ω, R23 = 60 Ω and R31 = 90 Ω, so the sum is 180 Ω.
- R1 = 30 × 90 / 180 = 15 Ω
- R2 = 30 × 60 / 180 = 10 Ω
- R3 = 60 × 90 / 180 = 30 Ω
- Back again: the sum of products is 900, so R12 = 900 / 30 = 30 Ω, R23 = 60 Ω and R31 = 90 Ω
Equal resistors: delta to star
| Delta resistor | Star resistor |
|---|---|
| 30 Ω | 10 Ω |
| 60 Ω | 20 Ω |
| 90 Ω | 30 Ω |
| 120 Ω | 40 Ω |
Frequently asked questions
Why is the star resistor one third of the delta resistor?
With equal values the sum in the denominator is 3R and the numerator is R², so each star resistor is R² / 3R = R / 3.
Does the neutral point of the star carry current?
The converted star has no neutral wire. It is an equivalent for the three terminals, not a physical neutral.
Can I use the same formulas for impedances?
Yes, with complex numbers for the impedances in place of the resistances.
When is this conversion needed?
When a network is not reducible by series and parallel rules, for example a bridge with a diagonal, and for balanced and unbalanced three-phase loads.
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