Delta-Star Resistor Conversion Formulas

Delta to star and star to delta resistor formulas: the equivalent star resistors of a delta network, the reverse conversion and the equal-resistor case.

The formulas

The delta network has the resistors Rab, Rbc and Rca between the three nodes a, b and c. The star network has Ra, Rb and Rc from each node to a common point. The two networks look the same from the outside when the resistors follow these formulas, so the one can replace the other when a circuit does not reduce by series and parallel. In each star formula the numerator holds the two delta resistors that meet at the node. The values here are examples.

Worked example: a delta of 30 Ω, 60 Ω and 90 Ω

1. Sum = 30 + 60 + 90 = 180 Ω
2. Ra = 30 × 90 / 180 = 15 Ω
3. Rb = 30 × 60 / 180 = 10 Ω
4. Rc = 60 × 90 / 180 = 30 Ω

Worked example: back from the star of 15 Ω, 10 Ω and 30 Ω

1. Ra × Rb + Rb × Rc + Rc × Ra = 900 Ω²
2. Rab = 900 / 30 = 30 Ω
3. Rbc = 900 / 15 = 60 Ω
4. Rca = 900 / 10 = 90 Ω

Equal resistors: star equivalent of an equal delta

Frequently asked questions

When do I need this?

For a bridge or a three-node network that has no series or parallel parts, such as the unbalanced Wheatstone bridge. Converting one triangle to a star gives a circuit that does reduce.

Why is the star resistor smaller?

Each star arm carries the effect of two delta arms in parallel with the third, so the star resistors are smaller, by a factor of 3 when all are equal.

Does the conversion hold for impedances?

Yes. The same formulas hold for complex impedances in AC circuits.

Is the common point of the star real?

No. It is a mathematical node, and it has no connection to the circuit, so it is not at ground potential.

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