How this instrument works
A delta network wires three resistors into a triangle across three terminals; a wye network wires three resistors into a star that meets at a hidden center point. Seen only from the three outer terminals, the two networks can be made electrically indistinguishable — and the arithmetic that makes that true is the delta-to-wye transform. Solve the three equations that force each pair of terminal-to-terminal resistances to match, and every wye leg falls out as the product of the two delta legs meeting at that corner, divided by the sum of all three delta legs.
The division by the full three-leg sum is what keeps the units honest: two resistances multiplied give ohms squared, and dividing by a third resistance brings the result back to plain ohms. It also explains the balanced-network shortcut worth memorizing — when all three delta legs are equal, the formula collapses to R_Δ times R_Δ over three R_Δ, which is simply R_Δ divided by three. Away from that symmetric case there is no shortcut; each of the three wye legs needs its own numerator and its own pass through the formula.
Set one delta leg to zero and the formula quietly degenerates into the familiar two-resistor parallel combination, because the shorted leg drops out of its own numerator pairing while it still counts once in the denominator. Power engineers reach for this conversion when an unbalanced three-phase delta load needs a per-phase wye equivalent for single-phase analysis, and circuit students use it to break a Wheatstone bridge or ladder network that will not reduce with plain series-parallel combination. The common error is applying the balanced-case shortcut — dividing by three — to a delta that is not actually symmetric.
- Enter the first delta leg in the "Delta resistance R_ab" field, in ohms or kiloohms.
- Enter the second leg in "Delta resistance R_bc" — it need not equal the first.
- Enter the third leg in "Delta resistance R_ca" to complete the triangle.
- Read "Equivalent Y resistance at node A" — the instrument applies R_Y = (R_a·R_b) ⁄ (R_a+R_b+R_c) automatically.
- Set any single leg to 0 to check that the network collapses into a plain two-resistor parallel combination.
Worked example — a balanced 10 Ω delta
Set all three delta legs to 10 Ω: R_ab = R_bc = R_ca = 10 Ω, the balanced case. The formula gives R_Y = (10 × 10) ⁄ (10 + 10 + 10) = 100 ⁄ 30 = 3.33333333333 Ω, and the instrument reports that repeating decimal rather than a rounded 3.33 Ω.
That figure is one-third of the 10 Ω leg value, exactly what the balanced-case shortcut R_Y = R_Δ ⁄ 3 predicts — a useful sanity check whenever all three delta resistances happen to match. Three 10 Ω heating elements wired in delta, for instance, behave for per-phase analysis like three 3.333 Ω elements wired in a star.
Questions
What is a delta-to-wye (Δ-Y) transformation?
It is an algebraic swap between two three-terminal resistor arrangements — a triangle (delta) and a star (wye) — that behave identically from outside the three terminals. Each wye leg equals the product of the two delta legs meeting at that terminal, divided by the sum of all three delta legs; the reverse wye-to-delta conversion uses a different formula built from pairwise products.
Why does a balanced delta divide by exactly three?
Because every delta leg is the same value R_Δ, the general formula R_Y = (R_a·R_b) ⁄ (R_a+R_b+R_c) becomes R_Δ² ⁄ 3R_Δ, and the R_Δ terms cancel to leave R_Δ ⁄ 3. It only works when all three legs are equal; feed an unbalanced delta into that shortcut and the answer will be wrong for at least two of the three wye legs.
What happens when one delta leg is 0 Ω?
The network reduces to two resistors in ordinary parallel. With R_ab = R_bc = 10 Ω and R_ca = 0 Ω, R_Y = (10 × 10) ⁄ (10 + 10 + 0) = 5 Ω — exactly the two-resistor parallel formula, because the shorted leg drops out of the numerator's pairing while still counting once in the denominator.
Does this formula work for AC impedances, not just DC resistors?
Yes, with complex impedance Z standing in for resistance R — the same product-over-sum pattern holds for capacitors, inductors, and mixed combinations, which is why the transform shows up constantly in three-phase power analysis where loads are rarely purely resistive. Substitute each Z for R and the algebra carries through unchanged; only the arithmetic becomes complex-number arithmetic.
How is delta-to-wye different from wye-to-delta?
This page runs delta-to-wye: three known triangle legs produce one star leg. Wye-to-delta runs the opposite direction with a different formula — each delta leg equals the sum of every pairwise product of the three wye legs, divided by the wye leg opposite that delta leg. The two conversions are inverses of each other but are not built from the same expression.
Where does this conversion actually get used?
Two common cases: reducing a Wheatstone bridge or ladder network that resists plain series-parallel combination down to something solvable step by step, and converting an unbalanced three-phase delta load to a wye equivalent so each phase can be analyzed with single-phase circuit tools. Power-system studies and PCB grounding-grid calculations both lean on it regularly.