Delta-star and star-delta conversion

A delta (Δ) network of three resistors can be replaced by an equivalent star (Y) network, and the reverse, so that the resistance between each pair of terminals stays the same. This lets bridge circuits be reduced by series-parallel rules.

        Delta                      Star
          A                          A
         / \                         |
    Rab /   \ Rca                    Ra
       /     \                       |
      B-------C                      O
         Rbc                       /   \
                                 Rb     Rc
                                 /       \
                                B         C

Delta to star

Let the delta resistances be R_AB, R_BC, R_CA. Each star resistor equals the product of the two delta resistors touching that terminal divided by the sum of all three:

• 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)

For equal delta resistors R_Δ, R_Y = R_Δ/3.

Star to delta

Let the star resistances be R_A, R_B, R_C. Let S = R_A R_B + R_B R_C + R_C R_A.

• R_AB = S/R_C
• R_BC = S/R_A
• R_CA = S/R_B

(Each delta resistor is the sum of products divided by the star resistor opposite to it.) For equal star resistors R_Y, R_Δ = 3 R_Y.

Derivation idea

Equate the resistance between each pair of terminals with the third terminal open. For example, between A and B: R_A + R_B = R_AB ∥ (R_BC + R_CA). Solving the three equations gives the formulae above.

Example

A delta of 30 Ω, 30 Ω, 30 Ω gives a star of 10 Ω, 10 Ω, 10 Ω.