Write formulas for Delta to Star and Star to Delta transformation.

Let the delta resistances be R₁₂, R₂₃, R₃₁ (between terminals 1-2, 2-3, 3-1) and the equivalent star resistances be R₁, R₂, R₃ (connected from terminals 1, 2, 3 to the star point N).

      1                    1
     / \                   |
 R12/   \R31              R1
   /     \                 N
  2-------3            R2/   \R3
     R23                2     3
   Delta                 Star

Delta to Star: each star arm = product of the two adjacent delta resistances / sum of the three delta resistances:

\[ R_1=\frac{R_{12}R_{31}}{R_{12}+R_{23}+R_{31}},\quad R_2=\frac{R_{12}R_{23}}{R_{12}+R_{23}+R_{31}},\quad R_3=\frac{R_{23}R_{31}}{R_{12}+R_{23}+R_{31}} \]

Star to Delta: each delta resistance = sum of products of star resistances taken two at a time / the star resistance of the opposite terminal:

\[ R_{12}=R_1+R_2+\frac{R_1R_2}{R_3},\quad R_{23}=R_2+R_3+\frac{R_2R_3}{R_1},\quad R_{31}=R_3+R_1+\frac{R_3R_1}{R_2} \]

i.e. \( R_{12} = \dfrac{R_1R_2+R_2R_3+R_3R_1}{R_3} \) etc. For equal resistances: R_Y = R_Δ/3 and R_Δ = 3R_Y.