What is Delta to Star Conversion?

What is Delta to Star Conversion? 

Consider a three terminal network in which the resistors are connected in the form ∆.
Such a network is known as delta network. Let the resistor values are R12, R23 and R31. Now we
find its equivalent Y-network such that both the circuits are identical as far as the terminals 1,
2 and 3 are concerned. Let the resistor values are R1, R2 and R3.

In Delta (Δ) Connection: 

                                                                    R12(R23 + R31)

Equivalent Resistance between 1 &2 =--------------------
                                                                    R12 + R23 +R31

In Star (Y) Connection: 

Equivalent Resistance between 1 &2 = R1 + R2
Resistance between terminal 1 & 2 in Y = Resistance between terminal 1 & 2 in Δ
delta

                  R12(R23 + R31)
R1 + R2 =----------------------------               --------------- (1)
                  R12 + R23 +R31


                                   R23(R31 + R12)
Similarly, R2 + R3 =---------------------------    ---------- (2)
                                   R12 + R23 +R31

                   R31(R12 + R23)
R3 + R1 =---------------------------                     ---------- (3)
                   R12 + R23 +R31

Subtract Equation (2) from Equation (3):


                   (R12R31 + R23R31 − R23R31 − R12R23)
R1 − R2 =------------------------------------------------------------------
                                     R12 + R23 +R31

                   R12(R31 − R23 )
R1 − R2 =-----------------------------              ---------- (4)
                   R12 + R23 +R31

Add Equation (1) and Equation (4):

            2 R12 R31
2R1 =---------------------------
          R12 + R23 +R31

          R12 R31
R1 =----------------------------
          R12 + R23 +R31


                           R23 R12
Similarly, R2 =------------------------------
                           R12 + R23 +R31

          R31 R23
R3 =------------------------------
          R12 + R23 +R31