Star - Delta transformation

A star network can be transformed to its equivalent delta connection and a delta can be transformed to its equivalent star connection. The star-delta transformation is
useful in simplifying the circuits while solving.

Delta connection

Star connection

In a star network the with resistances R_a, R_b and R_c the resistances at the terminals are as given below,

R_AB = R_a+R_b
R_BC = R_b+R_c
R_AC = R_c+R_a

For a delta network formed with resistors R_ab, R_bc, and R_ac the resistances at the terminals of the network are,

R_AB = [R_abR_ac+R_abR_bc] / [R_ab + R_bc + R_ac]
R_BC = [R_bcR_ac+R_bcR_ab] / [R_ab + R_bc + R_ac]
R_AC = [R_acR_ab+R_acR_bc] / [R_ab + R_bc + R_ac]

For equivalent transformation it is necessary that the terminal resistances should be equal,

i.e. R_AB (∆) = R_AB(Y), R_BC (∆) = R_BC(Y) and R_AC (∆) = R_AC(Y),

Substituting the values in the above we get the following equations,

R_a + R_b = [R_abR_ac+R_abR_bc] / [R_ab + R_bc + R_ac] ----- (1)

R_b + R_c = [R_bcR_ac+R_bcR_ab] / [R_ab + R_bc + R_ac] ----- (2)

R_c + R_a = [R_acR_ab+R_acR_bc] / [R_ab + R_bc + R_ac] ----- (3)

Delta to Star Transformation

Applying the below algebraic manipulations for the above three equations gives the equivalent star network values of R_a, R_b and R_c from the R_ab, R_bc and R_ac of the given delta network.
From (1) + (3) - (2) we get,

2R_a = [ R_abR_ac + R_abR_bc + R_acR_ab + R_acR_bc - R_bcR_ac - R_bcR_ab ] / [R_ab + R_bc + R_ac]

⇒ 2R_a = 2R_abR_ac / [R_ab + R_bc + R_ac]

R_a = R_abR_ac/ [R_ab + R_bc + R_ac]     ----- (4)

Similarly from (1) + (2) - (3) we get,

R_b = R_bcR_ab / [R_ab + R_bc + R_ac]    -----(5)

and from (2) + (3) - (1) we get,

R_c= R_acR_bc / [R_ab + R_bc + R_ac]     ----(6)

Star to Delta transformation

To obtain the equation for star to delta transformation the following manipulations can be done.

[ {(4)*(5)} + {(5)*(6)} + {(6)*(4)} ] / (4)

From above we get,
\[\frac{ (R_{a}R_{b} + R_{b}R_{c} + R_{a}R_{c})}{R_{a}} = \frac{[R_{ab}^{2}R_{bc}R_{ac}+R_{ab}R_{bc}^{2}R_{ac} + R_{ab}R_{bc}R_{ac}^{2}]}{ [R_{ab} + R_{bc} + R_{ac}] ^{2}}X \frac{ R_{ab} + R_{bc} + R_{ac} }{R_{ab}R_{ac}}\]

⇒ R_ab = (R_aR_b + R_bR_c + R_aR_c ) / R_c

Similarly ,

R_bc = (R_aR_b + R_bR_c + R_aR_c ) / R_a
R_ac = (R_aR_b + R_bR_c + R_aR_c ) / R_b

Also Read

Star connection
Delta connection