Star Delta transformation

STAR DELTA TRANSFORMATION 

                                   STAR                                                  DELTA

 Figure (a) shows a Y ( star or wye ) connected impedance circuit and figure (b) a delta (Δ) connected  impedance circuit , their Τ and π shape shape circuit also represented respectively in above figure 

The two system will be exactly equivalent if the impedance between any pair of terminal a . b and c in figure (a) for the star ,is same as that between the corresponding pairs for the delta connection in figure (b) .when the third terminal is isolated 

First take star connection : the impedance between the terminals a and b  is  .

Now take delta connection : between  the same terminal a and b there are two parallel path , one having a impedance  and other having an impedance of  

so equivalent impedance between terminals a and b for delta connected network will be 

   = 

For transformation, an impedance between terminals a and b has to be same in both star and delta connected network ,so 

  =                 eq(1)

similarly we can write between terminals b and c 

   =                eq(2)

in similar fashion between terminal c and a

      =               eq(3)

DELTA TO STAR TRANSFORMATION:

                                                                                        TO

For transformation from delta to star , impedance  ,, given in delta network , and we have to find equivalent impedance ,, between respective terminals in star network 

to find  , now subtracting eq(2) from eq(1) and adding the result to eq(3), we get 

      =                    eq(4)

in similar fashion 

       =                  eq(5)

       =                  eq(6)

NOTE:- from above eq(4), eq(5) ,eq(6) it is clear that equivalent impedance of each arm of the star is given by the product of the impedance of the two delta side that meet at its ends divide by the sum of three delta impedance

STAR TO DELTA TRANSFORMATION: 

 in this case impedance , and  of star connected network is given and we have to find impedance , and  in delta connected network as shown in figure above 

let us define

   =                  eq(7)

after substituting the respective values of  , , from eq(4), eq(5) , eq(6) in eq(7) we will get 

      =                                 eq(8)          

from eq(4) amd eq(8)

    =

  so                  

     =    =                 eq(9)

  in similar fashion

       =                             eq(10)

        =                             eq(11) 

NOTE:-  Therefore , the equivalent impedance of each arm of the delta is given by the some of star impedance between those terminals plus the product of these two star impedances divided by the third impedance