### Star Delta Transformation

Consider the three resistances R1 ,R2 and R3 connected in Star as shown in Fig. 1.

 Fig.  1
Now by Star-Delta conversion. it is always possible to replace these Star connected resistances by three equivalent Delta connected resistances R12 , R23 and R31, between the same terminals. This is called equivalent Delta of the given Star.
Now we are interested in finding out values of R12 ,R23 and R31 interms of R1 ,R2 andR3.
For this we can use set of equations derived in previous article. From the result of Delta-Star transformation we know that,

Now multiply (g) and (h), (h) and (i), (i) and (g) to get following three equations.
Now add equations (j) ,(k) and (l)
But
Substituting in above in R.H.S. we get,
...                                R1 R2 +R2 R3 +R3 R1 =R1 R23

Similarly substituting in R.H.S., remaining values, we can write relations for remaining tow resistances.
and
Easy way to remember the result :
The equivalent Delta connected resistance to be connected between any tow terminals is sum of the tow resistances connected between the same tow terminals and star point respectively in star, plus the product of the same two star resistances divided by the third star resistance.
 Fig. 2  Star and equivalent Delta
So if we want equivalent delta resistance between terminal (3) and (1), then take sum of the tow resistances connected between same tow terminals (3) and (1) and star point respectively i.e. terminal (3) to star point R3 and terminal (1) to star point i.e. R1. Then to this sum of R1 and R3, add the term which is the product of the same tow resistances i.e. R1 and R3 divided by the third star resistance which is R2.
...               We can write, R31 = R1 + R3 + (R1 R3 )/R which is same as derived above.
Note : If all three resistances in a star connection are of same magnitude say R, then its equivalent Delta contains all resistances of same magnitude of,
R12 = R31 = R23 =R +R + ((R x R)/R) = 3 R