Delta/Star* Transformation

Suppose we are given three resistances R12, R23 and R31 connected in delta fashion between terminals 1, 2 and 3 as in Fig. 2.185 (a). So far as the respective terminals are concerned, these three given resistances can be replaced by the three resistances R1, R2 and R3 connected in star as shown in Fig. 2.185 (b). These two arrangements will be electrically equivalent if the resistance as measured between any pair of terminals is the same in both the arrangements. Let us find this condition.

First, take delta connection : 

Between terminals 1 and 2, there are two parallel paths; one having a resistance of R12 and the other having a resistance of (R12 + R31). ∴ Resistance between terminals 1 and 2 is 

Now, take star connection : The resistance between the same terminals 1 and 2 is (R1 + R2). As terminal resistances have to be the same 

Similarly, for terminals 2 and 3 and terminals 3 and 1, we get

Now, subtracting (ii) from (i) and adding the result to (iii), we get

How to Remember ? 

It is seen from above that each numerator is the product of the two sides of the delta which meet at the point in star. Hence, it should be remembered that : 

''resistance of each arm of the star is given by the product of the resistances of the two delta sides that meet at its end divided by the sum of the three delta resistances.''

Star/Delta Transformation 

This transformation can be easily done by using equations (i), (ii) and (iii) given above. Multiplying (i) and (ii), (ii) and (iii), (iii) and (i) and adding them together and then simplifying them, we get

How to Remember ? 

''The equivalent delta resistance between any two terminals is given by the sum of star resistances between those terminals plus the product of these two star resistances divide by the third star resistances.''