# Delta - Star transformation | Star - Delta Transformation

## Delta - Star Transformation

The replacement of delta or mesh by equivalent star connection is known as**delta - star transformation**. The two connections are equivalent or identical to each other if the impedance is measured between any pair of lines. That means, the value of impedance will be the same if it is measured between any pair of lines irrespective of whether the delta is connected between the lines or its equivalent star is connected between that lines.

Consider a delta system that's three corner points are A, B and C as shown in the figure. Electrical resistance of the branch between points A & B, B & C and C & A are R_{1}, R_{2} and R_{3} respectively. The resistance between the points A & B will be,

Now, one star system is connected to these points A, B, and C as shown in the figure. Three arms R_{A}, R_{B} and R_{C} of the star system are connected with A, B and C respectively. Now if we measure the resistance value between points A and B, we will get,
Since the two systems are identical, resistance measured between terminals A and B in both systems must be equal.

Similarly, resistance between points B and C being equal in the two systems,

And resistance between points C and A being equal in the two systems,

Adding equations (I), (II) and (III) we get,

Subtracting equations (I), (II) and (III) from equation (IV) we get,

The relation of delta - star transformation can be expressed as follows.

The equivalent star resistance connected to a given terminal, is equal to the product of the two delta resistances connected to the same terminal divided by the sum of the delta connected resistances.

If the delta connected system has same resistance R at its three sides then equivalent star resistance r will be,

## Star - Delta Transformation

For**star - delta transformation**we just multiply equations (v), (VI) & (VI), (VII) & (VII),(V) that is by doing (v)X(VI) + (VI)X(VII) + (VII)X(V) we get,

Now dividing equation (VIII) by equations (V), (VI) and equations (VII) separately we get,