Star to delta conversion formula: Three branches in an electrical network can be connected in numbers of forms but most common among them is either star or delta form. In delta connection, three branches are so connect, that they form a closed loop. As these three branches is connect from nose to tail, they form a triangular close loop, this configuration is refer as delta connection.

On the other hand, when either terminal of three branches is connect to a common point to form a Y like pattern is call as star connection. But these star and delta connections can be transform from one form to another. For simplifying complex network, delta to star or **star to delta transformation** is often require.

**Delta To Star Conversion**

The replacement of delta or mesh by equivalent star connection is call as **delta – star transformation**. The two connections are equivalent or identical to each other if the impedance is measure between any pair of lines.

That means, the value of impedance will be the same. If it is measure between any pair of lines irrespective. Whether the delta is connect between the lines or its equivalent star is connect 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 and B, B and C and C and A are R_{1}, R_{2} and R_{3} respectively.

The resistance between the points A and B will be,

Now, one star system is connect 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 is connect 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 To Delta Conversion formula

For **star – delta transformation** we just multiply equations (v), (VI) and (VI), (VII) and (VII), (V) that is by doing (v) × (VI) + (VI) × (VII) + (VII) × (V) we get,

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

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