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# Delta-Wye Transformation

By BYJU'S Exam Prep

Updated on: September 25th, 2023

**Delta-Wye Transformation** technique is used to resolve complex electrical circuits with resistors. We know that the basic connections for connecting the resistors are series connections and parallel connections. Then, we can get the simplified equivalent circuit easily. But if the circuit is having multiple resistances, then sometimes, we may face the difficulty to simplify it. That time, wherever required we can use Delta – Wye Transformation so that we can simplify the given resistive circuit/network easily.

In this article, get an overview of Delta–the Wye Transformation technique and how to apply this technique while solving the resistive circuit/network problems. Here, you will get to know, what is Delta – Wye Transformation technique and the formulas that we will use in this technique.

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## What is Delta–Wye Transformation Technique?

We can easily find the equivalent resistance, whenever the resistors are connected in series or parallel. We can easily calculate the equivalent resistance, whenever the resistors are connected in bridge form and that the Wheatstone bridge is balanced.

It is difficult to calculate the equivalent resistance directly, whenever the resistors are connected in bridge form if that Wheatstone bridge is not balanced. That time we must convert the delta form into Wye form, and it is known as the conversion of delta to wye. Similarly, we can do the conversion of the wye to delta, wherever required. These two conversions together we can call Delta–Wye Transformation. Now, let’s discuss the following two cases of Delta–Wye Transformation.

- Conversion of Delta–Wye or Conversion of Delta-Star
- Conversion of Wye-Delta or Conversion of Star-Delta

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## Delta–Wye Transformation Formula

Depending upon the conversion of the network different delta-wye transformation formula is produced. Let us see the formula for finding the resistance for various cases of conversion.

### Conversion of Delta–Wye or Conversion of Delta – Star

If the resistors are connected in the form of a Delta or triangle (∆), then it is known as the Delta network. If we invert this delta symbol, then we will get the symbol of the del operator (). This del form, we can modify and represent in the form of the pi symbol (П). Then it is known as pi-network or П-network. Hence, these three networks (∆, ∇, П) are the same.

If the resistors are connected in the form of the letter Y, then it is known as the Wye network. If we invert the letter Y, then it is known as an inverted Y or star (⅄) network. The letter Y, can be modified into the letter T, then it is known as T-network. Hence, these three networks (Y, ⅄, T) are the same.

The above figure represents the conversion of the Delta network into a star network. Here, Delta networks consist of three resistors RA, RB, and RC. The resistors in the corresponding Wye network are R1, R_{2,}and R3. We can calculate the values of R_{1}, R_{2, }and R_{3 }by using the following formulae.

- R
_{1}=(R_{A}R_{C})/(R_{A}+R_{B}+R_{C}). - R
_{2}=(R_{A}R_{B})/(R_{A}+R_{B}+R_{C}) - R
_{3}=(R_{B}R_{C})//(R_{A}+R_{B}+R_{C})

### Conversion of Wye-Delta or Conversion of Star-Delta

Previously, we converted the Delta network into the Star network. Now, let’s see the conversion of Wye-Delta or Star–Delta. The following figure represents the conversion of the Star network into the Delta network.

Here, Star networks consist of three resistors R_{1}, R_{2, }and R_{3}. The resistors in the corresponding Delta network are R_{A}, R_{B, }and R_{C}. We can calculate the values of R_{A}, R_{B, }and RC by using the following formulae.

- R
_{A}=R_{1}+R_{2}+(R_{1}R_{2})/R_{3}. - R
_{B}=R_{2}+R_{3}+(R_{2}R_{3})/R_{1}. - R
_{C}=R_{1}+R_{3}+(R_{1}R_{3})R_{2}.

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## Delta – Wye Transformation Examples

Q1. In the Delta network, if all the resistances are having the same value of15 Ω, then each resistor of the corresponding star network will have the same value of _______Ω.

Ans: 5

Q2. In the Star network, if all the resistances are having the same value of15 Ω, then each resistor of the corresponding Delta network will have the same value of _______Ω.

Ans: 45