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The wye delta transformation is very useful to convert one combination of resistors to another combination to ease our calculation.

It is often in an electric circuit where resistors are connected neither in parallel nor in series.

This topic will help us understand about:

- Wye to delta transformation
- Delta to wye transformation

Before jumping on to the transformation and conversion formula, we need to understand:

- Wye network
- Delta network

The Wye network has the shape of wye (Y) or tee (T) as shown below respectively.

The Delta network has the shape of delta (Δ) or pi (π) as shown below respectively.

These networks are mainly used for three-phase circuits to achieve desired voltage and current. Not only that, we will likely find these networks in an electrical filter and matching networks.

## Delta to Wye Transformation

Since the wye network is more convenient to analyze over the delta network, we will start with the delta wye transformation.

We overlap the delta network with the wye network and analyze the equivalent resistance. Since we have to find the equivalent resistance, both wye and delta networks should have equal resistance to each other.

We will compare wye and delta networks and make sure the resistance in the wye (Y) network is equal to the resistance in the delta (Δ) network.

Observe the previous wye circuit below.

The left one is wye (Y) or tee (T) network and the right one is delta (Δ) or pi (π).

The resistance for terminal 1 and 2 for wye network is

The resistance for terminal 1 and 2 for delta network is

Since we assume both wye and delta networks have equal resistances then

The resistance for terminal 1 and 3 for wye network is

The resistance for terminal 1 and 3 for delta network is

Since we assume both wye and delta networks have equal resistances then

The resistance for terminal 3 and 4 for wye network is

The resistance for terminal 3 and 4 for delta network is

Since we assume both wye and delta networks have equal resistances then

We subtract (1c) from (1a) and we get

We add (1b) and (2) and we have

We subtract (2) from (1b) and we get

We subtract (4) from (1a) and we have

Actually, we only need the equations from (3) to (5).

Combining the wye network and delta network we have the network below. We just need to place the node “n” in the center.

The delta to wye conversion rule is

Each resistor in the Y network is the product of the resistors in the two adjacent Δ branches, divided by the sum of the three Δ resistors.

**Wye to Delta Transformation**

Looking back to the equations (3) to (5) we conclude that

Dividing the (6) by each equations (3) to (5) resulting in

From the equations (7) to (9) along with Figure.(1),

The wye to delta conversion rule is

Each resistor in the Δ network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor.

The Y and Δ networks are said to be balanced when

When Y and Δ networks are balanced, then the conversion formula will be much easier,

## Wye to Delta Formula

Below are the wye to delta transformation formula

## Delta to Wye Formula

Below are the delta to wye transformation formula

**Wye Delta Transformation Summary**

After learning a lot of wye delta transformation or delta wye transformation, we can conclude that:

- Each resistor in the Y network is the product of the resistors in the two adjacent Δ branches, divided by the sum of the three Δ resistors.
- Each resistor in the Δ network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor.
- Y network resistance, RY is always smaller than Δ because Y network is a series connection while Δ network is a parallel connection.

## Wye Delta Transformation Example

To get a better understanding, we will solve simple examples.

1. Observe the Δ network below and find its equivalent Y network.

First we draw the Y network inside the Δ network.

To find the R1,

To find the R2,

To find the R3,

Now we reverse the calculation above.

2. Observe the Y network below and find its equivalent Δ network.

First we draw the Δ network outside the Y network.

To find the Ra,

To find the Rb,

To find the Rc,

For both Y network and Δ network has the equal resistance regardless of the transformation.