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

## Wye-Delta Transformations

The example can be seen in Figure.(1). Many circuits have the type in Figure.(1) can be solved using three-terminal equivalent networks.

Figure 1. The bridge network |

The solution is the wye (Y) or tee (T) shown in Figures.(2a) and (2b) and the delta (Δ) or pi (Π) network in Figures.(3a) and (3b).

Figure 2. Form of network : (a) Y, (b) T |

Figure 3. Form of networks : (a)delta , (b) pi |

These networks occur as a part of a larger network. These are used in the three-phase network, an electrical filter, and matching networks.

Our objectives here are how to identify the type of the networks and how to apply wye-delta transformation in the circuit analysis.

## Delta to Wye Conversion

Let us assume the condition where the wye network is more convenient in a place with a delta configuration circuit.

We superimpose a wye network on the existing delta network and find the equivalent resistances in the wye network.

In order to get equivalent resistances in the wye network, we compare the two networks and make sure that the resistance between each pair of nodes in delta (Δ) or pi (Π) is the same with wye (Y) or tee (T) network.

For terminal 1 and 2 in Figures.(2) and (3) for example,

(1) |

Setting *R _{12}*(Y) =

*R*(Δ) makes

_{12}(2a) |

Equally,

(2b) |

(2c) |

Subtracting Equations.(2c) from (2a), we have

(3) |

Adding Equations.(2b) and (3) we get

(4) |

Subtracting Equations.(3) from (2b) gives

(5) |

Subtracting Equations.(4) from (2a) we get

(6) |

Actually, we do not need to memorize Equations.(4) to (6).

In order to transform a Δ network to Y, we create an extra node n as shown in Figure.(4).

Figure 4. Superposition of wye and delta network |

And the 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 ot the three Δ resistors.

One can follow this rule and obtain Equations.(4) to (6) from Figure.(4).

## Wye to Delta Conversion

In order to get the conversion formulas for transforming a wye network to an equivalent delta network, we note from Equations.(4) to (6) that

(7) |

Dividing Equation.(7) by each of Equations.(4) to (6) gives the following equations :

(8) |

(9) |

(10) |

From Equations.(8) to (10) and Figure.(4), the conversion rule for Y to Δ follows :

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

(11) |

Under these conditions, the conversion formula is

(12) |

For you who ask why R_{Y} is smaller than R_{Δ}. Looking from the connection. the Y network is like a “series” connection and Δ network is like a “parallel” connection.

The equation above is made from the Kirchhoff’s laws, node voltage analysis , and mesh current analysis.

## Wye-Delta Transformations Examples

For better understanding let us review the example below :

**1.Convert the Δ network in Figure.(5a) to an equivalent Y network.**

Figure 5 |

__Solution :__

Using Equations.(5) to (6) we get

The equivalent Y network is shown in Figure.(5b).

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