# STAR-DELTA CONVERSION

The

**Y-Δ transform**, also written**wye-delta**and also known by many other names, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter Δ. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899.^{[1]}It is widely used in analysis of three-phase electric power circuits.
The Y-Δ transform can be considered a special case of the star-mesh transform for three resistors. In mathematics, the Y-Δ transform plays an important role in theory of circular planar graphs.

^{[2}Basic symbolic representation |

## Basic Y-Δ transformation[edit]

The transformation is used to establish equivalence for networks with three terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming the impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equations given here are valid for complex as well as real impedances.

### Equations for the transformation from Δ to Y[edit]

The general idea is to compute the impedance at a terminal node of the Y circuit with impedances , to adjacent nodes in the Δ circuit by

where are all impedances in the Δ circuit. This yields the specific formulae,

### Equations for the transformation from Y to Δ[edit]

The general idea is to compute an impedance in the Δ circuit by

where is the sum of the products of all pairs of impedances in the Y circuit and is the impedance of the node in the Y circuit which is opposite the edge with . The formulae for the individual edges are thus

Or, if using admittance instead of resistance:

Note that the general formula in Y to Δ using admittance is similar to Δ to Y using resistance.

### Δ-load to Y-load transformation equations[edit]

To relate from Δ to from Y, the impedance between two corresponding nodes is compared. The impedance in either configuration is determined as if one of the nodes is disconnected from the circuit.

The impedance between

*N*_{1}and*N*_{2}with*N*_{3}disconnected in Δ:
To simplify, let be the sum of .

Thus,

The corresponding impedance between N

_{1}and N_{2}in Y is simple:
hence:

- (1)

Repeating for :

- (2)

and for :

- (3)

From here, the values of can be determined by linear combination (addition and/or subtraction).

For example, adding (1) and (3), then subtracting (2) yields

For completeness:

- (4)
- (5)
- (6)

### Y-load to Δ-load transformation equations[edit]

Let

- .

We can write the Δ to Y equations as

- (1)
- (2)
- (3)

Multiplying the pairs of equations yields

- (4)
- (5)
- (6)

and the sum of these equations is

- (7)

Factor from the right side, leaving in the numerator, canceling with an in the denominator.

- (8)

Note the similarity between (8) and {(1), (2), (3)}

Divide (8) by (1)

which is the equation for . Dividing (8) by (2) or (3) (expressions for or ) gives the remaining equations.

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