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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]

Δ and Y circuits with the labels which are used in this article.
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]

Δ and Y circuits with the labels that are used in this article.
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 N1 and N2 with N3 disconnected in Δ:
To simplify, let  be the sum of .
The corresponding impedance between N1 and N2 in Y is simple:
Repeating for :
and for :
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:

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

We can write the Δ to Y equations as
Multiplying the pairs of equations yields
and the sum of these equations is
Factor  from the right side, leaving  in the numerator, canceling with an  in the denominator.
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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