Wye - Delta Transformation

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.It is widely used in analysis of three-phase electric power circuits.        

The Y-Δ transform is known by a variety of other names, mostly based upon the two shapes involved, listed in either order. The Y, spelled out as wye, can also be called T or star; the Δ, spelled out as delta, can also be called triangle, Π (spelled out as pi), or mesh. Thus, common names for the transformation include wye-delta or delta-wye, star-delta, star-mesh, or T-Π.

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.

 Transformation from Delta-load to Wye-load 

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

where R are all impedances in the Δ circuit. This yields the specific formula

 Transformation from Wye-load to Delta-load 

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

where Rp = R1R2 + R2R3 + R3R1 is the sum of the products of all pairs of impedances in the Y circuit and Ropposite is the impedance of the node in the Y circuit which is opposite the edge with R. The formula for the individual edges are thus