Wye - Delta

Wye-Delta Transformations

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.

Delta and Wye Circuits

Basic Wye-Delta Trasformation

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-load to Y-load 3-phase circuit

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

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

Equations for the transformation from Y-load to Δ-load 3-phase circuit

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 formula for the individual edges are thus

Simplification of Networks

Resistive networks between two terminals can theoretically be simplified to a single equivalent resistor (more generally, the same is true of impedance). Series and parallel transforms are basic tools for doing so, but for complex networks such as the bridge illustrated here, they do not suffice.

The Y-Δ transform can be used to eliminate one node at a time and produce a network that can be further simplified, as shown.

The reverse transformation, Δ-Y, which adds a node, is often handy to pave the way for further simplification as well.