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.
Contents
- Names
- Basic Y transformation
- Equations for the transformation from to Y
- Equations for the transformation from Y to
- A proof of the existence and uniqueness of the transformation
- Simplification of networks
- Graph theory
- load to Y load transformation equations
- Y load to load transformation equations
- References
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.
Names
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.
Basic Y-Δ transformation
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
The general idea is to compute the impedance
where
Equations for the transformation from Y to Δ
The general idea is to compute an impedance
where
A proof of the existence and uniqueness of the transformation
The feasibility of the transformation can be shown as a consequence of the superposition theorem for electric circuits. A short proof, rather than one derived as a corollary of the more general star-mesh transform, can be given as follows. The equivalence lies in the statement that for any external voltages (
It can be readily shown by Kirchhoff's circuit laws that
Though usually six equations are more than enough to express three variables (
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.
Every two-terminal network represented by a planar graph can be reduced to a single equivalent resistor by a sequence of series, parallel, Y-Δ, and Δ-Y transformations. However, there are non-planar networks that cannot be simplified using these transformations, such as a regular square grid wrapped around a torus, or any member of the Petersen family.
Graph theory
In graph theory, the Y-Δ transform means replacing a Y subgraph of a graph with the equivalent Δ subgraph. The transform preserves the number of edges in a graph, but not the number of vertices or the number of cycles. Two graphs are said to be Y-Δ equivalent if one can be obtained from the other by a series of Y-Δ transforms in either direction. For example, the Petersen family is a Y-Δ equivalence class.
Δ-load to Y-load transformation equations
To relate
The impedance between N1 and N2 with N3 disconnected in Δ:
To simplify, let
Thus,
The corresponding impedance between N1 and N2 in Y is simple:
hence:
Repeating for
and for
From here, the values of
For example, adding (1) and (3), then subtracting (2) yields
thus,
where
For completeness:
Y-load to Δ-load transformation equations
Let
We can write the Δ to Y equations as
Multiplying the pairs of equations yields
and the sum of these equations is
Factor
Note the similarity between (8) and {(1),(2),(3)}
Divide (8) by (1)
which is the equation for