Y-Δ transform
In circuit design, 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.
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.
In mathematics, the Y-Δ transform plays an important role in theory of circular planar graphs.
The Y-Δ transform is known by a variety of other names, mostly based upon the two shapes involved, listed in either order.
Thus, common names for the transformation include wye-delta or delta-wye, star-delta, star-mesh, or T-Π.
Complex impedance is a quantity measured in ohms which represents resistance as positive real numbers in the usual manner, and also represents reactance as positive and negative imaginary values.
This yields the specific formula The general idea is to compute an impedance
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.
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.
According to the superposition theorem, the voltages can be obtained by studying the superposition of the resulting voltages at the nodes of the following three problems applied at the three nodes with current: The equivalence can be readily shown by using Kirchhoff's circuit laws that
Now each problem is relatively simple, since it involves only one single ideal current source.
To obtain exactly the same outcome voltages at the nodes for each problem, the equivalent resistances in the two circuits must be the same, this can be easily found by using the basic rules of series and parallel circuits: Though usually six equations are more than enough to express three variables (
), here it is straightforward to show that these equations indeed lead to the above designed expressions.
In fact, the superposition theorem establishes the relation between the values of the resistances, the uniqueness theorem guarantees the uniqueness of such solution.
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.
[3] 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.
The impedance in either configuration is determined as if one of the nodes is disconnected from the circuit.
During the analysis of balanced three-phase power systems, usually an equivalent per-phase (or single-phase) circuit is analyzed instead due to its simplicity.
For that, equivalent wye connections are used for generators, transformers, loads and motors.
The stator windings of a practical delta-connected three-phase generator, shown in the following figure, can be converted to an equivalent wye-connected generator, using the six following formulas[a]:
The neutral node of the equivalent network is fictitious, and so are the line-to-neutral phasor voltages.
If the actual delta generator is balanced, meaning that the internal phasor voltages have the same magnitude and are phase-shifted by 120° between each other and the three complex impedances are the same, then the previous formulas reduce to the four following:
