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In electrical engineering, 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]

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. 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.

Equations for the transformation from Δ to Y

The general idea is to compute the impedance ${\displaystyle R_{\text{Y}}}$ at a terminal node of the Y circuit with impedances ${\displaystyle R'}$, ${\displaystyle R''}$ to adjacent nodes in the Δ circuit by

${\displaystyle R_{\text{Y}}={\frac {R'R''}{\sum R_{\Delta }}}}$

where ${\displaystyle R_{\Delta }}$ are all impedances in the Δ circuit. This yields the specific formula

${\displaystyle {\begin{aligned}R_{1}&={\frac {R_{\text{b}}R_{\text{c}}}{R_{\text{a}}+R_{\text{b}}+R_{\text{c}}}}\\R_{2}&={\frac {R_{\text{a}}R_{\text{c}}}{R_{\text{a}}+R_{\text{b}}+R_{\text{c}}}}\\R_{3}&={\frac {R_{\text{a}}R_{\text{b}}}{R_{\text{a}}+R_{\text{b}}+R_{\text{c}}}}\end{aligned}}}$

Equations for the transformation from Y to Δ

The general idea is to compute an impedance ${\displaystyle R_{\Delta }}$ in the Δ circuit by

${\displaystyle R_{\Delta }={\frac {R_{P}}{R_{\text{opposite}}}}}$

where ${\displaystyle R_{P}=R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}$ is the sum of the products of all pairs of impedances in the Y circuit and ${\displaystyle R_{\text{opposite}}}$ is the impedance of the node in the Y circuit which is opposite the edge with ${\displaystyle R_{\Delta }}$. The formulae for the individual edges are thus

${\displaystyle {\begin{aligned}R_{\text{a}}&={\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{1}}}\\R_{\text{b}}&={\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{2}}}\\R_{\text{c}}&={\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{3}}}\end{aligned}}}$

Or, if using admittance instead of resistance:

Zdroj:https://en.wikipedia.org?pojem=Y-Δ_transform
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