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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 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 Basic symbolic representation

## 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 $R_{\text{Y}}$ at a terminal node of the Y circuit with impedances $R'$ $R''$ to adjacent nodes in the Δ circuit by
$R_{\text{Y}}={\frac {R'R''}{\sum R_{\Delta }}}$ where $R_{\Delta }$ are all impedances in the Δ circuit. This yields the specific formulae,

{\begin{aligned}R_{1}&={\frac {R_{\text{b}}R_{\text{c}}}{R_{\text{a}}+R_{\text{b}}+R_{\text{c}}}}\\[3pt]R_{2}&={\frac {R_{\text{a}}R_{\text{c}}}{R_{\text{a}}+R_{\text{b}}+R_{\text{c}}}}\\[3pt]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 $R_{\Delta }$ in the Δ circuit by
$R_{\Delta }={\frac {R_{P}}{R_{\text{opposite}}}}$ where $R_{P}=R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}$ is the sum of the products of all pairs of impedances in the Y circuit and $R_{\text{opposite}}$ is the impedance of the node in the Y circuit which is opposite the edge with $R_{\Delta }$ . The formulae for the individual edges are thus
{\begin{aligned}R_{\text{a}}&={\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{1}}}\\[3pt]R_{\text{b}}&={\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{2}}}\\[3pt]R_{\text{c}}&={\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{3}}}\end{aligned}} {\begin{aligned}Y_{\text{a}}&={\frac {Y_{3}Y_{2}}{\sum Y_{\text{Y}}}}\\[3pt]Y_{\text{b}}&={\frac {Y_{3}Y_{1}}{\sum Y_{\text{Y}}}}\\[3pt]Y_{\text{c}}&={\frac {Y_{1}Y_{2}}{\sum Y_{\text{Y}}}}\end{aligned}} Note that the general formula in Y to Δ using admittance is similar to Δ to Y using resistance.

To relate $\left\{R_{\text{a}},R_{\text{b}},R_{\text{c}}\right\}$ from Δ to $\left\{R_{1},R_{2},R_{3}\right\}$ from Y, the impedance between two corresponding nodes is compared. The impedance in either configuration is determined as if one of the nodes is disconnected from the circuit.
The impedance between N1 and N2 with N3 disconnected in Δ:
{\begin{aligned}R_{\Delta }\left(N_{1},N_{2}\right)&=R_{\text{c}}\parallel (R_{\text{a}}+R_{\text{b}})\\[3pt]&={\frac {1}{{\frac {1}{R_{\text{c}}}}+{\frac {1}{R_{\text{a}}+R_{\text{b}}}}}}\\[3pt]&={\frac {R_{\text{c}}\left(R_{\text{a}}+R_{\text{b}}\right)}{R_{\text{a}}+R_{\text{b}}+R_{\text{c}}}}\end{aligned}} To simplify, let $R_{\text{T}}$ be the sum of $\left\{R_{\text{a}},R_{\text{b}},R_{\text{c}}\right\}$ .
$R_{\text{T}}=R_{\text{a}}+R_{\text{b}}+R_{\text{c}}$ Thus,
$R_{\Delta }\left(N_{1},N_{2}\right)={\frac {R_{\text{c}}(R_{\text{a}}+R_{\text{b}})}{R_{\text{T}}}}$ The corresponding impedance between N1 and N2 in Y is simple:
$R_{\text{Y}}\left(N_{1},N_{2}\right)=R_{1}+R_{2}$ hence:
$R_{1}+R_{2}={\frac {R_{\text{c}}(R_{\text{a}}+R_{\text{b}})}{R_{\text{T}}}}$ (1)
Repeating for $R(N_{2},N_{3})$ :
$R_{2}+R_{3}={\frac {R_{\text{a}}(R_{\text{b}}+R_{\text{c}})}{R_{\text{T}}}}$ (2)
and for $R\left(N_{1},N_{3}\right)$ :
$R_{1}+R_{3}={\frac {R_{\text{b}}\left(R_{\text{a}}+R_{\text{c}}\right)}{R_{\text{T}}}}.$ (3)
From here, the values of $\left\{R_{1},R_{2},R_{3}\right\}$ can be determined by linear combination (addition and/or subtraction).
For example, adding (1) and (3), then subtracting (2) yields
{\begin{aligned}R_{1}+R_{2}+R_{1}+R_{3}-R_{2}-R_{3}&={\frac {R_{\text{c}}(R_{\text{a}}+R_{\text{b}})}{R_{\text{T}}}}+{\frac {R_{\text{b}}(R_{\text{a}}+R_{\text{c}})}{R_{\text{T}}}}-{\frac {R_{\text{a}}(R_{\text{b}}+R_{\text{c}})}{R_{\text{T}}}}\\[3pt]{}\Rightarrow 2R_{1}&={\frac {2R_{\text{b}}R_{\text{c}}}{R_{\text{T}}}}\\[3pt]{}\Rightarrow R_{1}&={\frac {R_{\text{b}}R_{\text{c}}}{R_{\text{T}}}}.\end{aligned}} For completeness:
$R_{1}={\frac {R_{\text{b}}R_{\text{c}}}{R_{\text{T}}}}$ (4)
$R_{2}={\frac {R_{\text{a}}R_{\text{c}}}{R_{\text{T}}}}$ (5)
$R_{3}={\frac {R_{\text{a}}R_{\text{b}}}{R_{\text{T}}}}$ (6)

Let
$R_{\text{T}}=R_{\text{a}}+R_{\text{b}}+R_{\text{c}}$ .
We can write the Δ to Y equations as
$R_{1}={\frac {R_{\text{b}}R_{\text{c}}}{R_{\text{T}}}}$ (1)
$R_{2}={\frac {R_{\text{a}}R_{\text{c}}}{R_{\text{T}}}}$ (2)
$R_{3}={\frac {R_{\text{a}}R_{\text{b}}}{R_{\text{T}}}}.$ (3)
Multiplying the pairs of equations yields
$R_{1}R_{2}={\frac {R_{\text{a}}R_{\text{b}}R_{\text{c}}^{2}}{R_{\text{T}}^{2}}}$ (4)
$R_{1}R_{3}={\frac {R_{\text{a}}R_{\text{b}}^{2}R_{\text{c}}}{R_{\text{T}}^{2}}}$ (5)
$R_{2}R_{3}={\frac {R_{\text{a}}^{2}R_{\text{b}}R_{\text{c}}}{R_{\text{T}}^{2}}}$ (6)
and the sum of these equations is
$R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}={\frac {R_{\text{a}}R_{\text{b}}R_{\text{c}}^{2}+R_{\text{a}}R_{\text{b}}^{2}R_{\text{c}}+R_{\text{a}}^{2}R_{\text{b}}R_{\text{c}}}{R_{\text{T}}^{2}}}$ (7)
Factor $R_{\text{a}}R_{\text{b}}R_{\text{c}}$ from the right side, leaving $R_{\text{T}}$ in the numerator, canceling with an $R_{\text{T}}$ in the denominator.
{\begin{aligned}R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}&={}{\frac {\left(R_{\text{a}}R_{\text{b}}R_{\text{c}}\right)\left(R_{\text{a}}+R_{\text{b}}+R_{\text{c}}\right)}{R_{\text{T}}^{2}}}\\&={}{\frac {R_{\text{a}}R_{\text{b}}R_{\text{c}}}{R_{\text{T}}}}\end{aligned}} (8)
Note the similarity between (8) and {(1), (2), (3)}
Divide (8) by (1)
{\begin{aligned}{\frac {R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}}{R_{1}}}&={}{\frac {R_{\text{a}}R_{\text{b}}R_{\text{c}}}{R_{\text{T}}}}{\frac {R_{\text{T}}}{R_{\text{b}}R_{\text{c}}}}\\&={}R_{\text{a}},\end{aligned}} which is the equation for $R_{\text{a}}$ . Dividing (8) by (2) or (3) (expressions for $R_{2}$ or $R_{3}$ ) gives the remaining equations.