Editing Y-Δ transform (section)

==Basic Y-Δ transformation==
[[Image:Wye-delta-2.svg|right|thumb|300px|Δ and Y circuits with the labels which are used in this article.]]
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 [[ohm]]s which represents resistance as positive real numbers in the usual manner, and also represents [[Electrical reactance|reactance]] as positive and negative [[imaginary value]]s.

===Equations for the transformation from Δ to Y===<!--This section is linked from [[Template:Network analysis navigation]]. Changing this heading will break the template unless updated there also.-->
The general idea is to compute the impedance <math>R_\text{Y}</math> at a terminal node of the Y circuit with impedances <math>R'</math>, <math>R''</math> to adjacent nodes in the Δ circuit by

:<math>R_\text{Y} = \frac{R'R''}{\sum R_\Delta}</math>

where <math>R_\Delta</math> are all impedances in the Δ circuit. This yields the specific formula

:<math>\begin{align}
  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{align}</math>

===Equations for the transformation from Y to Δ===<!--This section is linked from [[Template:Network analysis navigation]]. Changing this heading will break the template unless updated there also.-->
The general idea is to compute an impedance <math>R_\Delta</math> in the Δ circuit by

:<math>R_\Delta = \frac{R_P}{R_\text{opposite}}</math>

where <math>R_P = R_1 R_2 + R_2 R_3 + R_3 R_1</math> is the sum of the products of all pairs of impedances in the Y circuit and <math>R_\text{opposite}</math> is the impedance of the node in the Y circuit which is opposite the edge with <math>R_\Delta</math>. The formulae for the individual edges are thus

:<math>\begin{align}
  R_\text{a} &= \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_1} = R_2+R_3+\frac{R_2R_3}{R_1} \\[3pt]
  R_\text{b} &= \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_2} = R_1+R_3+\frac{R_1R_3}{R_2} \\[3pt]
  R_\text{c} &= \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_3} = R_1+R_2+\frac{R_1R_2}{R_3}
\end{align}</math>

Or, if using [[admittance]] instead of resistance:
:<math>\begin{align}
  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{align}</math>

Note that the general formula in Y to Δ using admittance is similar to Δ to Y using resistance.