Editing Y-Δ transform (section)

==Demonstration==
===Δ-load to Y-load transformation equations===
[[Image:Wye-delta-2.svg|right|thumb|325px|Δ and Y circuits with the labels that are used in this article.]]

To relate <math>\left\{R_\text{a}, R_\text{b}, R_\text{c}\right\}</math> from Δ to <math>\left\{R_1, R_2, R_3\right\}</math> 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 ''N''<sub>1</sub> and ''N''<sub>2</sub> with ''N''<sub>3</sub> disconnected in Δ:

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

To simplify, let <math>R_\text{T}</math> be the sum of <math>\left\{R_\text{a}, R_\text{b}, R_\text{c}\right\}</math>.
:<math> R_\text{T} = R_\text{a} + R_\text{b} + R_\text{c} </math>

Thus,

:<math>R_\Delta\left(N_1, N_2\right) = \frac{R_\text{c}(R_\text{a} + R_\text{b})}{R_\text{T}}</math>

The corresponding impedance between N<sub>1</sub> and N<sub>2</sub> in Y is simple:

:<math>R_\text{Y}\left(N_1, N_2\right) = R_1 + R_2</math>

hence:

:<math>R_1 + R_2 = \frac{R_\text{c}(R_\text{a} + R_\text{b})}{R_\text{T}}</math>   (1)

Repeating for <math>R(N_2,N_3)</math>:

:<math>R_2 + R_3 = \frac{R_\text{a}(R_\text{b} + R_\text{c})}{R_\text{T}}</math>   (2)

and for <math>R\left(N_1, N_3\right)</math>:

:<math>R_1 + R_3 = \frac{R_\text{b}\left(R_\text{a} + R_\text{c}\right)}{R_\text{T}}.</math>   (3)

From here, the values of <math>\left\{R_1, R_2, R_3\right\}</math> can be determined by linear combination (addition and/or subtraction).

For example, adding (1) and (3), then subtracting (2) yields

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

For completeness:

:<math>R_1 = \frac{R_\text{b}R_\text{c}}{R_\text{T}}</math>   (4)
:<math>R_2 = \frac{R_\text{a}R_\text{c}}{R_\text{T}}</math>   (5)
:<math>R_3 = \frac{R_\text{a}R_\text{b}}{R_\text{T}}</math>   (6)

===Y-load to Δ-load transformation equations===
Let 

:<math>R_\text{T} = R_\text{a} + R_\text{b} + R_\text{c}</math>.

We can write the Δ to Y equations as

:<math>R_1 = \frac{R_\text{b}R_\text{c}}{R_\text{T}} </math>   (1)
:<math>R_2 = \frac{R_\text{a}R_\text{c}}{R_\text{T}} </math>   (2)
:<math>R_3 = \frac{R_\text{a}R_\text{b}}{R_\text{T}}. </math>   (3)

Multiplying the pairs of equations yields

:<math>R_1 R_2 = \frac{R_\text{a}R_\text{b}R_\text{c}^2 }{R_\text{T}^2}</math>   (4)
:<math>R_1 R_3 = \frac{R_\text{a}R_\text{b}^2 R_\text{c}}{R_\text{T}^2}</math>   (5)
:<math>R_2 R_3 = \frac{R_\text{a}^2 R_\text{b}R_\text{c}}{R_\text{T}^2}</math>   (6)

and the sum of these equations is

:<math>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}^2R_\text{c} +
    R_\text{a}^2R_\text{b}R_\text{c}}
  {R_\text{T}^2}
</math>   (7)

Factor <math>R_\text{a}R_\text{b}R_\text{c}</math> from the right side, leaving <math>R_\text{T}</math> in the numerator, canceling with an <math>R_\text{T}</math> in the denominator.

:<math>\begin{align}
  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{align}</math>   (8)

Note the similarity between (8) and {(1), (2), (3)}

Divide (8) by (1)

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

which is the equation for <math>R_\text{a}</math>. Dividing (8) by (2) or (3) (expressions for <math>R_2</math> or <math>R_3</math>) gives the remaining equations.