Editing Y-Δ transform (section)

===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.