Editing Y-Δ transform (section)

==A proof of the existence and uniqueness of the transformation==
The feasibility of the transformation can be shown as a consequence of the [[superposition theorem|superposition theorem for electric circuits]]. A short proof, rather than one derived as a corollary of the more general [[star-mesh transform]], can be given as follows. The equivalence lies in the statement that for any external voltages (<math>V_1, V_2</math> and <math>V_3</math>) applying at the three nodes (<math>N_1, N_2</math> and <math>N_3</math>), the corresponding currents (<math>I_1, I_2</math> and <math>I_3</math>) are exactly the same for both the Y and Δ circuit, and vice versa. In this proof, we start with given external currents at the nodes. According to the superposition theorem, the voltages can be obtained by studying the superposition of the resulting voltages at the nodes of the following three problems applied at the three nodes with current:

# <math>  \frac{1}{3}\left(I_1 - I_2\right),   -\frac{1}{3}\left(I_1 - I_2\right), 0</math>
# <math>0,\frac{1}{3}\left(I_2 - I_3\right),   -\frac{1}{3}\left(I_2 - I_3\right)</math> and
# <math> -\frac{1}{3}\left(I_3 - I_1\right), 0, \frac{1}{3}\left(I_3 - I_1\right)</math>

The equivalence can be readily shown by using [[Kirchhoff's circuit laws]] that <math>I_1 + I_2 + I_3 = 0</math>. Now each problem is relatively simple, since it involves only one single ideal current source. To obtain exactly the same outcome voltages at the nodes for each problem, the equivalent resistances in the two circuits must be the same, this can be easily found by using the basic rules of [[series and parallel circuits]]:

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

Though usually six equations are more than enough to express three variables (<math>R_1, R_2, R_3</math>) in term of the other three variables(<math>R_\text{a}, R_\text{b}, R_\text{c}</math>), here it is straightforward to show that these equations indeed lead to the above designed expressions.

In fact, the superposition theorem establishes the relation between the values of the resistances, the [[electromagnetism uniqueness theorem|uniqueness theorem]] guarantees the uniqueness of such solution.