Understanding derivation of delta-y transform

[TheCanadian]

Hi,

I was just looking over my textbook, and it mentions a $\Delta$-y and y-$\Delta$ transformation that is helpful for dealing with circuits in these configurations. The equations can be found here: https://en.wikipedia.org/wiki/Y-Δ_t...xistence_and_uniqueness_of_the_transformation

After looking through the above link and searching for a proper derivation elsewhere, I simply don't seem to understand how the transform equations were derived. If I'm not mistaken, the whole purpose is to be able to evaluate a $\Delta$ circuit as a Y circuit, and vice versa. Thus, equivalent resistances must be found. But after looking at the derivation provided in the wikipedia link, I don't quite see how it is known that the impedance at a node is: $R = \frac {R'R''}{\sum R_{\Delta}}$. Maybe my understanding of nodal analysis is poor, but how is that expression specifically derived?

Any help would be greatly appreciated!

 

[ehild]

Given R_A, R_B, R_C, determine R1,R2,R3 so as the equivalent resistance between any two points is the same in both circuits.
For 1,2 it means :
$$\frac{1}{\frac{1}{R_C}+\frac{1}{R_A+R_B}}=R_1+R_2$$
Write up the other two equations for 2,3 and 3,1, and solve for R₁, R₂, R₃.Or watch the video

 

[TheCanadian]

View attachment 89767

Given R_A, R_B, R_C, determine R1,R2,R3 so as the equivalent resistance between any two points is the same in both circuits.
For 1,2 it means :
$$\frac{1}{\frac{1}{R_C}+\frac{1}{R_A+R_B}}=R_1+R_2$$
Write up the other two equations for 2,3 and 3,1, and solve for R₁, R₂, R₃.Or watch the video

Thank you! Putting the equation into that form really helped!