# Winding Resistance Challenge

#### xptpcrewx

##### Power System Engineer
Once again, thanks to everyone who participated and/or contributed to this challenge, and special thanks/recognition to those who solved it. Below is my version of the solution - which is the procedure to derive the necessary equations along with the final result. In the spirit of post #13, the variables are expressed in Z = {R + jX} notation.

Note: There may be a simpler derivation than the ones shown below, but herein I have employed the "brute-force" method.

Question #1 (20 points): What is the value of the single lowest Y connected phase winding?

Let ZW1, ZW2, and ZW3 be the winding impedances connected between terminals X1-X0, X2-X0, and X3-X0 respectively.
A system of equation describing a wye circuit topology can then be expressed as:

 ZX1-X2 = (ZW1 + ZW2)​
 ZX2-X3 = (ZW2 + ZW3)​
 ZX3-X1 = (ZW3 + ZW1)​

Rewriting these expressions gives:

 ZW1 = (ZX1-X2ZW2)​
 ZW2 = (ZX2-X3ZW3)​
 ZW3 = (ZX3-X1ZW1)​

Substituting  into  yields equation ,
Substituting  into  yields equation ,

Substituting  into  yields equation ,
Substituting  into  yields equation ,

Substituting  into  yields equation ,
Substituting  into  yields equation ,

Rearranging expressions , , and  yields:

ZW3 = (1/2)·[ ZX3-X1ZX1-X2 + ZX2-X3]
ZW1 = (1/2)·[ ZX1-X2ZX2-X3 + ZX3-X1]
ZW2 = (1/2)·[ ZX2-X3ZX3-X1 + ZX1-X2]

Plugging in RX1-X2 = 2.667·mΩ, RX2-X3 = 2.672·mΩ, RX3-X1 = 2.645·mΩ into the above equations gives:

ZW1 = 1.320·mΩ
ZW2 = 1.347·mΩ​
ZW3 = 1.325·mΩ​

Hence by inspection, Winding 1 is the single lowest Y connected phase winding.

Question #2 (80 points): What is the value of the single lowest Δ connected phase winding?

Let ZW1, ZW2, and ZW3 be the winding impedances connected between terminals H1-H2, H2-H3, and H3-H1 respectively.
A system of equation describing a delta circuit topology can then be expressed as:

 ZH1-H2 = [ZW1 || (ZW3 + ZW2)] = {[ZW1·(ZW3 + ZW2)]/[ZW1 + (ZW3 + ZW2)]}​
 ZH2-H3 = [ZW2 || (ZW1 + ZW3)] = {[ZW2·(ZW1 + ZW3)]/[ZW2 + (ZW1 + ZW3)]}​
 ZH3-H1 = [ZW3 || (ZW2 + ZW1)] = {[ZW3·(ZW2 + ZW1)]/[ZW3 + (ZW2 + ZW1)]}​

Subtracting  and  yields equation ,
Subtracting  and  yields equation ,
Subtracting  and  yields equation ,

Adding  and  yields equation ,
Adding  and  yields equation ,
Adding  and  yields equation ,

Dividing  by  yields equation ,
Dividing  by  yields equation ,
Dividing  by  yields equation ,

Substituting  and  into  yields equation ,
Substituting  into  yields equation ,

Substituting  and  into  yields equation ,
Substituting  into  yields equation ,

Substituting  and  into  yields equation ,
Substituting  into  yields equation ,

Rearranging/simplifying expressions , , and  yields:

ZW2 = {ZH2-H3 + (1/2)·[(ZH3-H1ZH1-H2 + ZH2-H3)·(ZH2-H3ZH3-H1 + ZH1-H2)]/(ZH1-H2ZH2-H3 + ZH3-H1)}
ZW3 = {ZH3-H1 + (1/2)·[(ZH3-H1ZH1-H2 + ZH2-H3)·(ZH1-H2ZH2-H3 + ZH3-H1)]/(ZH2-H3ZH3-H1 + ZH1-H2)}
ZW1 = {ZH1-H2 + (1/2)·[(ZH1-H2ZH2-H3 + ZH3-H1)·(ZH2-H3ZH3-H1 + ZH1-H2)]/(ZH3-H1 ZH1-H2 + ZH2-H3)}

Plugging in RH1-H2 = 2.428·mΩ, RH2-H3 = 2.315·mΩ, RH3-H1 = 2.445·mΩ into the above equations gives:

ZW1 = 3.688·mΩ
ZW2 = 3.363·mΩ
ZW3 = 3.743·mΩ​
Hence by inspection, Winding 2 is the single lowest Δ connected phase winding.