# 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:

[1] ZX1-X2 = (ZW1 + ZW2)​
[2] ZX2-X3 = (ZW2 + ZW3)​
[3] ZX3-X1 = (ZW3 + ZW1)​

Rewriting these expressions gives:

[4] ZW1 = (ZX1-X2ZW2)​
[5] ZW2 = (ZX2-X3ZW3)​
[6] ZW3 = (ZX3-X1ZW1)​

Substituting [4] into [3] yields equation [7],
Substituting [5] into [7] yields equation [8],

Substituting [5] into [1] yields equation [9],
Substituting [6] into [9] yields equation [10],

Substituting [6] into [3] yields equation [11],
Substituting [4] into [11] yields equation [12],

Rearranging expressions [8], [10], and [12] 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:

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

Subtracting [1] and [2] yields equation [4],
Subtracting [2] and [3] yields equation [5],
Subtracting [3] and [1] yields equation [6],

Adding [4] and [3] yields equation [7],
Adding [5] and [1] yields equation [8],
Adding [6] and [2] yields equation [9],

Dividing [7] by [8] yields equation [10],
Dividing [8] by [9] yields equation [11],
Dividing [9] by [7] yields equation [12],

Substituting [10] and [11] into [7] yields equation [13],
Substituting [10] into [13] yields equation [14],

Substituting [11] and [12] into [8] yields equation [15],
Substituting [11] into [15] yields equation [16],

Substituting [12] and [10] into [9] yields equation [17],
Substituting [12] into [17] yields equation [18],

Rearranging/simplifying expressions [14], [16], and [18] 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.