Winding Resistance Challenge

[xptpcrewx]

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.

Spoiler: Solution Question #1 (20 points)

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

Let Z_W1, Z_W2, and Z_W3 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] Z_X1-X2 = (Z_W1 + Z_W2)​

[2] Z_X2-X3 = (Z_W2 + Z_W3)​

[3] Z_X3-X1 = (Z_W3 + Z_W1)​

Rewriting these expressions gives:

[4] Z_W1 = (Z_X1-X2 – Z_W2)​

[5] Z_W2 = (Z_X2-X3 – Z_W3)​

[6] Z_W3 = (Z_X3-X1 – Z_W1)​

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:

Z_W3 = (1/2)·[ Z_X3-X1 – Z_X1-X2 + Z_X2-X3]​

Z_W1 = (1/2)·[ Z_X1-X2 – Z_X2-X3 + Z_X3-X1]​

Z_W2 = (1/2)·[ Z_X2-X3 – Z_X3-X1 + Z_X1-X2]​

Plugging in R_X1-X2 = 2.667·mΩ, R_X2-X3 = 2.672·mΩ, R_X3-X1 = 2.645·mΩ into the above equations gives:

Z_W1 = 1.320·mΩ​

Z_W2 = 1.347·mΩ​

Z_W3 = 1.325·mΩ​

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

Spoiler: Solution Question #2 (80 points)

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

Let Z_W1, Z_W2, and Z_W3 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] Z_H1-H2 = [Z_W1 || (Z_W3 + Z_W2)] = {[Z_W1·(Z_W3 + Z_W2)]/[Z_W1 + (Z_W3 + Z_W2)]}​

[2] Z_H2-H3 = [Z_W2 || (Z_W1 + Z_W3)] = {[Z_W2·(Z_W1 + Z_W3)]/[Z_W2 + (Z_W1 + Z_W3)]}​

[3] Z_H3-H1 = [Z_W3 || (Z_W2 + Z_W1)] = {[Z_W3·(Z_W2 + Z_W1)]/[Z_W3 + (Z_W2 + Z_W1)]}​

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:

Z_W2 = {Z_H2-H3 + (1/2)·[(Z_H3-H1 – Z_H1-H2 + Z_H2-H3)·(Z_H2-H3 – Z_H3-H1 + Z_H1-H2)]/(Z_H1-H2 – Z_H2-H3 + Z_H3-H1)}​

Z_W3 = {Z_H3-H1 + (1/2)·[(Z_H3-H1 – Z_H1-H2 + Z_H2-H3)·(Z_H1-H2 – Z_H2-H3 + Z_H3-H1)]/(Z_H2-H3 – Z_H3-H1 + Z_H1-H2)}​

Z_W1 = {Z_H1-H2 + (1/2)·[(Z_H1-H2 – Z_H2-H3 + Z_H3-H1)·(Z_H2-H3 – Z_H3-H1 + Z_H1-H2)]/(Z_H3-H1 – Z_H1-H2 + Z_H2-H3)}​

Plugging in R_H1-H2 = 2.428·mΩ, R_H2-H3 = 2.315·mΩ, R_H3-H1 = 2.445·mΩ into the above equations gives:

Z_W1 = 3.688·mΩ​

Z_W2 = 3.363·mΩ​

Z_W3 = 3.743·mΩ​

​

Hence by inspection, Winding 2 is the single lowest Δ connected phase winding.