#### xptpcrewx

##### Power System Engineer

- Location
- Las Vegas, Nevada, USA

- Occupation
- Licensed Electrical Engineer, Licensed Electrical Contractor, Certified Master Electrician

**Z**= {R +

**j**X} notation.

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

__What is the value of the single lowest Y connected phase winding?__

**Question #1 (20 points):**Let

**Z**

_{W1},

**Z**

_{W2}, and

**Z**

_{W3}be the winding impedances connected between terminals X

_{1}-X

_{0}, X

_{2}-X

_{0}, and X

_{3}-X

_{0}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,__

**W**

**inding 1**__is the single lowest Y connected phase winding.__

__What is the value of the single lowest Δ connected phase winding?__

**Question #2 (80 points):**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,__

**W**

**inding 2**__is the single lowest Δ connected phase winding.__