Star-Delta Transformation

1
Star-Delta Transformation
2
Examples

• Star-to-Delta
• Page 19
• Advance Electrical Eng
• by
• Morton

3
Mesh Transformation

• Examples
• From
• The application of Matrix Theory to EE
• By
• Lewis Pruce
• Chapter four page 95

4
Mesh Analysis
Mesh 1 25I1 60I1 60I2 - 25I3 10
0 85 I1 60 I2 - 25 I3 10 ..(1)
5
Mesh Analysis
Mesh 2 60 I2 40 I2 - 60 I1 - 40 I3 20 0 -
60 I1 100 I2 - 40 I3 - 20 .(2)
6
Mesh Analysis
Mesh 3 40 I3 25 I3 10 I3 25 I1 40 I2
0 -25 I1 - 40 I2 75 I3 0 . (3)
7
Mesh Analysis

• So we have our three mesh equations from the
  circulating currents in the three meshes. Note we
  write them down in a certain order
• A I1 B I2 C I3 D
• We can represent these simultaneous equations as
  a matrix equation as follows ZI E

8
Mesh Analysis

• ZI E
• Can be written with values

9
Mesh Analysis

• Z can written like

Where Z11 represents impedance of loop 1 and does
not share with any other loop. Z12 is an
impedance in loop 1 that is shared with loop
2. Z13 is an impedance of the loop 1 and shared
with loop 3.
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Mesh Analysis

• Similarly each row has impedances present in the
  respective loop but shared with mesh of suffix
  no. of the column.

11
Mesh Analysis

• The numbers on the diagonal of the matrix are
  positive. These are the mesh self-impedance and
  are just the sum of the impedances in each mesh.
• The numbers off the diagonal represent the
  total impedances from one mesh with respect to
  another i.e. Z12, Z13 etc. Note there are always
  two i.e. the impedance from mesh one to mesh two
  is the same as from mesh 2 to mesh 1. Hence for
  example Z31 is the same as Z13.
• The numbers from one mesh with respect to
  another are always negative.
• Note that the emf are positive when aiding a
  circulating current i.e. on the LHS of the mesh
  and negative when opposing a circulating current
  i.e. on the RHS of a mesh.

12
Solution using Cramers Rule

• Find the determinant of Z matrix

That could be done by expanding either one row or
one column For 1st element i.e 85 minors are
Is evaluated (100 x 75) (-40 x -40) 5900
and is called co-factor of 85
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Solution using Cramers Rule

• The procedure is repeated for each element of the
  row (column) chosen
• Another thing to note is that the sign of the
  number in the determinant has a pattern as
  follows

14
Solution using Cramers Rule

• ?R () (85) (100)(75) (-40)(-40)
• (-) (-60) (-60)(75) (-25)(-40)
• () (-25) (-60)(-40) (-25)(100)
• Note the signs ( - ) !!
• ?R 49000

15
Solution using Cramers Rule

• The next step is to replace the coefficients of
  I1 with the numbers on the right hand side of the
  equation, that is the column vector of applied
  e.m.f's, as follows and work out its determinant
  ?1 -51000
  ?

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Solution using Cramers Rule

• Then for I2 and I3 and workout their
  determinants. i.e
• ?2 -60000
• ?3 -49000 ?
  ?

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Solution using Cramers Rule

• Then to workout different branch currents
• I1 ?1/ ? R -1.04 A
• I2 ? 2/ ? R -1.22 A
• I3 ? 3/ ? R -1 A

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Solution by Matrix Inversion

• ZI E

I Z-1 E
Which involves finding the inverse matrix Z -1
There are several methods of doing this. One
method is as follows
Where CT is the adjoint of the matrix Z and Z
is the determinant of the matrix Z
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Solution by Matrix Inversion

• 
  Z 49000

Finally I Z-1 E
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Solution by Matrix Inversion
We can now label the currents
21
Assignment exercise

• Determine Loop branch currents using both
  methodsi.e. Cramers Matrix

22
Hyperlinks

• en.wikipedia.org/wiki/Mesh_analysis