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ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΤΟΜΕΑΣ ΗΛΕΚΤΡΙΚΗΣ ΕΝΕΡΓΕΙΑΣ ΕΡΓΑΣΤΗΡΙΟ ΣΥΣΤΗΜΑΤΩΝ ΗΛΕΚΤΡΙΚΗΣ ΕΝΕΡΓΕΙΑΣ ∆ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΥΠΟΛΟΓΙΣΜΟΣ ΑΝΤΙΣΤΑΣΕΩΝ ΣΤΡΩΜΑΤΟΠΟΙΗΜΕΝΗΣ ΓΗΣ ΑΠΟ ΜΕΤΡΗΣΕΙΣ ΑΝΤΙΣΤΑΣΗΣ ΣΤΗΝ ΕΠΙΦΑΝΕΙΑ Α∆ΑΜ ΧΡΥΣΟΥΛΑ , Α.Ε.Μ 4696 ΒΙΛΛΙΩΤΗΣ ΣΩΤΗΡΙΟΣ , Α.Ε.Μ 4632 ΕΠΙΒΛΕΠΟΝΤΕΣ: ΠΑΠΑΓΙΑΝΝΗΣ ΓΡΗΓΟΡΙΟΣ, ΕΠΙΚΟΥΡΟΣ ΚΑΘΗΓΗΤΗΣ ΠΑΠΑ∆ΟΠΟΥΛΟΣ ΘΕΟΦΙΛΟΣ, ΜΕΤΑΠΤΥΧΙΑΚΟΣ ΦΟΙΤΗΤΗΣ ΘΕΣΣΑΛΟΝΙΚΗ 2008

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  • , .. 4696 , .. 4632

    : , , 2008

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    ,

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    .

    Wenner Schlumberger.

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    MATLAB .

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  • 1. .5 1.1. .....5 1.2.

    ...5 2. ...9

    2.1. 9 2.2. .10 2.3. 12

    2.3.1. .12

    2.3.2. .13

    2.4. 17 2.5. ...27 2.6. ...29 2.7 ..33 2.8 ..35

    3 ..40 3.1. 43 3.2. .45

    3.2.1. .45

    3.2.2. Newton..46

    3.2.3. Levenberg-Marguardt.47

    3.2.4. .48

    3.2.5. Quasi-Newton..50

    3.3. ..51

    3.4. .59 3.5. .60

    4. 62 4.1. .62 4.2. .69

    5. 71 5.1. DEL ALAMO71 5.2. F. DAWALIBI76 5.3. HANS SEEDHER- ARORA78 5.4. I. GONOS..80 5.5. .81

    6. .82

    3

  • 7. ..84 8. ..85 9. ..90

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  • 1.

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    7,2 330

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    -5 23 79.000

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    3.

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  • 1.4

    (/m)

    3 x 10-1

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    1012 - 1013

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    9 x 1012 - 1014

    100 - 106

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    20 2 x 103

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    2.3 2.3.1

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    ( 2.5).

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    ohm/m.

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    41

  • D= m . ( ) ( )( ) ( )j m j j md r x,r / = jr

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    = = i (7)

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    = vxv

    = ( )F x , : ( )pF xx . vxv.

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    42

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    rj

    X

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    m x 1

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    ( )F x 2D& & D.

    44

  • ( )2

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    :

    Levenberg-Marguardt Newton Quasi-Newton

    3.2.1

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    , x px +1px

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    () px

    ( ) ( ) ( )p pt 1F X F X F X x xH x2 = + + (17)

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    , .

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    JtJ .

    46

  • R, .

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    3.2.3 Levenberg-Marguardt

    ,

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    ( )F x x ,

    px ,

    ( )F x tx x 1 = Lagrangian,

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    ,

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    . ,

    47

  • ( )p tF x J D = (23) ,

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    1, 2 z.

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    rj

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    p 1 px x x += ()

    rj

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    = (25)

    j=1.m i=13

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    48

  • tE D J x= +

    , :

    & &2E = & &2 0E

    x , j=1..m

    = + 210 2 2m a a a ai

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    x ix x x (26)

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    1p t tx J J J D = (27)

    ,

    D ,

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    Levenberg-Marguardt, JtJ .

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    .

    49

  • 3.2.5 Quasi-Newton

    px ( )pA F x , .

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    Quasi-Newton , -

    .

    Quasi-Newton,

    ( ) ( )( )p 1 p p p p 11x x x A F x F x+ + = = (28)

    ,

    :

    1. p tx J D =

    2. Newton 1p t tx J J R J D = +

    3. Levenberg-Marguardt 1p t tx J J I J D = +

    4. 1p t tx J J J D =

    5. Quasi-Newton

    ( ) ( )( )p 1 p p p p 11x x x A F x F x+ + = =

    .

    50

  • ,

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    2. 02 .

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    4. ,1 m . 0z

    51

  • ,

    F, , , 2D& & F( )F x& & .

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    1, 2 z

    1,

    Turbo C.

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    1-5 .

    ,

    10.000.

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    [] 1.005 = , 2.005 z ( )F X 1.E 3 , ,

    ,

    . 1. E 5

    52

  • 1.

    Guess a vector as starting point

    0 0 01 , ,z 0

    Compute ( )pF X

    Normalize it ( )pnF X

    FORM THE CHANGES

    ( )pnx F x =

    Searching the maximum decrease of ( )F x

    ( ) 1. 5pF X E

    Compute 1( )pF X +

    ( ) 1. 10F X E

    SOLVE

    1

    1 ( ) ( )p px x F x F Xx

    + =

    CORRECT the vector to the NEW value

    1p px x x = +

    53

  • 2. (FOGT)

    , k, z.

    1,

    Turbo C.

    , ,

    0.001 .

    , ,

    ,

    .

    ( )F X 1.E 3 , ,

    ,

    . 1. . E 5

    3. .

    (FOGT),

    1,k, z,

    2, Turbo C.

    , ( )F x

    . ,

    1.000.

    1.E 5

    IV, (0,1)

    .

    1. . E 3

    54

  • Build [ ] vector

    Compute 1 [ ] ( )p p ptk nX X F X+ =

    Compute Gradient s Components ( )DF X Normalize it ( ) ( ) ( )

    pp pnF X F X F X =

    Compute 1( )pF X +

    1( ) ( ) ( )p p pF X F X F X+ =

    ( )pF X