ET04.pdf

X the following results are obtained 108 109 by

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x , the following results are obtained: (108) (109) By subtracting Eq. 109 from Eq. 108 and rearranging the terms, the first derivative can be written as: (110) Then, neglecting terms with ( x ) 2 or higher powers of x , the expression in Eq. 107 is obtained. This technique is less intuitive than the one used to derive Eq. 107 but shows two important points. 1. The error produced by the approximation is on the order of ( x ) 2 . Thus, there is a simple way of estimating the error and, at the same time, the solution can be improved by reducing the size of x . 2. The finite difference formula in Eq. 110 was obtained by choosing an appropriate expansion to cancel specific terms of the expansions. This indicates that higher order derivatives and different approximations to the same derivatives can be obtained. Indeed, many useful difference formulas have been derived. 134 An approximation for the second derivative is obtained by adding the two expansions in Eqs. 108 and 109 and rearranging the terms: (111) This particular approximation also introduces an error on the order of ( x ) 2 . The finite difference expressions derived here use points on both sides of the point at which the derivatives are calculated. They are therefore called central difference expressions. Backward and forward difference formulas may also be used. 134 Finite Difference Formulation for Two-Dimensional and Axisymmetric Field Problems The first step in the field formulation consists of replacing the partial derivatives by a difference equation. Referring to the grid in Fig. 21, Eq. 19 reduces in the case of an axisymmetric geometry in cylindrical coordinates to: (112) Here the current density J s was replaced by an equivalent nodal current I i,j . A similar expression may be written for the two-dimensional field equation. If the grid is equally spaced in both directions (W r = W z = h ) and a constant 1 2 1 2 1 1 2 1 1 1 2 2 µ ωσ × + ( ) + + + ( ) = − + ( ) + ( ) + ( ) ( ) + ( ) A A A r r A A r A A A z A r I j A i j i j i j i j i j i j i j i j i j i j i j , , , , , , , , , , , ′′ = + ( ) ′′′′ ( ) + + ( ) ( ) y y y y x y x i i i i i 1 1 2 2 2 1 12 L = ′′′ ( ) + + ( ) ( ) y y y x y x i x x x x i i i 2 1 6 2 L y y y x y x y x x x i i i 1 2 3 2 3 ( ) = + ′′ ( ) ′′′ ( ) + ! ! L y y y x y x y x x x i i i i ( ) ! ! 1 2 3 2 3 + = + + ′′ ( ) + ′′′ ( ) + L = + ( ) ( ) y y y x i i i 1 1 2 = ( ) + ( ) ( ) y y y x x x x x x i i i 2 98 Electromagnetic Testing
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permeability can be assumed for all points, Eq. 112 is further simplified as: (113) Equation 112 forms the basis for solution of eddy current problems in axisymmetric geometries. In its present form, the equation is of little use because, besides applying only to regular grids, it can be used to describe only nonmagnetic media. This limitation can be seen by testing where the interface between two materials is shown (Fig. 22). When calculating the value of the magnetic
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  • Fall '19
  • Wind, The Land, Magnetic Field, Dodd, Modeling of Electromagnetic Testing

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