Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal

Curvilinear Coordinates Systems from the Covariant Form of Maxwell’s Equations

408

on the variable . The field vector t

3

x

and t

G are

not coupled since the operator t in Equation (20) is

null. The propagation Equation (38) is given by:

G

22233 21313

12331 1

23 23

322 0

c

kgg g

gg

(86)

This last equation may be compared to the equations

described by [22,23].

6. Conclusion and Future Work

In this paper, a generalized second order potential for-

mulation (SOVP) is proposed for solving scattering or

radiation problems described in an arbitrary non-or-

thogonal curvilinear coordinate system. This formulation

takes advantages from the tensor analysis but no exper-

tise is finally required for developing the expressions of

the electromagnetic field in terms of two scalar potentials,

usually the transverse electric potential and the transverse

magnetic potential. For writing the components of the

electrical field and the magnetic field for any curvilinear

coordinate system, it is necessary to write the metric

tensor which is easily defined in the paper and to use the

vector cross product as usual in a Cartesian coordinate

system. This SOVP formulation represents the key stone

for implementing new numerical models dedicated to

eddy current calculations based on the covariant form of

the Maxwell’ equations. By using a specific curvilinear

coordinate system matching the geometry of the bound-

ary surface, it is possible to write easily and analytically

the boundary conditions implying the covariant and con-

travariant components of the electromagnetic field. In

future work, a numerical method will be developed for

calculating eddy currents induced in a conducting work-

piece due to a 3D eddy current probe scanning the boun-

dary surface described by an arbitrary and irregular ge-

ometry.

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