INTRODUCTION
In discharge or partial discharge researches, electric field`s line has
very important role in the analysis of initiation and propagation process.
It`s often assumed that the discharge begins at a local point on an electrode
where the electrical field becomes enough strong and it follows the field`s
line going from that point (Bondiou and Gallimberti, 1994 ; Giralt and
Buret, 2000; Gallimberti et al., 2002). In these investigations,
the rod to plan geometry is generally used because the discharge`s line
and the point where it`s initiated are known. In a real configuration,
the starting point of the discharge and its trajectory are random.
Electric field calculation`s softwares have permitted a significant improvement
in the design of highvoltage apparatus by bringing a detailed knowledge
of the field in the actual geometry (Longeot et al., 1991; Meunier
et al., 1991; Zaho and Comber, 2000). However, the software often
used presents only some lines representing equal values, the field and
its coordinates at a point. Therefore, any tool giving access to field
line will be useful in discharge and partial discharge studies.
Finite elements method is used to solve the Laplace`s equation with a
scalar potential formulation (Chari and Salon, 2000). The electrical field
is then derivate. It allows the determination of the first point of the
field`s line. Then, using approximation methods, the propagation of the
field`s line is established. Both convergent and divergent cases are presented.
The calculation code is validated using the result of the rod to plane
structure studied by Ibrahim (1988). The field`s lines and what could
be a discharge`s line for an aerial insulator are also investigated.
The main aim of this study is to provide the electric field`s lines in
any kind of electrode gap. Thus, the electrical discharge can be initiated
at any point of the electrode. It may follow any field line. This project
poses the base of the development of a discharge model applicable to industrial
structure.
MATERIALS AND METHODS
Electric Field Calculation
Three dimensional tools must be used when solving electric discharge
problems. Nevertheless, most of the electrical devices can be modelled
using 2D or axisymmetric formulation. The electric field cartography is
determined by solving the Laplace`s equation (Chari and Salon, 2000; Yeo
et al., 1997):
where, V is the scalar electric potential, Îµ and Îµ_{0}
are the relative and absolute dielectric permittivities, respectively.
In order to have a unique solution, boundary and interface conditions
are necessary. They are of the type:
where,
is the normal derivative of the potential. Interfaces conditions express
the continuity of the potential on both sides of two mediums having different
permittivities.
Equation 1 is discretized by a finite element method
(Chari and Salon, 2000). Elements are linear triangles. The electric field
E can then be derivate from the electric potential.
This field is describe by its Xcoordinate E_{x} and Ycoordinate
E_{y}.

for axisymmetric problems 
Starting Point and Orientation of the Field`s Line
At every point of the field line, the electric field E is tangent
to the line. Let us call dl an elementary displacement of the field`s
line. Therefore:
The direction of the electric field makes an angle Î± with a horizontal
reference axis. dl is characterized by a Xcoordinate dx and a Ycoordinate
dy. They are expressed as:
dx = dl.cosÎ±anddy = dl.sinÎ± 
An electrode is described by a set of mesh nodes. Each of them can be
considered as a field`s line starting point.
The direction of the field is determined by its Xcoordinate and Ycoordinate.
The orientation of the field`s line differs, depending on whether the
field is divergent (going from electrode) or a convergent (oriented toward
electrode). As shown in Fig. 1, for divergent field,
Î± can be expressed as:

Fig. 1: 
Divergent field: The electric field and the electric
field`s line are oriented in the same direction 

Fig. 2: 
Convergent field: The direction of the electric field
is opposed to that of the electric field`s line 
Figure 2 shows that in convergent field, field`s line
orientation is opposed to the direction of the field. In this study, the
value of Î± must be corrected as follows:
Field Line Propagation
The field`s line evolves by successive jump according to the direction
of the electric field. The length dl of a jump is a parameter depending
on the geometry. A hundredth of the greatest dimension of the geometry
is a best compromise between the CPU time and the computation precision.
The evolution of the field line is calculated as shown by the following
flowchart (Fig. 3). The last point of a field`s line
is reached when it attains an other electrode or goes outside the domain
under consideration.
Approximation at a Point
The electric field`s line, while progressing, goes through points
other than the mesh`s nodes. For these points, it is necessary to make
an approximation of the geometrical and electric parameters. Let us consider
two finite elements, triangles of the first order (Fig.
4).

Fig. 3: 
Flowchart showing the stepwise propagation process of
a field`s line 

Fig. 4: 
Possible position of a point where the approximation
must be made: Point M (x, y) belongs to one triangle, Point P (x,
y) belongs to several triangles 
If the point belongs to only one triangle, electric parameters are obtained
by a polynomial approximation (Chari and Salon, 2000).
Coefficients a_{i}, b_{i} and c_{i} are determined
as follow:
Where:
When this point belongs to several triangles, as shown in Fig.
4, the approximation is made on each of theme and an average is calculated.
If H designates the potential, the Xcoordinate or the Ycoordinate of
the field, it can be calculated as:
where, N being the number of triangles to which belongs the point and
H_{k} being the approximation of H in the kth triangle.
Line of Discharge
The field`s lines previously given can be used in electric discharge`s
models. Indeed, it is commonly accepted that the electric discharge starts
on one high voltage electrode and propagates following a field`s line
toward one low voltage electrode (Giralt and Buret, 2000).
The discharge`s line is thus a field`s line which leaves from a point
where the electric field is sufficiently intense to start and maintain
an electric discharge.
RESULTS AND DISCUSSION
One difficulty, in discharge studies, is the validation of calculation
codes by an analytic result or by experimental values. These difficulties
have led most authors to use the rodtoplane geometry (Ortega et al.,
1994; Goelian et al., 1997). In this configuration, the discharge`s
line is simply the symmetry axis of the tip. Geometrical and electrical
parameters of the field`s line are reachable analytically. Photographs
submitted by Hayakawa et al. (2005) showed that the discharge spreads
along the field`s line. Pointplane structure has played and continues
to play an important role in the studies on the phenomena of electric
discharge.
Ibrahim (1988) has proposed a formula (Eq. 2) for the
rod plane gaps in the ranges 0.11 cm for the electrode radius and 212
cm for the gap spacing. If E(x) is the field per unit applied voltage
at a distance x cm along the gap axis from the rod tip, then:
Where:

Fig. 5: 
Comparison between theoretical and calculated field`s
line, (â€”) Theoretical field`s line and (.....) Simulated field`s
line 

Fig. 6: 
Electrical field comparison between simulation and Eq.
2, (â€”) Eq. 2 and (.....) Simulation 
First of all, let chose a geometry which observes these conditions in
order to check simulation results (Electrode radius = 0.7 cm and Gap spacing
= 8 cm). Figure 5 and 6 compare successfully
simulations results with those of Ibrahim (1988).
It is necessary to extend the well known theories in pointplane geometry
to intervals of any form. Indeed, most electrical devices can`t be classed
as rodtoplane geometry (i.e., transformer, aerial insulator, â€¦).
For an aerial insulator, a real industrial structure (Fig.
7), Fig. 8 shows all the field`s lines leaving from
the electrode. A discharge`s model can thus be built for this geometry.
The discharge would then be randomly initiated on a point of the electrode.
It would follow the associated line of field.

Fig. 7: 
A photographs showing a link of an insulator string 

Fig. 8: 
Field`s lines and definition of the geometry 
Figure 8 shows two types of electric field lines. Some
of them go through the dielectrics and others did not. These two types
of electric field`s lines, respectively, are useful in a perforation or
flashover model. The electric field`s lines are ranked in descending order
according to field modulus of the starting point. The first line is therefore
starting at the point where the field is most intense. It is the perforation
discharge (Fig. 9). The first line which does not cross
any dielectric will be the flashover line (Fig. 9).
No field`s line follows the shape of solid isolating material. The case
of slipping discharge in which the discharge follows the dielectric`s
shape was not investigated.
During a discharge event, Giralt and Buret (2000) distinguishes the potential
imposed by the sources and the potential due to space charges. Figure
10 and 11 show the potential and the field created
by the sources for the perforation and the flashover lines.

Fig. 9: 
The two field`s lines under consideration, (.....) Perforation
line and (â€”) Flashover line 

Fig. 10: 
Potential evolution along the lines, (â€”) Perforation
line and (.....) Flashover line 

Fig. 11: 
Field evolution along the lines, (â€”) Perforation
line and (.....) Flashover line 
CONCLUSION
This study presents the interest of proposing a presumed trajectory of
the discharge in high voltage devices and the possibility to expand the
discharge`s theory to realistic configuration. A new computer program,
with the ability to determine electric field line in any arbitrary electrode`s
shape, has been described. The technique of the progression of the field
line has been presented for divergent and convergent field. With this
new method, the electric discharge can be initiated at any point on the
high voltage electrode. That would make it possible to take into account
the randomness of the starting point of a discharge.