INTRODUCTION
The progress of technology and the need for increasingly growing electrical
energy at low frequencies (50 and 60 Hz) have led to the installation
of High Voltage (HV) and Ultra High Voltage (UHV) transmission lines.
These lines create electric and magnetic fields which have different properties
and different effects on the living cell. The electric field is quickly
attenuated by the conducting objects (buildings, human body, trees, etc.).
The magnetic field decreases with the distance which separates the observation
point from the source and is not attenuated by most materials.
Many epidemiological and experimental studies have been interested by
the interactions between the magnetic field and diseases such as breast
cancer (Feychting et al., 1998; Erren, 2001; Scott et al.,
2002; London et al., 2003; Sorahan and Nichols, 2004; Kliukiene
et al., 2004) and children leukemia (Skinner et al., 2002;
Foliart et al., 2006; Blaasaas et al., 2003). These experimental
researches only measure the magnetic field in the surroundings of the
populations. Typical values are often 0.1 to 0.2 μT (Heroux, 1987;
Qiu, 2004; Havas, 2002).
When the field`s source is relatively simple, as in threephase lines
structure, a rigorous calculation of the field`s module is possible. In
this document, we presented the calculation of the magnetic field produced
by air threephase lines (three or six conductors). To illustrate present
calculation, we use the pylons defined by Shaler et al. (2003).
We define the security distance for the studied pylons. This study can
also be used in a pylon optimization procedure.
MATERIALS AND METHODS
Field created by one conductor: Let us consider a conductor placed
at the point P(x_{0},y_{0}) and traversed by a current
;
f being the frequency expressed in Hertz. According to the ampere`s theorem,
this conductor creates a magnetic field
at a point M(x, y), with module
This field is oriented so that constitute
a direct trihedron. The line`s length is supposed to be infinite. Thus,
we can limit the study in the plan perpendicular to the conductor.
The direction of PM is given by an angle β expressed as:
The direction of the magnetic field (Fig. 1, 2)
is perpendicular to that of PM and its direction depends on the sign of
the current. It is defined by an angle α expressed as:

Fig. 1: 
Orientation of the field for an outgoing current 

Fig. 2: 
Orientation of the field for an incoming current 
This field is characterized by a horizontal component B_{x} and
a vertical one B_{y}.
Field created by a threephase line: Assuming that the currents
form a balanced system, the intensity of the instantaneous current in
each conductor is given by:
The magnetic permeability is practically constant in the area under consideration
(air) and we can apply the superposition theorem. The resulting field
created by a set of conductors is the vectorial sum of the field created
by each conductor. Thus, we have:
B_{x} and B_{y} depend on distances and thus on the structure
of the pylons. Calculation is organized according to the following flowchart
(Fig. 3). It makes it possible to determine the field
at a point at a given moment.

Fig. 3: 
Flowchart showing the magnetic field calculation procedure.
The space point and the moment of calculation are input parameters 
Definition of the security distance: The magnetic fields cross
all materials used in the dwellings construction. It is then imperative
to establish security distances on both sides of the equipment generating
these fields. A reference value of the magnetic field being fixed, one
seeks the distance from which the field becomes lower than this reference.
In the case of three phase lines, several points answer the question.
But if one limits the investigation on a plan parallel to the ground,
only two points, located on both sides, will be retained. We will use
0.2 μT as the reference value for the calculation.
RESULTS AND DISCUSSION
The security distance depends on the pylon`s geometry i.e. the position
of the conductors the ones compared to the others and to the ground. We
built a computer program which takes as input parameters the geometry
of the pylon and the line`s load. To illustrate our calculation, we use
pylon`s structure defined by Shaher et al. (2003) recalled here
after in Table 1.
Four parameters can describe the system: the magnetic field (module and
phase), the time t, the space coordinates (x, y) and the line load I.

Fig. 4: 
Field`s modulus as a function of time at M(0,0) and
I = 1000 A, Red: Pylon 1; Green dash: Pylon 2; Blue dashdot: Pylon
3 
Table 1: 
Description of the geometry of conductors on the pylons 

Thus, the study of the magnetic field amounts studying the functions of
type:
We propose, in the following figures,
• 
B = f(t): Evolution of the field with time (one period of a 50 Hz
sinusoidal wave) at fixed load and fixed point 
• 
B = f(M): Evolution of the field along a straight line parallel
to the ground. The line load and the time are fixed 
• 
B = f(I) : Evolution of the field with the load at a given moment
and at a fixed point. 
• 
Calculation of the security distance with respect to 0.2 μT
as reference value. 
Figure 4 and 5 shows the module and
the phase of the magnetic field produced by pylons at M(0, 0) according
to time. One period of a sinusoidal voltage of 50 Hz was used. The line`s
load is fixed at I = 1000 A. It can be noted that the field is maximum
at t = 6 msec for pylon 1 and 2. For pylon 3, the field is maximum at
t = 10 msec.

Fig. 5: 
Field`s orientation as a function of time at M(0, 0)
and I = 1000 A, Red: Pylon 1; Green dash: Pylon 2; Blue dashdot :
Pylon 3 

Fig. 6: 
Field`s modulus along a straight line (y = 0; t = 6
msec), Red: Pylon 1; Green dash: Pylon 2; Blue dashdot: Pylon 3 
The disposition of the conductors on the pylons may explain
this difference. The variations of the module and the phase might create
induced currents in conducting objects (Dawson et al., 1999).
Figure 69 shows the fast decrease, as 1/r, of the
field`s module and the orientation of the field along a line. The orientation
of the field changes especially in the vicinity of the line.
As the field is a function of time and load, we compute a security distance
for t = 6 msec, function of the load (Fig. 10) and a security
distance for I = 1000 A, function of time (Fig. 11).
It can be noted that for very weak loads, the maximum field is lower
than the reference value. For high loads, the security distance can be
approximated by a polynomial of the first degree.

Fig. 7: 
Field`s orientation along a straight line (y = 0; t
= 6 msec), Red: Pylon 1; Green dash: Pylon 2; Blue dashdot: Pylon
3 

Fig. 8: 
Field`s modulus along a straight line (y = 0; t = 10
msec), Red: Pylon 1; Green dash: Pylon 2; Blue dashdot: Pylon 3 

Fig. 9: 
Field`s orientation along a straight line (y = 0; t
= 10 msec), Red: Pylon 1; Green dash: Pylon 2; Blue dashdot: Pylon
3 

Fig. 10: 
Evolution of security distance with line`s loadt =
6 msec, Red: Pylon 1; Green dash: Pylon 2; Blue dashdot: Pylon 3 

Fig. 11: 
Evolution of security distance with time −I =
1000 A, Red: Pylon 1; Green dash: Pylon 2; Blue dashdot: Pylon 3 
CONCLUSION
We built a computer program in order to carry out accurate calculations
of the magnetic field according to time, space and the load of a threephase
line (3 or 6 conductors). This calculation has enabled us to define the
security distance as being the distance beyond which the field is lower
than a critical value. This distance depends on the type of the pylon,
the time and the line load. The number of parameters did not allow us
to calculate standard security distances. For the tested pylons, if reference
value is 0.2 μT, the security distance can exceed several hundred
meters. This study can also be used in a pylon optimization procedure.