Download Electric field quantities in bundle and stranded conductors of

Electric field quantities in bundle and stranded conductors
of overhead transmission lines
Bruno Assunção Cardador
Instituto Superior Técnico, Lisbon, Portugal
October 2015
Abstract
Electricity is an essential commodity for the comfort and convenience of population. Therefore it is necessary to preserve and maintain
structures that allow the electric power generation, transmission and consumption. For the power transmission, overhead power lines are commonly
used. When sizing very high voltage transmission lines it is essential to know the maximum value of the electric field at the conductors surface, to
prevent the corona effect. To limit this maximum electrical field bundle conductors are currently used.
This article reports the study of the electric field of bundle and stranded conductors in overhead transmission lines. The multipole method is
applied for the accounting of the proximity effect. Results are obtained regarding the electric field on the conductor surfaces and the radius of the
equivalent conductor concerning the electric potential point of view. Results are compared with usual approximate expressions in the literature,
including comparison with the well-known equivalent geometric mean radius (G.M.R.). Results took into consideration the sub-conductor radius,
the number of sub-conductors per phase as well as the proximity effect between them.
As expected, it was possible to verify that there is a deviation in the results of both the electric field at the conductor surfaces and the equivalent
conductor radius when calculated by the multipole method and by the approximate expressions. Results show that these deviations tend to increase
when the proximity effect is made worse.
Keywords: electric field, geometric mean radius, bundle conductors, stranded conductors.
1.
surface of the conductors and a decrease of losses caused by the
corona effect.
Introduction
Electric field evaluation is crucial for the design of overhead
power lines. As the voltage increases, so does the electric field.
This may add relevance to the phenomenon of the corona effect.
To avoid such phenomenon, it is necessary that the phase
conductors are correctly dimensioned.
For systems with one sub-conductor per phase, the electric
field on its periphery is uniform. The same is not verified when
there are several sub-conductors per phase, since the electric
field lines between sub-conductors are affected due to the
proximity effect. This non uniformity of the electric field
increases with the number of sub-conductors per phase for
bundle conductors with sub-conductors having the same
dimensions.
The corona effect occurs when the electric field at the surface
of the conductors exceeds the dielectric strength [4] (about 30
kV/cm in dry air). This effect is characterized by the partial
dielectric breakdown, which causes audible noise, radio
interference, conductor vibration and power losses. The electric
field that originates the corona effect is related with the diameter
of the conductors, the configuration of the lines and the weather
conditions, such as temperature, humidity and air pressure.
Considering that the dielectric involving the sub-conductors
is not charged, then the electric potential satisfies the Laplace’s
equation as a consequence of the fundamental Maxwell’s
equations. In such conditions, the multipole method may be used
to calculate the electric potential under the presence of the
proximity effect amongst sub-conductors. The solution is built
by overlapping solutions with singularities of all entire orders
(multipoles) at the axis of the various cylindrical sub-conductor
that compose the conductor structure. The solution is then
obtained by imposing the appropriate boundary conditions over
each sub-conductor surface.
To avoid the corona effect, it is necessary to keep the value
of the maximum electric field within acceptable limits, without
having to increase the transverse section of the conductors. For
this aim, several sub-conductors per phase are used.
The main advantages of using sub-conductors in
transmission lines are: a decrease of the electric field at the
1
Where ε represents the dielectric constant of the medium.
Relating the fundamental equations of the electric field (2.2) and
(2.3), with (2.5), whenever the medium is homogeneous or
sectionally homogeneous the Poisson’s equation is obtained:
The most used conductors in overhead transmission lines are
bundle and stranded conductors. In such structures the distance
from the conductors to the ground is considered much greater
than the radius of the bundle, being the proximity of the ground
usually dismissed. Consequently, the analysis of the proximity
is exclusively focused on the interaction between the subconductors.
𝜌
lap 𝑉 = − ⁄𝜀
(2.6)
For the case with null electric charge in the dielectric, the
Laplace’s equation is reached:
The sub-conductors that compose the bundle are cylindrical
and solid. They are composed by linear, isotropic and
homogeneous materials. The insulating medium involving the
bundle is taken as a perfect dielectric.
lap 𝑉 = 0
(2.7)
2.1 Solution of 2D Laplace’s equation in cylindrical
coordinates
The bundle and stranded conductor configurations are
geometrically quite similar. The sub-conductors are
symmetrically disposed over a circumference (bundle).
However, in the case of stranded conductors, the sub-conductors
are closely packed together against each other and twisted where
in the last case only the external layer of sub-conductors is
considered. For the stranded conductor case, the influence of the
inner layers is neglected.
Laplace’s equation is solved by defining the electric
potential or the normal derivative of electric potential in all
points of the conductor boundary, thus ensuring the existence
and the uniqueness of the solution.
As the conductors have a cylindrical geometry, the Laplace’s
equation is solved in cylindrical coordinates.
A group of conductors can be represented by only one
equivalent cylindrical conductor from the perspective of its
electric potential function. The seeking of such equivalent
conductor radius constitutes also an objective of this paper.
Comparison is done with a quite common approximation where
the equivalent conductor is located in the geometric centre of the
bundle with an equivalent radius equal to the geometric mean
radius.
Figure 2.1 – Representation of the coordinates of a cylinder.
The solution of Laplace’s equation is achieved by
considering the transverse coordinates as follows:
2. Electric field of cylindrical conductor systems in
proximity
1 ∂
𝜕𝑉
1 ∂2 𝑉
(𝑟 ) + 2 2 = 0
𝑟 ∂𝑟 𝜕𝑟
𝑟 ∂𝜑
The electromagnetic field of cylindrical transmission lines
for the perfect conductor assumption has the configuration of the
stationary electric and magnetic fields on transverse plans. In
particular the circulation of the electric field in a closed path (s)
in transverse plans is equal to zero
∮ 𝐄 ds⃗ = 0
Through the method of separation of variables we get the
multiplication of two functions:
𝑉(𝑟, 𝜑) = 𝑅(𝑟)𝛷(𝜑)
To calculate the electric field, the fundamental Maxwell’s
equations are taken into account:
rot 𝐄 = 0
(2.2)
𝛷(𝜑) = ej𝑝𝜑
div 𝐃 = 𝜌
(2.3)
For the expressions of zero order (p=0):
1
R(𝑟) = ln ( )
𝑟
With (2.2), it is possible to define the electric field as a
gradient:
(2.4)
𝑝 ∈ ℤ
(2.10)
(2.11)
For the remaining terms where p assumes an integer value
different from zero, the solution to the R function is described
by:
The electric displacement field D depends on the electric
field E as follows:
𝐃 = 𝜀𝐄
(2.9)
For each r value, the function of V along the azimuthal
coordinate has to be periodical with period 2π. To represent
periodic functions, a development of Fourier series is used,
where the base function of the development in the exponential
form is given by:
(2.1)
s⃗⃗
𝐄 = −grad 𝑉
(2.8)
𝑅(𝑟) = 𝑟 ±𝑝
(2.5)
2
(2.12)
The form of the solution of the potential function is a linear
combination of the solutions presented in (2.10), (2.11) and
(2.12), characterized by having singularities of p order, with
𝑝 = −∞, . . . ,0, . . . , +∞:
̅ (𝑤
̅ (𝑤
𝑊
̅) = 𝑊
̅ 𝑘𝑖 ) +
∞
∞
+∑
𝑚=1
+∞
1 1
𝑉 = 𝐶0 ln + ∑ 𝑐𝑝 𝑟 −|𝑝| ej𝑝𝜑
𝑟 2 𝑝=−∞
(−1)𝑚 (𝑖)
(𝑖)
−𝑝
̅𝑘𝑖 ] 𝑤
̅𝑚
𝑚 [𝐶0 + ∑ 𝑐−𝑝 𝒞(𝑝, 𝑚) 𝑤
𝑚𝑤
̅ 𝑘𝑖
(2.19)
𝑝=1
(2.13)
where
𝑝≠0
(2.20)
𝑤
̅ = 𝑟 ej𝜑
Where cp and C0 are the coefficients of the solution of the
potential vector of p and zero order, respectively.
(𝑖)
The normalization of the coefficients 𝑐𝑝 and the distance r
is achieved according to the radius of the sub-conductor at the
axis Ok.
(𝑖)
The normalization of 𝑐𝑝 coefficients is defined by the
relationship between these and the radius of i sub-conductor:
The solution of (2.13) can be obtained from a complex
function of complex variable W such that:
∞
1
̅ (𝑦̅) = 𝐶0 ln ( ) + ∑ 𝑐−𝑝 𝑦̅ −𝑝
𝑊
𝑦̅
(2.14)
𝑝=1
where:
(𝑖)
𝑐𝑝
(𝑖)
𝑦̅ =
𝑟𝑒 𝑗𝑝𝜑
𝐶𝑝 =
(2.15)
(2.21)
𝑝
𝑟𝑖
The relation between r distance with the sub-conductors
radius represents the normalization of r distance
and
̅ (𝑦̅)}
𝑉 = Re{𝑊
(2.16)
𝑅=
To set the boundary conditions on the cylindrical surfaces of
several massive sub-conductors, a Taylor series has to be
considered to develop (2.13) around the axis of the conductor
where the boundary conditions are going to be considered. This
way, it is possible to get the solution with singularities in the
axis of an i conductor, but focused on the k conductor, as
represented in Fig. 2.2:
𝑟
𝑟𝑘
(2.22)
Given the normalizations mentioned above, the following
expression represents the result of the sum of (2.19):
∞
̅ (𝑘) = 𝐶0(𝑖) ln
𝑊
𝑖
1
𝑤
̅ 𝑘𝑖 −𝑝
+ ∑ 𝐶−𝑝 ( ) +
(𝑤
̅ 𝑘𝑖 ⁄𝑟𝑖 )
𝑟𝑖
𝑝=1
∞
∞
𝑚=1
𝑝=1
1
(𝑖)
(𝑖)
+ ∑ 𝑅 𝑚 [𝐶0 𝒰𝑘𝑖 (𝑚, 0) + ∑ 𝐶−𝑝 𝒰ki (𝑚, 𝑝)] ej𝑚𝜑
𝑚
|(2.23)
Where 𝒰𝑘𝑖 is obtained by the expansion in Taylor series of
the terms of p order through the normalization mentioned before
[5].
𝑤
̅ 𝑘𝑖 −𝑚 𝑤
̅ 𝑘𝑖 −𝑝
𝒰ki (m, p) = (−1)𝑚 ( ) ( ) 𝒞(𝑝, 𝑚)
𝑟𝑘
𝑟𝑖
Considering only the real part of (2.24), is obtained a
solution that satisfies the Laplace’s equation, with singularities
at the axis of the i sub-conductor, but focused on the axis of the
k sub-conductor [5]:
Figure 2.2 – Representation of the solution focused on the k conductor [5].
As we develop (2.13) as a Taylor series, it can be noted that
the solution is convergent when the 𝑟 distance is shorter than the
modulus of the distance between the centres of the subconductors |𝑤
̅𝑘𝑖 |, which gets the following expression [5]:
1
+
|𝑤
̅ 𝑘𝑖 ⁄𝑟𝑖 |
∞
−𝑝
∗
1
𝑤
̅ 𝑘𝑖 −𝑝
𝑤
̅ 𝑘𝑖
+ ∑ [𝐶−𝑝 ( ) + 𝐶𝑝 ( ) ] +
2
𝑟𝑖
𝑟𝑖
(𝑘)
𝑉𝑖
̅
1 𝑑𝑚 𝑊
[
]
=
𝑚! d 𝑦̅ 𝑚 𝑦̅=𝑤̅
𝑘𝑖
=
(−1)𝑚
𝑚
𝑚𝑤
̅ 𝑘𝑖
(𝑖)
(𝑖)
(𝑘)
𝑅|𝑚|
(𝑖)
[𝐶0 {
2|𝑚|
𝑚=−∞
−𝑝
𝑚≠0
𝑝=1
∞
where
(𝑖)
] = 𝐶0 ln
+ ∑
[𝐶0 + ∑ 𝑐−𝑝 𝒞(𝑝, 𝑚) 𝑤
̅𝑘𝑖 ]
(𝑝 + 𝑚 − 1)!
𝒞(𝑝, 𝑚) =
(𝑝 − 1)! (𝑚 − 1)!
̅𝑖
= Re [𝑊
𝑝=1
∞
(2.17)
∞
((2.24)
𝒰𝑘𝑖 (𝑚, 0)
}+
∗
𝒰𝑘𝑖
(−𝑚, 0)
(2.25)
(𝑖)
𝐶−𝑝 𝒰𝑘𝑖 (𝑚, 𝑝)
+∑{
}] ejmφ
𝑝=1 𝐶 (𝑖) 𝒰 ∗ (−𝑚, 𝑝)
𝑘𝑖
𝑝
(2.18)
In the previous expression, the top line is for m>0 values and
the lower line for m<0 values, which gets:
Thus, the final expression that represents the solutions with
singularities in the axis of an i sub-conductor, but centred on the
k sub-conductor is achieved:
(𝑖)
(𝑖) ∗
𝐶𝑝 = (𝐶−𝑝 )
3
(2.26)
Finally, it is possible to build the complete solution of 𝑉 (𝑘) ,
centred on the k sub-conductor with the singularities of the 𝑁
sub-conductors in proximity of each other. This way, it is
possible to associate the terms of zero order and m order of the
development of Fourier series.
The expression of the potential of each sub-conductor is
function of the normalized distance of R and the azimuthal
coordinate 𝜑 [5]:
distance r and the radius of each sub-conductor are the same,
meaning that the normalized distance R assumes the unitary
value.
In the boundary conditions all terms of the Fourier series are
defined as zero, with the exception of the constant terms, to
ensure that the potential function at the surface of each subconductor is constant. The constant terms are equal to the value
of the potential of the corresponding sub-conductor:
+∞
(𝑘)
𝑉 (𝑘) (𝑅, 𝜑) = ∑ 𝐴𝑚 (𝑅)𝑒 j𝑚𝜑
(2.27)
𝑉 (𝑘) (1, 𝜑) = 𝑉𝑘
𝑚=−∞
This way, the expression of the potential for the zero order
terms through (2.28) is obtained:
Associating the terms of zero order of (2.13) and (2.25), the
following linear combination is obtained [5]:
N
1
(𝑘)
(𝑘)
𝐴0 (𝑅) = 𝐶0 ln ( ) +
𝑅
𝑟𝑖
(𝑖)
𝑉𝑘 = ∑ 𝐶0 ln (
) + 𝑃𝑘
|𝑤
̅ 𝑘𝑖 |
(𝑘)
The 𝐶0 coefficient of (2.28) can be related to the electric
charge from the actual sub-conductor k as:
𝑖=1
(𝑖≠𝑘)
(𝑘)
𝐶0
Where:
∞
𝑖=1
(𝑖≠𝑘)
𝑝=1
]
((2.29)
(2.35)
(𝑘)
𝐴𝑚 (𝑅) = 0
(2.36)
The previous consideration establishes the system of
(𝑘)
equations that allows to get the solutions of the coefficients 𝐶𝑚
in the boundary conditions [5]:
In the same way, the m order terms are associated. To
achieve this, it is used the second term of (2.13) and the terms
of the sum in m of (2.25). The solution is formed by the overlay
of the solution of sub-conductor k and the solution of the other
sub-conductors centred on the sub-conductor k [5]:
(𝑘)
𝑞𝑘
2π𝜀
Given the boundary conditions, the terms of m order are null:
The first term of (2.28) corresponds to the solutions of subconductor k. The remaining terms correspond to the solutions of
the terms associated to the other sub-conductors, but centred on
the axis of sub-conductor k.
𝑅|𝑚|
(𝑘)
[𝛽
2|𝑚| 𝑚
=
Where 𝑞𝑘 represents the charge per unit length of the subconductor k and 𝜀 the dielectric constant of the medium.
−𝑝
∗
−𝑝
1
(𝑖) 𝑤𝑘𝑖
(𝑖) 𝑤𝑘𝑖
𝑃𝑘 = ∑ ∑ [𝐶−𝑝 ( ) + 𝐶𝑝 ( )
2
𝑟𝑖
𝑟𝑖
(𝑘)
𝐴𝑚 (𝑅) =
(2.34)
𝑖=1
(𝑖≠𝑘)
(2.28)
𝑁
𝑟𝑖
(𝑖)
+ ∑ 𝐶0 ln (
) + 𝑃𝑘
|𝑤
̅ 𝑘𝑖 |
𝑁
(2.33)
(𝑘)
(𝑘)
(𝑘)
𝛾𝑚 + |𝑚|𝐶𝑚 = −𝛽𝑚 ,
(2.37)
𝑚 = ±1, ±2, … , 𝑘 = 1,2, … 𝑁
2.2 Electric field on a sub-conductor surface
The electric field at the surface of the sub-conductors is a
function of the azimuthal coordinate 𝜑 and the distance R. The
value of the electric field at the surface of the sub-conductors is
calculated through the gradient of the potential obtained in
(2.27), which gets the following expression:
(2.30)
(𝑘)
+|𝑚|𝐶𝑚 𝑅−2|𝑚| + 𝛾𝑚
Where
𝑁
(𝑘)
𝛽𝑚
= ∑
(𝑖)
𝐶0 {
𝑖=1
(𝑖≠𝑘)
𝐄 (𝑘) = −grad 𝑉 (𝑘) (𝑅, 𝜑)
𝒰𝑘𝑖 (𝑚, 0)
}
∗
𝒰𝑘𝑖
(−𝑚, 0)
(2.31)
The solution of the electric field expression in cylindrical
coordinates [4]:
and
𝑁
(𝑘)
𝛾𝑚
𝑚𝑖
(2.38)
∂𝑉 (𝑘)
1 ∂𝑉 (𝑘)
𝐄 (𝑘) = − (
𝑒⃗𝑟 +
𝑒⃗ )
∂𝑟
𝑟 ∂𝜑 𝜑
(𝑖)
𝐶−𝑝 𝒰𝑘𝑖 (𝑚, 𝑝)
= ∑ ∑{
}
𝑖=1 𝑝=1 𝐶 (𝑖) 𝒰∗ (−𝑚, 𝑝)
𝑘𝑖
𝑝
(2.32)
(2.39)
To obtain the expression that allows to observe the evolution
of the intensity of the electric field at the surface of the subconductors, the boundary conditions are taken into account, so,
the normalized distance R is equal to one and the electric
potential is constant at the surface of the sub-conductors. In this
condition, the component of the electric field according to the
azimuth component is equal to zero.
(𝑖≠𝑘)
In the previous expressions, the first line is for m>0 values
and the second line for m<0 values.
2.1.1 Boundary conditions
(𝑘)
To obtain the solution of the coefficients 𝐶𝑚 , thus obtaining
the solution of the potential vector, boundary conditions are
imposed to the surface of the sub-conductors. In this case, the
4
(𝑘)
𝐸𝜑 = −
1 ∂𝑉 (𝑘)
|
=0
𝑟 ∂𝜑 𝑟=𝑟
The results are presented as a function of the proximity
parameter 𝑅0 defined by
(2.40)
𝑘
𝑅0 =
The zero order terms of Fourier series development do not
depend on the azimuthal coordinate, and the non-zero order
terms assume the value of zero in the boundary conditions.
(𝑘)
(𝑘)
=−
+∞
𝜋
𝑅0𝑚𝑎𝑥 = sen ( )
𝑁
(2.41)
𝑚≠0
The evolution of the intensity of the electric field along the
surface of a sub-conductor according to 𝜑 is presented in Fig.
3.2 for N=3 and different values of 𝑅0 , where the normalized
electric field intensity is
Finally, solving the previous expression, is obtained the
expression for the calculation of the electric field intensity at the
surface of the sub-conductors for R=1:
(𝑘)
(𝑘)
𝐸𝑟 |
𝑟=𝑟𝑘
=
+∞
(𝑘)
𝐶0
𝐶𝑚
{1 + ∑ |𝑚| (𝑘) 𝑒 𝑗𝑚𝜑 }
𝑟𝑘
𝐶
𝑚=−∞
𝑚≠0
(3.2)
The maximum value of 𝑅0 , 𝑅0𝑚𝑎𝑥 , is verified when the subconductors are packed against each other (stranded conductors).
(𝑘)
1 ∂𝐴0 (𝑅)
∂𝐴𝑚 (𝑅) 𝑗𝑚𝜑
+ ∑
𝑒
{
}
𝑟𝑘
∂𝑅
∂𝑅
𝑚=−∞
(3.1)
where
Considering the zero order terms and the non-zero order ones
of Fourier series development according the radial component
of the electric field, is verified that:
𝐸𝑟
𝑟𝑐
𝑎
(2.42)
𝐸𝑛 =
0
𝑅𝑒
𝑅0 𝑁
+∞
{1 + ∑ |𝑚|
𝑚=−∞
𝑚≠0
(𝑘)
𝐶𝑚
(𝑘)
𝐶0
𝑒𝑗𝑚𝜑 }
(3.4)
where
The first term of (2.42) corresponds to the electric field on
the surface of sub-conductors without the influence of the other
sub-conductors, neglecting the proximity effect, denominated
average electric field. The parameters which influences the
electric field at sub-conductor surface are the radius of sub-
𝑅𝑒 =
𝑟𝑒
= 1 + 𝑅0
𝑎
(3.5)
and 𝑟𝑒 is the external cable radius represented in (Fig. 3.1).
(𝑘)
conductor k, 𝑟𝑘 , the coefficients 𝐶𝑚 and the 𝜑 coordinate.
The developed algorithm is general where the conductors
can be arranged in any manner with different radius, since they
have parallel axes and are circular cylindrical.
3. Numerical results
To validate the algorithm developed, several results were
obtained for the electric field and for the equivalent radius for
the potential, so as to compare them with the published results.
The obtained results are very close to the results in published
articles.
Figure 3.2 – Distribution of the electric field in a three sub-conductor
bundle.
3.2 Intensity of the electric field at the surface of a cable
3.1 Electric field distribution on a sub-conductor surface
Whenever the charge of all sub-conductors of the bundle has
the same polarity, the electric field is more intense in the
peripheral region of the sub-conductors. It is verified that the
maximum intensity of the electric field at the surface of a subconductor when 𝜑 = 0.
In this section is represented the distribution of the electric
field at the surface of sub-conductor 1 for a bundle of N
cylindrical sub-conductors of the same radius 𝑟𝑐 , symmetrically
disposed over a circumference of radius 𝑎.
For the stranded sub-conductor the evolution of the
normalized electric field at the sub-conductor versus the
azimuthal coordinate 𝜑 is represented in Fig. 3.3, corresponding
to different values of N. The obtained results are in complete
agreement with [6].
Figure 3.1 – Configuration of a bundle of N cylindrical sub-conductors
The electric field at the surface of sub-conductor 1, is
presented according to the function of the angle 𝜑, Fig. 3.1.
5
of the same ratio. The values were obtained for bundles of two,
three, four and six sub-conductors.
Figure 3.3 – Relation between normalized electric field and azimuthal
coordinate
The maximum value of the electric field at the surface of the
sub-conductor is given by (2.42) with 𝜑 = 0 which for the
normalized values is formulated using:
+∞
𝐸𝑛 𝑚𝑎𝑥 =
The approximate expression leads to a linear behaviour of
the maximum electric field, usually with values above the ones
of the exact expression. Comparing the evolution of the two
expressions with an increase in the radius of the sub-conductors,
it is possible to verify that for sub-conductors with lower radius
values, whenever proximity amongst them is too small, there is
no difference between the values of the approximate expression
and those of the exact expression. The ratio between the
maximum electric field and the average electric field increases
with the increase of the radius of the sub-conductors, and it also
increases with the number of sub-conductors in the bundle,
being the maximum value of this ratio reached when the subconductors are packed together, side by side.
(𝑘)
𝑅𝑒
𝐶𝑚
{1 + ∑ |𝑚| (𝑘) }
𝑅0 𝑁
𝐶
𝑚=−∞
𝑚≠0
Figure 3.4 – Relation between the maximum and average electric field at
the surface of a sub-conductor.
(3.4)
0
The maximum value of the electric field may also be given
by an approximate expression described in the literature [1]:
𝐸𝑚𝑎𝑥 =
𝐶0
[1 + (𝑁 − 1)𝑅0 ]
𝑟𝑐
(3.6)
Disregarding the proximity between sub-conductors, the
electric field at the surface of the sub-conductors would be
constant, making it possible to define it as the average electric
field
𝐸𝑎𝑣 =
𝐶0
𝑟𝑐
As the sub-conductors increase in size, the results of the
exact expression assume inferior values to those of the
approximate one. Thus, we verify that, by using the approximate
expression, the conductors will be oversized for the value of
maximum electric field which must not be exceeded, since the
maximum electric field calculated by the approximate
expression is higher than the value of the exact expression.
(3.7)
Connecting the approximate equation of the maximum
electric field (3.6) with (3.7):
𝐸𝑚𝑎𝑥
(
)
= 1 + (𝑁 − 1)𝑅0
𝐸𝑎𝑣 𝑎𝑝𝑝𝑟𝑜𝑥
(3.8)
3.3 Deviation between the approximate and the exact value
of the ratio between maximum and average electric field
With (3.8) it is possible to get the approximate value of the
ratio between maximum and average electric field.
A deviation between the values of the two expressions is
verified, essentially, to the presence of various sub-conductors
in the bundle and to their proximity effect. The deviation is
calculated with the following equation:
On the other hand, (3.4) allows to get the exact value of the
maximum electric field through the multipole method. Thus, we
get the expression to calculate the exact value of the ratio
between the maximum electric field and the average electric
field.
Emax
ϵE = ((
)
Eavg
approx
+∞
(𝑘)
𝐸𝑛
𝐶𝑚
( 𝑚𝑎𝑥 )
= 1 + ∑ |𝑚| (𝑘)
𝐸𝑎𝑣 𝑒𝑥𝑎𝑐𝑡
𝐶
𝑚=−∞
𝑚≠0
Emax
−(
)
Eavg
) × 100
(3.10)
exact
The y-axis in Fig. 3.5, corresponds to the deviation as a
percentage, and the x-axis, 𝑅0 . .
(3.9)
0
The evolution of the exact and approximate expressions is
obtained as a function of 𝑅0 . The size of the sub-conductors are
between zero and their maximum value. The maximum value is
achieved when all of the sub-conductors in the bundle are in
contact with each other. The evolution of the ratio of the
maximum and average electric field was verified to a different
number of sub-conductors.
In the y-axis is represented the value of the ratio between the
maximum electric field and the average electric field at the
surface of the sub-conductors, and in the x-axis, 𝑅0 . The solid
lines represent the evolution of the expression of the exact value
of the ratio between the maximum and average electric field; the
dotted lines represent the values of the approximate expression
Figure 3.5 – Deviation between the approximate and the exact value of the
ratio of maximum and average electric field at the surface of the subconductors.
Results show that the approximate expression to calculate
the maximum electric field has to be restricted to the cases where
6
the distance between sub-conductors is very large enough
compared to their transverse dimension.
1
𝑅𝑉 = (𝑁𝑅0 )𝑁 𝑒
3.4 Equivalent radius for the potential of a bundle conductor
The first factor of (3.18) refers to the approximate
calculation of the equivalent radius according to the expression
of the geometric mean radius. The second factor refers to the
terms of the multipole, which takes the singularities of the
various sub-conductors.
The representation of the evolution of (3.11) and (3.18) is
produced observing the ratio between the radius of the subconductors rc and the radius of the bundle 𝑎, 𝑅0 .
The approximate expression used to calculate the equivalent
radius for the potential is referred to as geometric mean radius
(G.M.R.) [1].
1⁄
𝑁
Fig. 3.5 attempts to verify the influence of the proximity of
the sub-conductors and the influence of sub-conductors number
in the bundle in the evolution of the equivalent radius, calculated
through the exact and approximate expressions. The dotted lines
represent the evolution in relation to the geometric mean radius
calculated by the approximate expression (3.11). The solid lines
represent the evolution of the equivalent radius for the potential
calculated by the exact expression according to the multipole
method (3.18).
(3.11)
The expression of the geometric mean radius is applied in
the situations where the ratio between the radius of the subconductors and the distance amongst them is small.
The exact expression of the equivalent radius for the
potential is obtained through the difference of potential between
a point at the surface of the sub-conductors and a point at a very
large distance from them. Given a very large distance from the
surface of the sub-conductors 𝑅∞ , the potential at this point gets
the following form:
1
𝑅∞
𝑉∞ = 𝑁𝐶0 ln
(3.12)
The potential at the surface of sub-conductor 1 (Fig. 3.2)
according to (2.34) and considering all sub-conductors with
equal charge q
𝑟𝑖
=
|𝑤
̅ 𝑘𝑖 |
𝑅0
2𝜋
|1 − 𝑒 𝑗 𝑁 (𝑖−1) |
,
Figure 3.6 - Equivalent radius for the potential
The values of the equivalent radius for the potential increase
as the radius of the sub-conductors increase, both in the exact
expression as in the approximate expression. The maximum
value for different number of sub-conductors occurs when the
ratio between the radius of the sub-conductors and the radius of
the bundle assumes its maximum value.
(3.13)
the following result is obtained
𝑉1 = 𝐶0 ln
1
+ 𝑃𝑘
𝑁𝑅0
(3.14)
3.5 Deviation between the approximate value and the exact
value of the equivalent radius
The difference of potential between the point at the surface
of the sub-conductor 1 and the point at distance 𝑅∞ is described
as follows:
𝑉1 − 𝑉∞ = 𝐶0 ln
(𝑅∞ )𝑁
+ 𝑃𝑘
𝑁𝑅0
It it verify a deviation in the values of the equivant radius
between the two expressions. Fig. 3.7 represents the evolution
of the deviation (in percentage) between the values obtained by
the geometric mean radius and those obtained by the exact
expression of the equivalent radius.
(3.15)
Considering an equivalent cylindrical conductors with radius
𝑅𝑉 , the difference of potential between the point at the surface
of the conductor and the point at distance 𝑅∞ is given by:
𝑉1 − 𝑉∞ = 𝑁𝐶0 ln
𝑅∞
𝑅𝑉
ϵR (%) =
1
𝑅𝑉
(𝐺. 𝑀. 𝑅. −𝑅𝑉 )
× 100
𝑅𝑉
(3.19)
In Fig. 3.7, the y-axis represents the value of the deviation in
a percentage, and the x-axis, 𝑅0 .
(3.16)
where
𝑉1 = 𝑁𝐶0 ln
(3.18)
When 𝑃𝑘 = 0 it is obtained (3.11) for the equivalent radius.
The equivalent radius for the potential is defined as the
radius of a coaxial cylindrical conductor, with the same charge
per unit length, whose potential, would be equal to the potential
of the bundle in relation to the same point with the same
reference at a large distance compared to the transverse
dimension of the bundle. This means that it is possible to
represent a bundle of conductors by one equivalent cylindrical
conductor only.
𝑅. 𝑀. 𝐺. = 𝑎(𝑁𝑅0 )
𝑃
− 𝑘
𝑁𝐶0
(3.17)
Equalling (3.14) and (3.17), we get the expression that
allows to calculate the equivalent radius for the potential:
7
To obtain the electric field in conductors with several subconductors it is considered the value of the equivalent radius for
the potential, which permits to achieve an equivalent conductor
with the same radius of the group of sub-conductors from the
potential point of view. Considering the results obtained, it can
be concluded that the values of the equivalent radius for the
potential increase as the radius of the sub-conductors increase,
both in the exact expression as in the approximate expression
(G.M.R.). The maximum value of the equivalent radius for the
potential, for different numbers of sub-conductors, is achieved
when the ratio between the radius of the sub-conductors and the
radius of the bundle reaches its maximum value. The maximum
value of the equivalent radius for the potential decreases with
the number of sub-conductors in the bundle.
Figure 3.7 – Deviation between the approximate value and the exact value
of the equivalent radius.
The deviation increases in modulus as the 𝑅0 increases,
when the proximity between sub-conductors increases. In the
cases where the proximity between sub-conductors is too small,
we verify that there is no deviation between the approximate
expression and the exact expression of the equivalent radius.
With the increase of the number of sub-conductors in the bundle
the deviation tends to become higher in absolute value.
The results of the two expressions for the values of the radius
of small sub-conductors are the same and that, as the radius of
the sub-conductors increase, the difference between the two also
increases. Such difference is caused by the proximity effect
between sub-conductors. It should be noted that the difference
between the two expressions is more relevant for the same radius
of the bundle for a larger number of sub-conductors, since the
proximity effect between sub-conductors is also higher.
4. Conclusions
The algorithm developed to validate the results is general
where the conductors can be arranged in any manner with
different radius, since they have parallel axes and are circular
cylindrical. The results obtained by this algorithm are consisted
with the results of published articles
The values of the exact expression of the equivalent radius
for the potential for the different numbers of sub-conductors are
always equal to or higher than the values obtained through the
approximate expression of the geometric mean radius. When a
deviation between the results of the two expressions occurs, it
can be verified that, for a given value of equivalent radius, the
radius of the sub-conductors is lower when the exact expression
is applied.
Regarding the evolution of the intensity of the electric field
at the surface of the sub-conductors it can be concluded that,
because of the sub-conductors proximity, the electric field at
their surface is not homogeneous. As we reduce the proximity
effect, the electric field at the surface of the sub-conductors
tends to be constant along the azimuthal coordinate of the subconductor. By the analysis of the results obtained, it can also be
concluded that the maximum value of the electric field at the
surface of the sub-conductors is achieved in a point of the subconductor belonging to the periphery of the cable.
As for the results obtained for the deviation between the
equivalent radius for the potential calculated with the exact
expression and through the geometric mean radius expression,
this deviation increases in modulus as the ratio between the
radius of the sub-conductors and the radius of the bundle
increases, that is, whenever the proximity of the sub-conductors
increases. Whenever the proximity between sub-conductors is
too small, it can be verified that there is no deviation between
the approximate expression and the exact expression of the
equivalent radius, which makes it possible to conclude that the
expression of the geometric mean radius will be valid to size the
conductors only when the proximity between sub-conductors
has a lower value.
Considering the results obtained regarding the ratio between
the maximum and average electric fields, according to the ratio
between the radius of the sub-conductors and the radius of the
bundle, it can be concluded that, as the ratio between the radius
of the sub-conductors and the radius of the bundle increases, the
ratio between the maximum electric field and the average
electric field increases too. The maximum of the ratio between
the maximum electric field and the average electric field is
verified in cases where the sub-conductors are all packed
together (stranded conductor), this maximum value increases
along with the number of sub-conductors present in the bundle.
It can be concluded that the results obtained by the approximated
expression leads to a linear behaviour of the maximum electric
field, which does not apply to the results obtained through the
exact expression. Comparing the results obtained by the
approximate expression and the exact expression, it can be
concluded that the values obtained as a result of the approximate
expression are equal to or higher than the values obtained by the
exact expression. As the proximity of the sub-conductors
increases, the deviation gets higher. Therefore, it is possible to
conclude that the use of the approximate expression to calculate
the maximum electric field should be used when the distance
between sub-conductors is too large when compared to their
transverse dimension.
8
References
[1] M. T., Correia de Barros, Cálculo aproximado do
campo eléctrico de condutores em feixe Teoria e
erros, Electricidade, nº151, pp. 1-12, Maio 1980.
[2] Software Matlab,
(http://www.mathworks.com/products/matlab).
[3] A. S. Timascheff, Field Patterns of Bundle
Conductors and Their Electrostatic Properties, AIEE
Trans. pt. III, Vol. 80, pp. 590-597, October 1961.
[4] J. A. Brandão Faria, Electromagnetic Foundations of
Electrical Engineering, Wiley, 2008.
[5] V. Maló Machado, M. Eduarda Pedro, J. Brandão
Faria, D. Van Dommelen, Magnetic field analysis of
three-condutor bundles in flat and triangular
configurations with the inclusion of proximity and skin
effects, Electric Power Systems Research, Vol. 81, pp.
2005-2014, November 2011.
[6] J. F. Borges da Silva: “The electrostatic field problem
of stranded and bundle conductors solved by the
multipole method”; Electricidade, nº142, pp. 1-11,
Março – Abril 1979.
[7] J. P. Sucena Paiva, Redes de Energia Elétrica, Sucena
Paiva, IST Press, Abril 2005.
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