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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. 9