Analysis of Capacitance to Ground Formulas for Different High-Voltage Electrodes

Stray capacitance can seriously affect the behavior of high-voltage devices, including voltage dividers, insulator strings, modular power supplies, or measuring instruments, among others. Therefore its effects must be considered when designing high-voltage projects and tests. Due to the difficulty in measuring the effects of stray capacitance, there is a lack of available experimental data. Therefore, for engineers and researchers there is a need to revise and update the available information, as well as to have useful and reliable data to estimate the stray capacitance in the initial designs. Although there are some analytical formulas to calculate the capacitance of some simple geometries, they have a limited scope. However, since such formulas can deal with different geometries and operating conditions, it is necessary to assess their consistency and applicability. This work calculates the stray capacitance to ground for geometries commonly found in high-voltage laboratories and facilities, including wires or rods of different lengths, spheres and circular rings, the latter ones being commonly applied as corona protections. This is carried out by comparing the results provided by the available analytical formulas with those obtained from finite element method (FEM) simulation, since field simulation methods allow solving such problem. The results of this work prove the suitability and flexibility of the FEM approach, because FEM models can deal with wider range of electrodes, configurations and operating conditions.


Introduction
The calculation of capacitance formulas has received little attention compared to the analysis of inductance calculation formulas [1,2].Nearby surfaces separated by an insulating medium such as air, subjected to different electric potentials, induce a stray or parasitic capacitance, and therefore this configuration acts as a capacitor.High voltages and high frequencies tend to amplify the effects of the unwanted stray capacitance.The analysis of stray capacitance effects is of interest in different disciplines, including electrical engineering, high-voltage applications, radio engineering or physical sciences, among others [3].
Different studies prove that the stray capacitance produces an uneven voltage distribution across each insulator unit in a high-voltage insulator string [4,5].The effect of the stray capacitance is to reduce the efficacy of each additional insulator unit due to the non-linear voltage distribution [6].This is because the capacitive current and the corresponding voltage drop across each insulator unit are greater in the insulator units closer to the conductors, due to the effect of the distributed stray capacitances to ground [7].The string elements closer to the line are subjected to a higher electrical stress than those closer to the tower.Grading rings can be used at both ends of the string to homogenize the voltage drop across each insulator unit [8].A similar effect occurs in high-voltage switching mode power supplies composed of several modules in series, since in [9] it is proved that the stray capacitance to ground has a significant impact on the individual voltage of each module.In other high-voltage applications, including high-voltage transformers [10] or high-voltage motors, parasitic capacitances have a key role to predict the frequency behavior of such machines.
Accurate methods to calculate the capacitance are based on the calculation of the electrostatic field generated by the system of charged objects under consideration [3].The capacitance of basic isolated geometries, such as very long horizontal cylindrical conductors, coaxial cylindrical conductors, concentric spheres, or spheres well above ground, can be easily deduced theoretically.However, such formulas have very restricted practical use.The calculation of the capacitance of conductive objects which are close to ground leads to challenging mathematical problems, even for simple geometries.Therefore, analytical solutions for capacitance only exist for a limited number of electrode geometries and configurations, which have almost no practical applications [11], and often only contemplate the stray capacitance to ground, thus disregarding the effects of nearby grounded electrodes, structures or walls [3].
As a consequence, computational methods are increasingly being applied to solve such problem, although most of the published works deal with very particular problems, such as insulator strings [4,5,7], transformer windings [10] or voltage dividers [12], among others.FEM is perhaps the most applied computational technique to calculate the effects of capacitance, since it allows dealing with complex three-dimensional geometries, as reflected in several works [13][14][15][16][17][18][19].
Since stray capacitances are not easily measurable, because of the low immunity to noise of the small signal to be acquired [20], results provided by numerical methods are a good alternative during the design stage of high-voltage devices and instruments.Therefore, the capacitance between energized electrodes or between electrodes and ground is a factor to be considered when designing and planning high-voltage projects and tests [11].
Most works analyze specific problems related to the unwanted effects of stray capacitance, such as in transformer windings [10], motor windings [21] or insulator strings [7], among others.However, there are no recent works providing a systematic account of the problems to calculate the stray capacitance to ground for geometries found in high-voltage laboratories and high-voltage installations such as substations.These geometries include wires, rods, spheres, and circular rings, the latter ones being commonly applied as corona protections.The stray capacitances due to these high-voltage electrodes can have a non-negligible impact on the measurement results and behavior of the devices involved.
This paper is focused to review and analyze the accuracy of different formulas found in the technical literature to calculate the stray capacitance to ground of various high-voltage electrodes.To this end, due to the lack of available experimental data because of experimental difficulties related to the small signal to be acquired and noise immunity, the results provided by the formulas are compared with the results provided by FEM simulations.It is noted that regardless the impact of the stray capacitance in high-voltage applications, at our knowledge, there are no published technical works assessing the accuracy of such formulas; thus this work contributes in this area.Results presented prove that FEM models offer flexibility, simplicity and accuracy to analyze the effects of stray capacitance in high-voltage systems with different geometries.

The FEM Approach to Analyze the Stray Capacitance
The stray capacitance to ground is directly related to the distribution of the electric field around high-voltage electrodes [22].It is a recognized fact that the effects of stray capacitance can be determined by means of FEM-based approaches [12,23].
The capacitance can be calculated from the ratio C = Q/U, defined by the charge Q stored in the system and the electric potential U, supposing that the system under analysis is far from other charged bodies [3].Therefore, the stray capacitance concept arises between any two charged bodies subjected to different electric potentials, and can be important in high-frequency and high-voltage applications.For low-frequency applications, the capacitance can be calculated by analyzing the energy related to the electrostatic field, thus disregarding the displacement current.To evaluate the capacitance of a given geometry, it is necessary to calculate the electric potential on the surface of the analyzed conductor.Next, the outer electric potential and the electric field are calculated within all points surrounding the conducting electrodes, by applying the potential as boundary condition [14,24].Finally, the capacitance of the analyzed system is calculated by applying: W E being the stored electrostatic energy.
The next paragraphs detail the process to follow for determining the stored electrostatic energy.
From the Gauss law that relates the distribution of the electric charge density (C/m 3 ) to the electric field and the relationship between the electric field and the electric potential U, Equation ( 3) allows solving for the electric potential and field in all points of the domain [26].Next, the energy density in any point of the air domain is calculated as: The electrostatic energy stored in the domain can be calculated by integrating the energy density over the volume of the domain outside the conductive high-voltage electrode [14,27]: Finally, the capacitance is calculated by applying Equation (1) [24].Therefore, the capacitance is calculated from the electrostatic energy stored in the air because of the incitation of U.
Figure 1 shows the surface meshes applied to some of the geometries analyzed in this paper, including cylindrical conductors, circular rings and spheres.For low-frequency applications, the capacitance can be calculated by analyzing the energy related to the electrostatic field, thus disregarding the displacement current.To evaluate the capacitance of a given geometry, it is necessary to calculate the electric potential on the surface of the analyzed conductor.Next, the outer electric potential and the electric field are calculated within all points surrounding the conducting electrodes, by applying the potential as boundary condition [14,24].Finally, the capacitance of the analyzed system is calculated by applying: WE being the stored electrostatic energy.
The next paragraphs detail the process to follow for determining the stored electrostatic energy.
From the Gauss law that relates the distribution of the electric charge density (C/m 3 ) to the electric field and the relationship between the electric field and the electric potential U, = - EU , the Poisson's equation for electrostatics arises [25],

-/ U  
(3) Equation ( 3) allows solving for the electric potential and field in all points of the domain [26].Next, the energy density in any point of the air domain is calculated as: The electrostatic energy stored in the domain can be calculated by integrating the energy density over the volume of the domain outside the conductive high-voltage electrode [14,27]: Finally, the capacitance is calculated by applying Equation (1) [24].Therefore, the capacitance is calculated from the electrostatic energy stored in the air because of the incitation of U.
Figure 1 shows the surface meshes applied to some of the geometries analyzed in this paper, including cylindrical conductors, circular rings and spheres.Figure 2 shows the blocks and meshes types applied in the simulations.Whereas a sufficiently large outer block was used to set the boundary conditions, a much smaller inner block was used in order to apply a much finer tetrahedral mesh to ensure improved accuracy.Figure 2 shows the blocks and meshes types applied in the simulations.Whereas a sufficiently large outer block was used to set the boundary conditions, a much smaller inner block was used in order to apply a much finer tetrahedral mesh to ensure improved accuracy.
Both three-dimensional (3D) and two-dimensional (2D) FEM simulations were carried out to simulate all geometries analyzed in this work.Whereas 3D-FEM simulations were applied to wires of finite length, circular rings and spheres, 2D-FEM simulations were applied to analyze wires of infinite length.Parametric simulations were carried out to automatically change the value of different parameters, such as the height about ground level or the curvature radius of the different analyzed geometries.Both three-dimensional (3D) and two-dimensional (2D) FEM simulations were carried out to simulate all geometries analyzed in this work.Whereas 3D-FEM simulations were applied to wires of finite length, circular rings and spheres, 2D-FEM simulations were applied to analyze wires of infinite length.Parametric simulations were carried out to automatically change the value of different parameters, such as the height about ground level or the curvature radius of the different analyzed geometries.
Simulations conducted in this work were performed by means of the Comsol Multiphysics package using the electrostatics module.They consist of approximately 0.5-9.5 million tetrahedral elements, 66-590 thousand triangular elements, 0.5-73 thousand edge elements, and 22-96 vertex elements, depending on the specific geometry analyzed.

The Analyzed Geometries
This section analyzes the accuracy of formulas to calculate the capacitance between different electrode geometries and an infinite ground plane, some of them being based on the method of images.

Finite-Length Straight Round Wire Which is Parallel to the Ground Plane
Figure 3 shows the layout of a straight conductor of round section which is parallel to a conducting plane.According to [3], the capacitance of a straight conductor of round section which is parallel to a conducting plane can be approximated as: being the permittivity of air, 8.85 × 10 −12 F/m, l the length of the conductor, h the height above ground and D1 a coefficient depending on l/(2h), which is obtained by interpolating the values given in Table 1.Simulations conducted in this work were performed by means of the Comsol Multiphysics package using the electrostatics module.They consist of approximately 0.5-9.5 million tetrahedral elements, 66-590 thousand triangular elements, 0.5-73 thousand edge elements, and 22-96 vertex elements, depending on the specific geometry analyzed.

The Analyzed Geometries
This section analyzes the accuracy of formulas to calculate the capacitance between different electrode geometries and an infinite ground plane, some of them being based on the method of images.Both three-dimensional (3D) and two-dimensional (2D) FEM simulations were carried out to simulate all geometries analyzed in this work.Whereas 3D-FEM simulations were applied to wires of finite length, circular rings and spheres, 2D-FEM simulations were applied to analyze wires of infinite length.Parametric simulations were carried out to automatically change the value of different parameters, such as the height about ground level or the curvature radius of the different analyzed geometries.
Simulations conducted in this work were performed by means of the Comsol Multiphysics package using the electrostatics module.They consist of approximately 0.5-9.5 million tetrahedral elements, 66-590 thousand triangular elements, 0.5-73 thousand edge elements, and 22-96 vertex elements, depending on the specific geometry analyzed.

The Analyzed Geometries
This section analyzes the accuracy of formulas to calculate the capacitance between different electrode geometries and an infinite ground plane, some of them being based on the method of images.According to [3], the capacitance of a straight conductor of round section which is parallel to a conducting plane can be approximated as:

Finite-Length Straight Round Wire Which is Parallel to the Ground Plane
being the permittivity of air, 8.85 × 10 −12 F/m, l the length of the conductor, h the height above ground and D1 a coefficient depending on l/(2h), which is obtained by interpolating the values given in Table 1.According to [3], the capacitance of a straight conductor of round section which is parallel to a conducting plane can be approximated as: being the permittivity of air, 8.85 × 10 −12 F/m, l the length of the conductor, h the height above ground and D 1 a coefficient depending on l/(2h), which is obtained by interpolating the values given in Table 1.Table 2 compares the capacitance of a finite-length straight round wire, which is parallel to the ground plane, provided by Equation ( 6) with those obtained by means of FEM simulations.Data presented in Table 2 show a good agreement between the results provided by Equation ( 6) and those from FEM simulations.

Finite-Length Straight Wire of Round Section Which Is Perpendicular to the Ground Plane
According to [3], the capacitance of a straight conductor of round section which is perpendicular to a conducting plane, (see Figure 4) can be approximated as: D 2 being a coefficient, which is calculated by interpolation of the values given in  2 show a good agreement between the results provided by Equation ( 6) and those from FEM simulations.

Finite-Length Straight Wire of Round Section Which is Perpendicular to the Ground Plane
According to [3], the capacitance of a straight conductor of round section which is perpendicular to a conducting plane, (see Figure 4) can be approximated as: D2 being a coefficient, which is calculated by interpolation of the values given in Table 3, as a function of h/l.     4 compares the capacitance of a finite-length straight round wire, which is perpendicular to the ground plane, provided by Equation ( 7) with those obtained by means of FEM simulations.Results presented in Table 4 show a good match between the data provided by Equation (7), with the data obtained from FEM simulations, especially when the height h of the conductor above the ground plane increases.

Infinite-Length Straight Circular Wire Which Is Parallel to a Conducting Plane
By applying the images method, the capacitance per unit length of a round straight conductor of infinite-length, which is parallel to a conducting plane, as shown in Figure 5, can be calculated as [3]: where a is the radius of the conductor, and h is the distance between its center and the ground plane.In [28], an equivalent formula is proposed: Figure 5 shows an infinite straight conductor of radius a, which is parallel to the ground plane: Table 5 summarizes the results attained by means of Equation ( 8), Equation ( 9) and FEM simulations for different values of the radius a, and the height h.It shows an excellent agreement between the results provided by Equation ( 8), Equation ( 9) and FEM simulations.By applying the method of the mirror images, the capacitance per unit length of an infinitelength straight conductor of square section, which is parallel to a conducting plane, can be calculated as [3]: where x = 3.39h/a, a being the side length of the square, and h the distance between the geometrical center of the wire and the ground plane.Figure 6 shows the layout of the analyzed conductor.Table 5 summarizes the results attained by means of Equations ( 8), ( 9) and FEM simulations for different values of the radius a, and the height h.It shows an excellent agreement between the results provided by Equations ( 8), ( 9) and FEM simulations.By applying the method of the mirror images, the capacitance per unit length of an infinite-length straight conductor of square section, which is parallel to a conducting plane, can be calculated as [3]: where x = 3.39 h/a, a being the side length of the square, and h the distance between the geometrical center of the wire and the ground plane.Figure 6 shows the layout of the analyzed conductor.Table 6 shows the results attained by applying Equation (10) and FEM for different values of the side length a, and the height h.

Sphere Over a Conducting Plane
Spheres are frequently applied as corona protections in high-voltage applications, although their stray capacitance can affect the behavior of sensitive high-voltage devices.
Figure 7 shows the layout of a sphere, where r is its radius, and h the distance between the lowest point of the sphere and the ground plane.Table 6 shows the results attained by applying Equation (10) and FEM for different values of the side length a, and the height h.

Sphere Over a Conducting Plane
Spheres are frequently applied as corona protections in high-voltage applications, although their stray capacitance can affect the behavior of sensitive high-voltage devices.
Figure 7 shows the layout of a sphere, where r is its radius, and h the distance between the lowest point of the sphere and the ground plane.By applying the method of the images, the capacitance of a conductive sphere of radius r placed at a height h above a ground plane is given by [29]: where A simplified expression for this arrangement is given in [29] as: An equivalent formulation to Equation ( 11) was proposed by Snow [28]: being defined as: and h* = h + r.
Table 7 shows the results obtained by applying Equation (11), Equation (12), Equation ( 13) and FEM, for different values of the radius r and the height h.By applying the method of the images, the capacitance of a conductive sphere of radius r placed at a height h above a ground plane is given by [29]: where η 0 = acos h(1 + h/r).
A simplified expression for this arrangement is given in [29] as: An equivalent formulation to Equation ( 11) was proposed by Snow [28]: being defined as: and h* = h + r.
Table 7 shows the results obtained by applying Equations ( 11)-( 13) and FEM, for different values of the radius r and the height h.Results presented in Table 7 show an excellent match between Equations ( 11), (13) and FEM results, although a greater difference when applying Equation ( 13), as expected.

Circular Ring or Toroid Which Plane Is Parallel to a Conducting Plane
Toroidal rings are commonly used as corona protections in high-voltage applications.However, the rings introduce a non-negligible capacitance in the system, whose impact cannot be neglected depending on the application.
Figure 8 shows a toroid of round section which is parallel to a conducting plane and different toroids lying in parallel planes.
By applying the images method, the capacitance of a circular ring or toroid, which is parallel to a conducting plane, can be calculated as [3]: where a and R are, respectively, the minor and major radiuses, h is the distance between the geometrical center of the toroid and the ground plane, k 2 = R 2 R 2 +h 2 and K is the complete elliptic integral of first kind with modulus k 2 , which can be computed by means of the ellipke function of MATLAB (2017, The MathWorks Ltd., Natick, MA, USA).
Table 8 summarizes the results obtained by applying Equation ( 15) and FEM, for different values of the radiuses a, R and the height h.It shows a good agreement between the results of both systems.

Circular Ring or Toroid Which Plane is Parallel to a Conducting Plane
Toroidal rings are commonly used as corona protections in high-voltage applications.However, the rings introduce a non-negligible capacitance in the system, whose impact cannot be neglected depending on the application.
Figure 8 shows a toroid of round section which is parallel to a conducting plane and different toroids lying in parallel planes.By applying the images method, the capacitance of a circular ring or toroid, which is parallel to a conducting plane, can be calculated as [3]:  FEM allows the simple simulation of several concentric corona rings sharing the same vertical axis as shown in Figure 8b.This configuration common applied in high-voltage generators, voltage dividers, or capacitors, among others.However, there are no analytical formulas to deal with this geometry.
Table 9 presents the results obtained by means of FEM simulations, when dealing with a row of five circular rings which are placed as shown in Figure 8b.According to [3], the capacitance of n parallel round conductors of finite length lying in the same plane, which is parallel to the ground plane, can be calculated as: a being the radius of the conductors, l the length of any conductor, h the distance between the center of conductor and the ground plane, b the distance between the centers of two adjacent conductors, and B n a coefficient, which is calculated as: The values of D 1 (0.5l/h) are given in Table 1.It is noted that Equation ( 16) is not a general formula, since it is only valid when b ≤ l/(n−1).
When the distance d k (k = 1, 2, . . ., n−1) between any two wires is inferior than the mean height above the ground plane, that is, d k << h, [3] suggests to apply: F 1 being calculated as: Figure 9 shows the layout of the n parallel round wires of finite length, lying in a plane parallel to the ground plane.
Table 10 summarizes the results obtained by applying Equations ( 16), (18) and FEM, for different values of the radius a, the length l, and the height h.Table 10 summarizes the results obtained by applying Equation ( 16), Equation ( 18) and FEM, for different values of the radius a, the length l, and the height h.According to the results summarized in Table 10, FEM results are more general results.FEM simulations avoid the inherent limitations and applicability of Formulas Equations ( 16) and ( 18), since they are not general, and only provide accurate results when fulfilling some constraints.According to [3], the capacitance of n identical straight wires of finite length parallel to the ground plane, and arranged on the surface of a circular cylinder, is given by Equation (18).It is noted that the coefficients D 1 (0.5l/h) and d k (distance between the centers of any two wires) included in Equation ( 19) are obtained from Table 1, and by applying d k = 2R sin(kπ/n) with k = 1, 2, 3, . . ., n−1, respectively.It is known that Equation ( 18) is not accurate when the distance d k (k = 1, 2, . . ., n−1) between any two wires is inferior than the height of any conductor above the ground plane, that is, d k << h.
Figure 10 shows n straight round conductors of finite length (radius a, length l) parallel to the ground plane, which are placed on the surface of a circular cylinder of radius R whose center is placed at a height h above the ground plane.
According to the results summarized in Table 10, FEM results are more general results.FEM simulations avoid the inherent limitations and applicability of Formulas Equation (16) and Equation (18), since they are not general, and only provide accurate results when fulfilling some constraints.

n Identical Straight Wires of Finite Length Parallel to the Ground Plane and Arranged on the Surface of a Circular Cylinder
According to [3], the capacitance of n identical straight wires of finite length parallel to the ground plane, and arranged on the surface of a circular cylinder, is given by Equation (18).It is noted that the coefficients D1 (0.5l/h) and dk (distance between the centers of any two wires) included in Equation ( 19) are obtained from Table 1, and by applying 2 sin( π / ) respectively.It is known that Equation ( 18) is not accurate when the distance dk (k = 1, 2, …, n−1) between any two wires is inferior than the height of any conductor above the ground plane, that is, dk << h.
Figure 10 shows n straight round conductors of finite length (radius a, length l) parallel to the ground plane, which are placed on the surface of a circular cylinder of radius R whose center is placed at a height h above the ground plane.Table 11 summarizes the results obtained by applying Equation (20) and FEM, for different values of the radius a, the length l, and the height h.Table 11 summarizes the results obtained by applying Equation (20) and FEM, for different values of the radius a, the length l, and the height h.Results summarized in Table 11 prove that the accuracy of Equation ( 18) lowers when reducing the height of the conductors above the ground plane.

Discussion
This section summarizes the results attained in this work.As deduced from Section 3, FEM simulations offer more flexibility, generalization capability, and the possibility to deal with more complex geometries than analytical formulas for calculating the stray capacitance of high-voltage electrodes to ground.Although analytical formulas only are available for some simple geometries, they can be effectively applied to determine the straight capacitance to ground of high-voltage electrodes with simple geometries.
Tables 12-14 summarize the main results attained in this work.
Results summarized in Table 12 clearly show a very good agreement between the results provided by analytical formulas and FEM simulation for wires of infinite length, whereas in the case of straight wires of finite length, maximum differences around 1-5% are found.
Results presented in Table 13 clearly show that in the case of spheres, analytical and FEM results are almost the same, whereas in the case of circular rings, maximum differences around 5-8% are found.

Geometry Analyzed Mean Difference Maximum Difference
Finite-length straight round wire which is parallel to the ground plane: Equation ( 6 When dealing with multi-wire configurations, the differences are much higher than in the geometries above, since differences around 5-56% can be found.In addition, for some geometries, the analytical formulas available for multi-wire geometries cannot be applied, since some of the input parameters are out of the applicability limits of such formulas.

Conclusions
This work has analyzed the behavior of several approximate and exact formulas to calculate the capacitance to ground of high-voltage electrodes with different geometries.Due to experimental difficulties involved, there is a lack of experimental data, so it is necessary to develop numerical methods to infer the value of the capacitance when designing high-voltage tests and projects.The analyzed geometries found in high-voltage applications include different combinations of wires and rods, spheres and circular rings, the latter two applied as corona protections in high-voltage projects.The results provided by the analyzed formulas have been compared against the results provided by FEM simulations, since FEM simulations are internationally recognized as a valid means of obtaining accurate and reliable data.The analysis performed in this work has proved the inherent limitations of the analyzed formulas.It has been proved that although in some configurations the results are very accurate (single straight wires or single sphere configurations), while in other cases important differences arise, especially when dealing with multi-wire configurations.The stray capacitance of a multi-toroid geometry has been also analyzed, which cannot be calculated by means of analytical formulas.Therefore, this work has proven that analytical formulas can only deal with a narrow number of geometries, and has also proven the enhanced performance and flexibility of FEM simulations, due to their suitability, flexibility, and the wider range of geometries that FEM simulations allow to be evaluated.

Figure 1 .
Figure 1.Surface meshes of some of the geometries analyzed in this paper.(a) Single-and multiconductor arrangements; (b) Circular ring or toroid; (c) Sphere.

Figure 1 .
Figure 1.Surface meshes of some of the geometries analyzed in this paper.(a) Single-and multi-conductor arrangements; (b) Circular ring or toroid; (c) Sphere.

Figure 2 .
Figure 2. Blocks and meshes applied in the finite element method (FEM) simulations.

Figure 3 .
Figure 3. Finite-length straight round wire of radius a and length l. which is parallel to the ground plane.

Figure 2 .
Figure 2. Blocks and meshes applied in the finite element method (FEM) simulations.

3. 1 .Figure 2 .
Figure3shows the layout of a straight conductor of round section which is parallel to a conducting plane.

Figure 3 Figure 3 .
Figure3shows the layout of a straight conductor of round section which is parallel to a conducting plane.

Figure 3 .
Figure 3. Finite-length straight round wire of radius a and length l. which is parallel to the ground plane.

Figure 4 .
Figure 4. Finite-length straight round wire of radius a, and length l, which is perpendicular to a conducting plane.

Figure 4 .
Figure 4. Finite-length straight round wire of radius a, and length l, which is perpendicular to a conducting plane.

Figure 5 .
Figure 5. Infinite-length straight round wire of radius a, which is placed parallel to a conducting plane at a height h.

Figure 5 .
Figure 5. Infinite-length straight round wire of radius a, which is placed parallel to a conducting plane at a height h.

Figure 6 .
Figure 6.Infinite-length straight wire of square section of side length a. which is placed parallel to a conducting plane at a height h.

Figure 6 .
Figure 6.Infinite-length straight wire of square section of side length a. which is placed parallel to a conducting plane at a height h.

Figure 7 .
Figure 7. Sphere of radius r placed at a height h over a conducting plane.

Figure 7 .
Figure 7. Sphere of radius r placed at a height h over a conducting plane.

Figure 8 .
Figure 8.(a) Toroid of circular cross section which plane is parallel to a conducting plane at a height h, with inner radius a, and outer radius R; (b) Different toroids lying in parallel planes.

Figure 8 .
Figure 8.(a) Toroid of circular cross section which plane is parallel to a conducting plane at a height h, with inner radius a, and outer radius R; (b) Different toroids lying in parallel planes.

Figure 9 .
Figure 9. n parallel round wires of finite length lying in a plane parallel to the ground plane.

Figure 9 .
Figure 9. n parallel round wires of finite length lying in a plane parallel to the ground plane.

3. 8 .
n Identical Straight Wires of Finite Length Parallel to the Ground Plane and Arranged on the Surface of a Circular Cylinder

Figure 10 .
Figure 10.n straight round conductors of finite length parallel to the ground plane and arranged on the surface of a circular cylinder.

Figure 10 .
Figure 10.n straight round conductors of finite length parallel to the ground plane and arranged on the surface of a circular cylinder.

Table 1 .
Values of coefficient D 1 as a function of l/(2h).

Table 2 .
Capacitance of a straight round wire of finite length which is parallel to a conducting plane.

Table 3
, as a function of h/l.

Table 3 .
Values of coefficient D2 as a function of h/l.

Table 4
compares the capacitance of a finite-length straight round wire, which is perpendicular to the ground plane, provided by Equation (7) with those obtained by means of FEM simulations.

Table 4 .
Capacitance of a straight round wire of finite length which is perpendicular to a conducting plane.

Table 3 .
Values of coefficient D 2 as a function of h/l.

Table 4 .
Capacitance of a straight round wire of finite length which is perpendicular to a conducting plane.

Table 5 .
Capacitance of a straight round wire of infinite length which is parallel to a conducting plane.

Table 5 .
Capacitance of a straight round wire of infinite length which is parallel to a conducting plane.

Table 6 .
Capacitance of a straight wire of infinite length and square section which is parallel to a conducting plane.

Table 6 .
Capacitance of a straight wire of infinite length and square section which is parallel to a conducting plane.

Table 7 .
Capacitance of a sphere over a conducting plane.

Table 7 .
Capacitance of a sphere over a conducting plane.

Table 8 .
Capacitance of a toroid which is parallel to a conducting plane for different configurations.

Table 9 .
Capacitance of a row of five circular rings which are parallel to a conducting plane.Parallel Round Wires of Finite Length Lying in a Plane Parallel to the Ground Plane

Table 10 .
Capacitance of n round straight wires of finite length lying in a plane which is parallel to the ground plane.

Table 10 .
Capacitance of n round straight wires of finite length lying in a plane which is parallel to the ground plane.

, l = 10 m, b = 3•a, h = 1 m
* Does not fulfill the requirements of the formula.

Table 11 .
Capacitance of n infinitely long parallel wires of round section lying in a plane parallel to a conducting plane.

Table 11 .
Capacitance of n infinitely long parallel wires of round section lying in a plane parallel to a conducting plane.

Table 12 .
Single straight wires.Comparative results between analytical formulas and FEM simulations.