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Fringing in a parallel-plate capacitor --- The Fringing Effect is a physical phenomenon that occurs within a parallel-plate capacitor. When the plate separation of a parallel-plate capacitor is a large percentage of the dimensions of the plates themselves, the electric field that is normally concentrated between the plates becomes a non-uniform charge distribution around the plates.

Fringing Fields
Consider a parallel-plate capacitor shown in Fig. 1 composed of infinitely-thin, ideally-conducting square plates with side lengths of a and area, separated by distance d. The upper plate has a total charge +Q, the lower plate has the opposite charge –Q; the net charge of the capacitor is zero. The potential difference (voltage) between plates is V. Feeding conductors are implied but excluded from consideration.

Assuming that the entire electric field is concentrated only within the capacitor and that the field is uniform in space, the approximate capacitance, $$ C_0 $$, is given by



$$ (1) \ \ \ \ C_0 = \frac{A\epsilon_0}{d} $$

where $$ \epsilon_0 $$ is the dielectric permittivity of vacuum if the capacitor is situated in vacuum. If the capacitor does not have a high-dielectric inside, Eq. (1) is a good approximation only if d is very small compared to the dimensions of the plates. Otherwise, the fringing effect must be taken into account, which is illustrated in Fig. 2, obtained numerically for a capacitor with. Two 1m by 1m plates separated by 0.2m are subject to potentials ±0.5V. The electric field at the capacitor center and in its volume is still approximately 5.0V/m, but it extends outside the capacitor, mostly in the lateral direction. For example, the field in the center plane of the capacitor at 5cm from its edge is 2.6V – see Fig. 2. The field also extends in the transverse direction. At the capacitor axis and at 10 cm from the upper plate of the capacitor, the field magnitude is still 0.8V/m as seen in Fig. 2. The electric field outside the capacitor possesses energy and this increases the capacitor value given in Eq. (1).

The fringing effect is accompanied by a non-uniform charge distribution on the plates also seen in Fig. 2. In fact, the surface charge density tends to infinity when approaching the edges of an infinitely-thin metal conductor of arbitrary shape. A relevant analytical example is an infinitely-thin metal disk of radius R with the total charge +Q. The surface charge density, $$\sigma$$, in $$\frac{C}{m^2}$$, is given as

$$ (2) \ \ \ \ \sigma = \frac{Q}{2 \pi r} \frac{1}{\sqrt(R^2 - r^2)}, 0 \leq r \leq R $$

and tends to infinity at the edges. Similarly, the electric field close to the metal edges increases – see Fig. 2 – compared to the values within the capacitor. In reality, every physical edge has a finite curvature which results in a very large but still finite charge density. From a numerical point of view, after integration over a surface, the singular charge distribution always gives a finite value. Therefore, numerical methods with standard non-singular elements are still capable of accurately solving for the net charge and the capacitance provided enough elements close to the boundary and an adaptive mesh refinement procedure.

For a two-dimensional parallel-plate capacitor formed by two infinitely-long metal strips, an analytical solution for the capacitance exists (see W. R. Smythe, Static and Dynamic Electricity, McGraw Hill, New York, 1950, problems 58, 59 on p. 109). Analytical solutions for two-dimensional metal strips have been also extended toward an electromagnetic case. For a three-dimensional rectangular parallel-plate capacitor shown in Fig. 1, there seems to be no analytical solution available in the literature. Therefore, numerical solutions using different methods have been employed. In 1959, D. K. Reitan applied the method of subareas, which is an early version of the MoM, to this problem. Every plate was divided into 36 square subareas. Reitan’s solution was later cited by R. Harrington. Next, M. Iskander presented similar data, with the same 36 subareas per plate. His result accurately scanned from the original text is given in Fig. 3, in terms of the dimensionless ratio  where C is the computed capacitance value, and C0 is given by Eq. (1). Since then, the capacitance curve from Fig. 3 has been used in computational classes.



The major task of this study is to revise the curve given in Fig. 3. Following Refs. [6]-[8], a MoM (or Boundary Element Method (BEM)) numerical procedure developed by S. Rao, T. Sarkar, and R. Harrington has been chosen for this purpose utilizing unstructured adaptive triangular meshes. The number of the triangular patches in the final mesh ranges from 52,000 to 62,000. In contrast to the finite-element method, the MoM solution does not require an artificial boundary around the capacitor. It is capable of handling infinitely-thin plates. It also does not require an explicit wire attachment since the electric potential is directly applied to the plate conductors through an appropriate boundary condition. The MoM algorithm has a great potential for capacitance calculation of complex geometries including fast iterative solvers. The details of the numerical procedure used in this study are described in the following section.

Figure 4 gives the computed capacitance values (curve 1) compared to those from Fig. 3 (curve 2). A significant difference in capacitance of up to 16% is observed. On the other hand, a comparison with more modern data for the capacitance of the parallel-plate capacitor shows the maximum deviation of 2%, with the same parameter range.



The computed capacitance values are summarized in Table 1. Based on comparison with the analytical solutions and experiments described in the next section, we believe that the capacitance values given in Table 1 have an accuracy of 0.1% or better for all separation distances.

Calculation of Potential Integrals
The MoM with pulse (piecewise-constant) or other basis functions requires the calculation of potential integrals of the type

$$ (3) \ \ \ \ INSERT \ EQUATION \ 3 \ HERE $$ $$ \ \int \cos \theta \,d\theta = \sin \theta.\ $$

over the area $$S_n$$ of triangle n. The careful calculation of potential integrals is critical for obtaining an accurate and reliable numerical solution. The first integral in Eq. (3) was found using analytical formulas derived in. The second integral has been found using analytical formulas derived in. The analytical formulations make it possible to find the potential and the tangential electric field at any point of the boundary. Outer (non-singular) integrals in the Galerkin method have been found using Gaussian quadratures of variable order.

Adaptive Mesh Refinement
The accurate capacitance calculations, with the error of 0.1-0.001% imply adaptive meshes with greatly different triangle sizes; smaller triangles close to the edges of the conductors should resolve the charge singularity. An initially uniform triangle mesh is refined adaptively, by subdividing edges of those triangles where the local solution error is the largest. In the collocation method , the electric potential on the object surface is matched to the given potential value, e.g. 1V  only for the collocation nodes (positions) – typically triangle centers or vertices. A relative potential mismatch for other positions could be used as a local error indicator. In the Galerkin method, there are no collocation nodes. However, once the numerical solution is available, the potential and the field may still be recalculated at any point of the boundary. Consider a metal object with the constant impressed potential of 1V. Using the existing MoM solution, the electric potential values are recalculated at every triangle center separately. Those values are never 1V but rather vary slightly about this value. The corresponding absolute difference is an error. A similar treatment applies to the tangential electric field, which must be zero at the triangle centers, but is never exactly zero. The local error indicator used for adaptive mesh refinement thus has a form

$$ equation \ \ 4a \ here$$

$$ equation \ \ 4b \ here$$

where $$A_n$$ is a triangle area, and $$0 \leq a \leq 1$$ is a parameter.

Adaptive Mesh Generation
After the edges of triangles in question have been subdivided, Delaunay triangulation is applied followed by Laplacian smoothing of the resulting mesh. We apply a weighted centroid of circumcenters (WCC) smoothing [20] that relocates each vertex to the (weighted) average of circumcenters for attached triangles. The smoothing is done several times while the mesh quality improves. Care must be taken for nodes which tend to cross the mesh boundary. Figure 5 shows the process of adaptive mesh generation for the capacitor from Fig. 2 with $$d/a=.2$$. At every iteration step, 15% of triangles having largest solution errors have been refined.

Control of Mesh Quality during Adaptive Refinement
Despite very different sizes, all triangles in the mesh should be good quality, i.e. close to an equilateral triangle in shape. Long and narrow triangles are not desirable since the electric charge distribution may significantly vary along their lengths. We use a simple yet effective measure of the mesh quality as the ratio between twice radius of the largest inscribed circle, $$r_in$$, and the radius of the smallest circumscribed circle, $$r_out$$ ,

$$ equation \ \ 5 \ \ here $$

where $$ \alpha \, \beta \, \gamma $$ are triangles

Solution Improvement and Convergence
Along with the adaptive mesh refinement, an existing MoM solution is improved at every iteration step in the following way. The electric potential is recalculated at the centers of all faces. A local potential error is now the right-hand side of MoM equations. The MoM equations are solved a second time, which gives the local charge error, $$dc_n$$. The solution for the local charge is then corrected as $$c_n$$ becomes $$c_n - dc_n$$. This method may be applied with or without adaptive mesh refinement. At every adaptive iteration step, the numerical capacitance is found as the ratio of the total charge on the upper plate to the difference in averaged numeric potentials recalculated for every plate. Figure 6 (b) shows the resulting capacitance values for the capacitor from Fig. 2 with $$d/a = 0.2$$ as functions of the mesh size (iteration number). Simultaneously, the minimum triangle quality is shown in Fig. 6a.





Results for two different error indicators
The capacitance value can be computed using the either of the two error indicators derived from the Galerkin method that is, using potential error or the error in tangential electric field. The potential error function (4a) is the local error indicator found from potential error and tangential electric field error function (4b) is the error in tangential electric field.

The Table II summarizes the normalized capacitance values using potential error function and tangential electric field error function at $$\alpha = 1.0$$

Table III summarizes the results for α = 0.5. The values of capacitance obtained from the tables are averaged for the respective configurations and are summarized in Table I.

Solution Test and Applications
The numerical algorithm has been tested by comparison with various electrostatic analytical solutions including self- and mutual capacitances. For the self-capacitance of an infinitely-thin metal circle, our methodology generated a relative numerical error of $$10^{-5}$$ for final meshes with 30,000-50,0000 triangles (after adaptive mesh refinement). Similar data was observed for the self-capacitance of a sphere and the capacitance of two spheres. The algorithm was also tested by comparison with the measurements of small capacitance changes of a parallel-plate capacitor with a sphere or cylinder inside and generated an excellent agreement. Several tests of the local error indicator in Eq. (4) and different capacitance calculation methods have been performed. Finally, the algorithm was tested against non-adaptive high-quality non-inform meshes with triangle size ratios of 1:10 to 1:20 generated following Refs. [21] - [22]. A very good agreement was observed.

Similar to the previous work of one of the authors, the entire numerical algorithm has been implemented in basic MATLAB®. It includes a Graphical User Interface (GUI) and, along with some other electrostatic modules,, is available for download through the Web.

Conclusions
The present study reviews the prior results and re-computes the normalized capacitance values at different separation distances for an isolated parallel-plate charged capacitor with infinitely-thin metal plates. It uses the MoM formulation augmented with adaptive mesh refinement and accurate calculations of potential integrals. The data for the normalized capacitance is given in several forms and compares well with available source in the literature. We believe that our data is accurate. We hope that the capacitance results may be of interest to a wide educational community including Electrical Engineering studies, Physics classes, and Electromagnetic studies.