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This paper presents a numerical analysis for computation of free space capacitance of different arbitrarily shaped conducting bodies based on the finite element method with triangular subsection modeling. Evaluation of capacitance of different arbitrary shapes is important for the electrostatic analysis. Capacitance computation is an important step in the prediction of electrostatic discharge which causes electromagnetic interference. We specifically illustrated capacitance computation of electrostatic models like unit cube, rectangular plate, triangular plate, T-shaped plate, sphere and two touching spheres. Numerical data on the capacitance of conducting objects are presented. The results are compared with other available results in the literature. We used the COMSOL Multiphysics software for the simulation. The models are designed in three-dimensional form using electrostatic environment and can be applied to any spacecraft circuit modeling design. The findings of this study show that the finite element method is a more accurate method and can be applied to any circuit modeling design.

The calculation of electrical capacitance of different arbitrary shapes, like unit cube, rectangular plate, triangular plate, T-shaped plate, sphere and two touching spheres, can be considered as significant objects for spacecraft surface charging design. The capacitance is a very important circuit parameter needed for the analysis of electrostatic discharge (ESD). Generally, the spacecraft geometry is considered as a square and the capacitance is being computed approximately. The external surface of the spacecraft design depends on how efficiently a physical structure has been modeled. A well-designed model not only enables conducting a potential study, but also reduces the number of iterations associated with the model. This study gives a complete insight into the properties of devices and circuits, including transmission, emission, electrostatic effects, etc. The problems related to electromagnetic field do not have a systematic solution, and a mathematical approach is essential. In addition, the studies involving electromagnetic field are usually complex and require a very good working knowledge.

The evaluation of capacitance of different arbitrary shapes is important in Computational Electromagnetics (CEM). It deals with the modeling of the interaction between the electromagnetic fields and the physical objects [

In this paper, the capacitance of the different geometrical assemblies was achieved by subdividing the structure into triangular subsections. The disadvantage of rectangular subsections is that it will not exactly fit into the any arbitrarily shaped geometry. In order to avoid the disadvantage, triangular patch modeling had been in use to perfectly model the arbitrarily shaped surfaces encountered in practical situations [

The FEM is a simpler and easier method compared to other techniques. This method is suitable for solving differential equations and utilizes a more powerful and useful numerical technique for handling the electromagnetic analysis including difficult geometries and inhomogeneous media. An efficient and exact computer model of various electromagnetic field problems, including spacecrafts, is made possible using modern high-speed computers and well-developed mathematical techniques. This model enables a spacecraft designer to visualize the targeted spacecraft on the desktop, thereby providing more information in many cases than can ever be measured in the laboratory. The turnaround time required to obtain the spacecraft properties after varying the spacecraft shape is usually calculated in minutes or hours by computer model. The designer can adjust the spacecraft by modifying certain specific parameters of the simulation model. The precision of the existing mathematical model is often such that only a small degree of adjustment is required. However, an accurate understanding of the computed results is essential. It is more important to ensure that the implemented models can be applied to the actual problem to be solved, and the results can be obtained with sufficient speed and accuracy [

FEM is well suitable for arbitrary shapes. The simple model of the FEM is based on the behavior of a function, which may be complex when viewed from an enormous region while a simple evaluation may be appropriate for a minor subregion. The entire region is separated into non overlapping subregions called as finite element, and the function of each element is approximated by the algebraic expression [

The expression of capacitance can be introduced by applying the FEM, i.e., charges and potentials in any system of conductors that create an electric field. Depending on the nature of the system of conductors measured, the capacitance of a solitary conductor, the capacitance between two conductors, and the capacitance in a system of many conductors can be distinguished. The capacitance of the surface can be computed from

where Q is the charge of the conductor (Coulomb),

V is the potential of the conductor (Voltage).

Calculating the capacitance of a simple system like a sphere is important in spacecraft design. Analytical expression for the sphere is denoted by the following equation [

The capacitance of the sphere can be computed from

where ε_{0} is the permittivity of free space (=8.854 × 10^{−}^{12} F/m),

a is the radius of the sphere (meter)

The potential V_{e} is calculated by

For a triangular element, a, b, and c are constants.

A typical triangular element is putting in place of the element governing equations.

The potential V_{e}_{1}, V_{e}_{2} and V_{e}_{3} are calculated by

Similar transformation matrices can be achieved when higher order plane elements are used. The shape functions are calculated in truss element nodes with known coordinates. Typically, these coordinates are agreed in the global coordinates while the shape functions of the plane elements are agreed in the natural coordinate systems. The surface to be analyzed is divided into N number of triangular subsections. The geometry of the reference element is mapped into the geometry of the source triangle using geometrical transformation functions. The parametric coordinates of ξ and η in the reference triangle can be mapped into a global coordinates of x and y. For triangular elements, the global coordinates (x, y) and the natural coordinates (x, h) are given by

where N_{1}, N_{2} and N_{3} are the geometrical transformation function, in which x_{i}, y_{i} (i = 1,2,3) represents the coordinates of the triangular element nodes and the shape functions have the following expressions represented in the natural coordinate system as:

For the known x, y coordinates, the corresponding natural coordinates can be obtained by solving the following system of equations. Using the above expressions, free space capacitance can be calculated [

There is no analytical expression for calculating the electrical capacitance of a unit cube. In this section, the finite element method is used for calculating unit cube [

measuring 1.0 m. The model is designed in three-dimensional modeling using electrostatic environment. In the boundary condition of model design, we used ground boundary which is zero potential. For the unit cube, the bottom of the cube is specified as ground with a voltage of 0 V. The top of the cube has a specified voltage of 1 V. It is assumed that the unit cube is made of highly conductive material in which the total resistive value is much lower. When the unit voltage is applied to the object the charge densities near the edge of the bodies [

From the unit cube model, we produced more number of subsections and 2300 domain element in the finite element mesh shown in

In some cases, the capacitance values are calculated with respect to variation in the number of domain elements and boundary elements. The results are tabulated in

In this section, the simulation of the unit cube is analyzed. The findings of this study are in accordance with the available results [

In this section, the modeling of rectangular plate by determining capacitance using finite element method is discussed. The capacitance values are calculated for certain cases with respect to the difference in the number of domain elements and boundary elements in rectangular plate.

Geometry | Proposed Method Capacitance | Method [ |
---|---|---|

Van Bladel [ | Variation method | 65.56 [pF] |

Read [ | Refined boundary element method | 66.07 [pF] |

Wintle [ | Random walk method with variance reduction | 66.06 [pF] |

Mascagni [ | Random walk on the boundary | 66.07 [pF] |

Hwang [ | Walk on planes | 66.07 [pF] |

Proposed method | 66.05 [pF] |

The simulation produces the finite element mesh with triangular subsections and 120 boundary elements as shown in

The modeling of an equilateral triangle is an important geometrical model for spacecraft circuit modeling design. The capacitance values are calculated with respect to the difference in the number of domain elements and boundary elements in equilateral triangle plate.

The modeling resulted in a finite element mesh with triangular subsections and 80 boundary elements as shown in

Ghosh and Chakrabarty [

In this section, the utility of FEM used to calculate the electrical capacitance of T-shaped plate is discussed.

In this section, the FEM is used to calculate the electrical capacitance of sphere.

Under the boundary condition of model design, it is possible to produce a number of subsections from the sphere model like 2058 domain element of the finite element mesh and the potential distribution simulations help to better understand the potential distribution of the metallic object. From the model, the finite element mesh with triangular subsections and 504 boundary elements were produced.

The capacitance values are calculated for some cases such as with respect to variation in a number of domain elements and boundary elements. The capacitance value obtained, 111.47 pF, is equated with the value obtained from the earlier results. The results tend to converge and the deviation in analytical and numerical results is reduced.In this study, the values matched with the available results. In this section, the sphere simulations with different subsections are analyzed. The results obtained in this study correspond with the available results [

In this section, FEM used for calculating the capacitance of two touching spheres is discussed. Touching spheres are important in circuit modeling design. When the two spheres touch, they can be regarded as being fused into one single complex conductor.

From the model, the finite element mesh with triangular subsections and 504 boundary elements were produced as shown in

Geometry | Proposed Method Capacitance | Method [ |
---|---|---|

Rectangular plate | 54.62 [pF] | 54.73 [pF] |

Triangular plate | 23.35 [pF] | 23.13 [pF] |

T-shaped plate | 98.43 [pF] | 98.53 [pF] |

Sphere | 111.47 [pF] | 111.8 [pF] |

Two touching spheres | 154.57 [pF] | 154.8 [pF] |

In the present study, the capacitance of a unit cube was 66.05 pF, which is similar to that obtained in other studies. The capacitance of the rectangular plate was found to be 54.62 pF, which is similar to capacitance obtained in other studies, i.e., 54.73 pF [

The results of five geometries are summarized in

However, this study has certain limitations that have been acknowledged. The simulation was confined to other methods like random walk method and method of moments. Nevertheless, in spite of these limitations, this study provided new insights into the capacitance computation of different conducting bodies.

FEM has been found to be the most accurate method for evaluating free space capacitances. In the present study, different arbitrary shapes are analysed for electrostatic modelling. The capacitance of different arbitrary shaped conductors like unit cube, rectangular plate, triangular plate, T-shaped plate, sphere and two touching spheres was calculated. Some of the simulations obtained in the study show the usage of FEM with COMSOL Multiphysics software. The results derived from using the software correspond with the results of previous studies. Thus, the method is more suitable for various shapes involved in spacecraft circuit modeling design. This approach is simple and can be applied to any practical shapes of the metallic objects.

Dhamodaran Muneeswaran,Dhanasekaran Raghavan, (2016) Evaluation of the Capacitance and Charge Distribution for Conducting Bodies by Circuit Modelling. Circuits and Systems,07,280-291. doi: 10.4236/cs.2016.74025