Kratos Electrostatic Application

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General Description

The KRATOS Electrostatic Application deals with the Finite Element Analysis of problems in electrostatics. It can be considered the most basic problem in electromagnetism and it uses to be the first chapter in the introductory books about electromagnetism.

Therefore, the following description of the Kratos Electrostatic Application can be used as an introduction to:

Electrostatic establishes the most basic principles of the Maxwell equations and it is excellent to work at academic level. Nevertheless, some serious industrial applications can be solved by using just these simplest equations and simulation program (circuit components such as condensers and bipolar diodes, electrical field distribution for high voltage systems, etc).

Theory

Electrostatics[1][2][3] refers to the physical phenomena related to the presence of electric charges in the objects.

Basic principles

Our first Maxwell's equation:

$\vec{\nabla}\cdot\vec{D} = \rho$

where:

$\vec{D} = \varepsilon\vec{E} = \varepsilon_0\varepsilon_r\vec{E}$

with: $\varepsilon, \varepsilon_0, \varepsilon_r$ the permittivity of the medium, of the free space and relative permittivity, respectively.

The electric field is the gradient of an electrostatic potential ($V [Volts]\,$):

$\vec{E}=-\vec{\nabla} V$

Our second Maxwell's equation:

$\vec{\nabla}\times\vec{E} = 0$

By combining these Maxwell's equations, we can easily can obtain the Poisson's equation:

$\vec{\nabla} \varepsilon \vec{\nabla} V + \rho_v = 0$

Basic principles extended

Poisson's Equation in Electrostatics

The detailed form of the Poisson's Equation[9] in Electrostatics is:

$\frac{\partial}{\partial x}\cdot \left( \varepsilon_{x} \cdot \frac{\partial V(x,y,z)}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(\varepsilon_{y} \cdot \frac{\partial V(x,y,z)}{\partial y} \right) + \frac{\partial}{\partial z}\cdot \left(\varepsilon_{z} \cdot \frac{\partial V(x,y,z)}{\partial z} \right) + \rho_v(x,y,z)=0$

Poisson's Equation extended

Electrostatic Boundary Conditions

For a domain ${\Omega} \,$, we should consider three different boundary conditions:

• Dirichlet boundary condition:
$\left . V - \bar V = 0 \right |_{\Gamma_{V}}$
• Neumann boundary condition:
$\left . \hat n \vec{D} - \bar D_n = 0 \right |_{\Gamma_{q}}$
• Infinit condition (when no physical boundary are presents -free space-):
$\left . \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{V}{r}$

Boundary Conditions extended