Kratos Electrostatic Application

From KratosWiki

(Difference between revisions)

Latest revision as of 18:35, 1 February 2010

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:

general concepts in electromagnetism;
the Finite Element Method applied to the electromagnetism;
the use of Kratos for basic problems;

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 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

References

The Finite Element Method for Electrostatics

Using the Application

Revision as of 17:30, 1 February 2010 (view source) JMora (Talk \| contribs) (→References) ← Older edit		Latest revision as of 18:35, 1 February 2010 (view source) JMora (Talk \| contribs) (→Basic principles)
Line 41:		Line 41:


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

	:<math>\vec{\nabla} \varepsilon \vec{\nabla} V + \rho_v = 0</math>		:<math>\vec{\nabla} \varepsilon \vec{\nabla} V + \rho_v = 0</math>

Kratos Electrostatic Application

Latest revision as of 18:35, 1 February 2010

Contents

General Description

Theory

Basic principles

Poisson's Equation in Electrostatics

Electrostatic Boundary Conditions

References

The Finite Element Method for Electrostatics

Using the Application

Views

Personal tools

Navigation

Search

Toolbox

Categories