Kratos Magnetostatic Application

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Latest revision as of 11:12, 12 January 2011

General Description

The KRATOS Magnetostatic Application deals with the Finite Element Analysis of problems in magnetostatics. Though the mathematics and general formulation is quite similar to Electrostatics, Magnetostatics presents important differences at a conceptual and a programming level. Therefore, if you are a beginner in electromagnetism or programming electromagnetic applications, we strongly recommend you that you first start with the Electrostatic application.

The following description of the Kratos Magnetostatic Application takes part of:

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

At a first glance, it could be said that Electrostatics works with static charges whereas Magnetostatics does it with dynamic charges (or static currents). That's partially true and we avoid by now to argue about it. On the opposite, we are going to consider Magnetostatics as a completely new phenomenon instead of thinking in a kind of 'dynamic' electrostatic.

As Electrostatics, Magnetostatics establishes very basic principles of the Maxwell equations, it is an excellent work at academic level, but it is also a powerful conceptual (and eventually the resulting computing tool) to solve important industrial applications (electrical machines such as motors and transformers, circuit components such as coils, permanent magnets, sensors, magnetic field distribution, etc).

Theory

Magnetostatics[1][2][3] refers to the physical phenomena related to the presence of stationary electric currents in the objects (magnetic materials can be considered as those with internal microcurrents). That means that we ignore any electrostatic charge, the electric field and we presume a constant magnetic field with respect to time.

Basic principles

Our third Maxwell's equation:

$\vec{\nabla}\cdot\vec{B} = 0$

where:

$\vec{B} = \mu \vec{H} = \mu_0\mu_r\vec{H}$

with: $\mu, \mu_0, \mu_r \,$ the (magnetic) permeability of the medium, of the free space and relative permeability, respectively.

The magnetic field is the curl of a magnetic vector potential ( $\vec{A} [Volt-seconds \; per \; metre]\,$ ):

$\vec{B}=\vec{\nabla} \times \vec{A}$

Our fourth Maxwell's equation:

$\vec{\nabla} \times \vec{H} = \vec{J}$

By combining these Maxwell's equations, we obtain:

$\vec{\nabla} \times \left ( \frac{1}{\mu} \vec{\nabla} \times \vec{A} \right ) - \vec{J} = 0$

Basic principles extended

Poisson's Equation in Magnetostatics

For 2D domains, we can reduce the Magnetostatic equation to the Poisson's Equation[8]. If there is no changes in the Z-direction and Z-component of the magnetic field, then $\frac{\partial}{\partial z} (·)=0$ and $\vec{A}=A_z \hat z$ and therefore:

$\vec{B}=\vec{\nabla} \times A_z \hat z = \frac{\partial}{\partial y} A_z \hat x - \frac{\partial}{\partial x} A_z \hat y$

$\vec{H}=\frac{1}{\mu} \vec{B} = \frac{1}{\mu_x} \frac{\partial}{\partial y} A_z \hat x - \frac{1}{\mu_y} \frac{\partial}{\partial x} A_z \hat y$

$-\frac{\partial}{\partial x}\cdot \left( \frac{1}{\mu_y} \frac{\partial A_z(x,y)}{\partial x}\right) - \frac{\partial}{\partial y}\cdot \left( \frac{1}{\mu_x} \frac{\partial A_z(x,y)}{\partial y}\right) - J_z(x,y)=0$

Poisson's Equation extended

Magnetostatic Boundary Conditions

For a 2D domain ${\Omega} \,$ , we should consider three different boundary conditions:

Dirichlet boundary condition:

$\left . A_z - \bar A_z = 0 \right |_{\Gamma_{A_z}}$

Neumann boundary condition:

$\left . \hat n \vec{H} - \bar H_n = 0 \right |_{\Gamma_{q}}$

Infinit condition (when no physical boundary are presents -free space-):

$\left . \frac{\partial A_z}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{A_z}{r}$

Boundary Conditions extended

References

The Finite Element Method for Magnetostatics

Using the Application

@@ Line 47: / Line 47: @@
 === Poisson's Equation in Magnetostatics ===
-For 2D domains, we can reduce the Magnetostatic equation to the [[Poisson's Equation]][9]. If there is no Z-component of the magnetic field, then <math>\vec{A}=A_z \hat(z)</math>
+For 2D domains, we can reduce the Magnetostatic equation to the [[Poisson's Equation]][8]. If there is no changes in the Z-direction and Z-component of the magnetic field, then &nbsp; <math>\frac{\partial}{\partial z} (·)=0</math> &nbsp; and &nbsp; <math>\vec{A}=A_z \hat z</math> &nbsp;  and therefore:
+::<math>\vec{B}=\vec{\nabla} \times A_z \hat z = \frac{\partial}{\partial y} A_z \hat x - \frac{\partial}{\partial x} A_z \hat y</math>
-The detailed form of the [[Poisson's Equation]][9] in Magnetostatics is:
+::<math>\vec{H}=\frac{1}{\mu} \vec{B} = \frac{1}{\mu_x} \frac{\partial}{\partial y} A_z \hat x - \frac{1}{\mu_y} \frac{\partial}{\partial x} A_z \hat y</math>
-::<math>\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)
+::<math>-\frac{\partial}{\partial x}\cdot \left( \frac{1}{\mu_y} \frac{\partial A_z(x,y)}{\partial x}\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</math>
+ - \frac{\partial}{\partial y}\cdot \left( \frac{1}{\mu_x} \frac{\partial A_z(x,y)}{\partial y}\right) - J_z(x,y)=0</math>
@@ Line 67: / Line 68: @@
-For a domain <math>{\Omega} \,</math>, we should consider three different boundary conditions:
+For a 2D domain <math>{\Omega} \,</math>, we should consider three different boundary conditions:
 * Dirichlet boundary condition:
-::<math>\left . V - \bar V = 0 \right |_{\Gamma_{V}}</math>
+::<math>\left . A_z - \bar A_z = 0 \right |_{\Gamma_{A_z}}</math>
 * Neumann boundary condition:
-::<math>\left . \hat n \vec{D} - \bar D_n = 0 \right |_{\Gamma_{q}} </math>
+::<math>\left . \hat n \vec{H} - \bar H_n = 0 \right |_{\Gamma_{q}} </math>
 * Infinit condition (when no physical boundary are presents -free space-):
-::<math>\left . \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{V}{r}</math>
+::<math>\left . \frac{\partial A_z}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{A_z}{r}</math>

Kratos Magnetostatic Application

Latest revision as of 11:12, 12 January 2011

Contents

General Description

Theory

Basic principles

Poisson's Equation in Magnetostatics

Magnetostatic Boundary Conditions

References

The Finite Element Method for Magnetostatics

Using the Application

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