Kratos Magnetostatic Application

(Difference between revisions)
 Revision as of 18:59, 1 February 2010 (view source)JMora (Talk | contribs) (→Poisson's Equation in Magnetostatics)← Older edit Revision as of 19:03, 1 February 2010 (view source)JMora (Talk | contribs) (→Poisson's Equation in Magnetostatics)Newer edit → Line 56: Line 56: − ::$\frac{\partial}{\partial x}\cdot \left( \varepsilon_{x} \cdot \frac{\partial V(x,y,z)}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(\varepsilon_{y} \cdot \frac{\partial V(x,y,z)}{\partial y} \right) + ::[itex]\frac{\partial}{\partial x}\cdot \left( \frac{1}{\mu_y} \frac{\partial A_z(x,y)}{\partial x}\right) − + \frac{\partial}{\partial z}\cdot \left(\varepsilon_{z} \cdot \frac{\partial V(x,y,z)}{\partial z} \right) + \rho_v(x,y,z)=0$ + + \frac{\partial}{\partial y}\cdot \left( \frac{1}{\mu_x} \frac{\partial A_z(x,y)}{\partial y}\right) \right) - J_z(x,y)=0[/itex]

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:

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[9]. 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:

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

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

Failed to parse (syntax error): \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) \right) - J_z(x,y)=0

Poisson's Equation extended

Magnetostatic 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