Kratos Magnetostatic Application

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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[9]. If there is no changes in the Z-direction and Z-component of the magnetic field, then &nbsp $\frac{\partial}{\partial z} (·)=0$ &nbsp and &nbsp $\vec{A}=A_z \hat z$ &nbsp and therefore:

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

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

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