# Kratos Magnetostatic Application

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 Revision as of 16:30, 1 February 2010 (view source)JMora (Talk | contribs)← Older edit Revision as of 18:39, 1 February 2010 (view source)JMora (Talk | contribs) Newer edit → Line 7: Line 7: * the use of Kratos for basic problems; * 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 [http://en.wikipedia.org/wiki/Electric_current static currents]). That's partially true and we avoid by now to + At a first glance, it could be said that Electrostatics works with static charges whereas Magnetostatics does it with dynamic charges (or [http://en.wikipedia.org/wiki/Electric_current 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. − , but we are going to consider Magnetostatics as a completely new phenomena instead of thinking in a kind of electrodynamics. + 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). − + − + − + − Magnetostatics + − + − 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 == == Theory == − Magnetostatics[1][2][3] refers to the physical phenomena related to the presence of electric charges in the objects. + 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 === === Basic principles === Line 25: Line 19: ::[[Image:Cargas.jpg]] ::[[Image:Cargas.jpg]] − Our first Maxwell's equation: + Our third Maxwell's equation: − :$\vec{\nabla}\cdot\vec{D} = \rho$ + :$\vec{\nabla}\cdot\vec{B} = 0$ where: where: − :$\vec{D} = \varepsilon\vec{E} = \varepsilon_0\varepsilon_r\vec{E}$ + :$\vec{B} = \mu \vec{H} = \mu_0\mu_r\vec{H}$ − + − 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 ($V [Volts]\,$): + with: $\mu, \mu_0, \mu_r \,$ the (magnetic) permeability of the medium, of the free space and relative permeability, respectively. − :$\vec{E}=-\vec{\nabla} V$ + The magnetic field is the curl of a magnetic vector potential ($\vec{A} [Volt-seconds \; per \; metre]\,$): − Our second Maxwell's equation: + :$\vec{B}=\vec{\nabla} \times \vec{A}$ − :$\vec{\nabla}\times\vec{E} = 0$ + Our fourth Maxwell's equation: + :$\vec{\nabla} \times \vec{H} = \vec{J}$ − By combining these Maxwell's equations, we can easily can obtain the Poisson's equation: − :$\vec{\nabla} \varepsilon \vec{\nabla} V + \rho_v = 0$ + By combining these Maxwell's equations, we obtain: + :$\vec{\nabla} \times \frac{1}{\mu} \vec{A} - \vec{J} = 0$ + that can be reduced to the Poisson's equation for two dimension domains. ::[[Magnetostatics Basic principles | '''''Basic principles extended''''']] ::[[Magnetostatics Basic principles | '''''Basic principles extended''''']] Line 92: Line 85: == References == == References == − # [http://www.answers.com/topic/electrostatics Sci-Tech Encyclopedia: Electrostatics] + # [http://www.answers.com/topic/magnetostatics Reference Answers: Magnetostatics] − # [http://en.wikipedia.org/wiki/Electrostatics wikipedia Electrostatics] + # [http://en.wikipedia.org/wiki/Magnetostatics wikipedia Magnetostatics] − # [http://en.wikipedia.org/wiki/Static_electricity wikipedia Static electricity] + # [http://en.wikipedia.org/wiki/Magnetism wikipedia Static Magnetism] − # [http://en.wikipedia.org/wiki/Electric_charge wikipedia Electric charge] + # [http://en.wikipedia.org/wiki/Electric_currents wikipedia Electric currents] − # [http://en.wikipedia.org/wiki/Electric_force Electric force] + # [http://en.wikipedia.org/wiki/Lorentz_force Magnetic force] − # [http://www.colorado.edu/physics/2000/waves_particles/wavpart2.html Electric force animation] + # [http://en.wikipedia.org/wiki/Magnetic_flux Magnetic flux] − # [http://en.wikipedia.org/wiki/Electric_flux Electric flux] + # [http://en.wikipedia.org/wiki/Magnetic_field Magnetic field] − # [http://en.wikipedia.org/wiki/Electric_displacement_field Electric displacement field] + # [http://en.wikipedia.org/wiki/Poisson%27s_equation Poisson's Equation] # [http://en.wikipedia.org/wiki/Poisson%27s_equation Poisson's Equation] − # [http://electron6.phys.utk.edu/phys594/Tools/e&m/summary/electrostatics/electrostatics.html Electrostatics Summary] == The Finite Element Method for Magnetostatics == == The Finite Element Method for Magnetostatics ==

## 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:

$\vec{\nabla} \times \frac{1}{\mu} \vec{A} - \vec{J} = 0$

that can be reduced to the Poisson's equation for two dimension domains.

Basic principles extended

### Poisson's Equation in Magnetostatics

The detailed form of the Poisson's Equation[9] in Magnetostatics 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

### 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