Difference between revisions of "Lecture 3."
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According to <math>\nabla\cdot\vec{B}(\vec{r},t)=0</math> the magnetic flux density is conservative and therefore can be described by the '''curl''' of a vector | According to <math>\nabla\cdot\vec{B}(\vec{r},t)=0</math> the magnetic flux density is conservative and therefore can be described by the '''curl''' of a vector | ||
| − | <math> \vec{B} = \nabla\times\vec{A}</math>, | + | ::<math> \vec{B} = \nabla\times\vec{A}</math>, |
where <math>\vec{A}</math> is the magnetic vector potential [Wb/m]. Substituting this expression into Faraday's law results in | where <math>\vec{A}</math> is the magnetic vector potential [Wb/m]. Substituting this expression into Faraday's law results in | ||
| − | <math>\nabla\times\vec{E}=-\frac{\partial}{\partial t} \left(\nabla\times\vec{A}\right)=-\nabla\times\left(\frac{\partial\vec{A}}{\partial t}\right) \to \nabla\times\left(\vec{E}+\frac{\partial\vec{A}}{\partial t}\right)=\vec{0}</math>, | + | ::<math>\nabla\times\vec{E}=-\frac{\partial}{\partial t} \left(\nabla\times\vec{A}\right)=-\nabla\times\left(\frac{\partial\vec{A}}{\partial t}\right) \to \nabla\times\left(\vec{E}+\frac{\partial\vec{A}}{\partial t}\right)=\vec{0}</math>, |
because rotation (i.e. derivative by space) and derivation by time can be replaced. The curl-less vector field <math>\vec{E}+\partial\vec{A}/\partial t</math> can be derived from the so-called electric scalar potential <math>V</math> (<math>\nabla\times\nabla\varphi\equiv0</math>, for any scalar function <math>\varphi=\varphi(\vec{r})</math>, or <math>\varphi=\varphi(\vec{r},t)</math>), | because rotation (i.e. derivative by space) and derivation by time can be replaced. The curl-less vector field <math>\vec{E}+\partial\vec{A}/\partial t</math> can be derived from the so-called electric scalar potential <math>V</math> (<math>\nabla\times\nabla\varphi\equiv0</math>, for any scalar function <math>\varphi=\varphi(\vec{r})</math>, or <math>\varphi=\varphi(\vec{r},t)</math>), | ||
| − | <math>\vec{E}+\frac{\partial\vec{A}}{\partial t}=-\nabla V</math>, | + | ::<math>\vec{E}+\frac{\partial\vec{A}}{\partial t}=-\nabla V</math>, |
and the <math>\vec{E}</math> electric field intensity vector can be described by two potentials as | and the <math>\vec{E}</math> electric field intensity vector can be described by two potentials as | ||
| − | <math>\vec{E}=-\frac{\partial\vec{A}}{\partial t}-\nabla V</math>. | + | ::<math>\vec{E}=-\frac{\partial\vec{A}}{\partial t}-\nabla V</math>. |
Substituting expression of <math>\vec{B}</math> and <math>\vec{E}</math> into Faraday's law leads to the partial differential equation | Substituting expression of <math>\vec{B}</math> and <math>\vec{E}</math> into Faraday's law leads to the partial differential equation | ||
| − | <math>\nabla\times\left(\frac{1}{\mu}\nabla\times\vec{A}\right)=\vec{J}_{S}-\frac{\partial\vec{A}}{\partial t}-\nabla V</math>. | + | ::<math>\nabla\times\left(\frac{1}{\mu}\nabla\times\vec{A}\right)=\vec{J}_{S}-\frac{\partial\vec{A}}{\partial t}-\nabla V</math>. |
</blockquote> | </blockquote> | ||
== Irodalom == | == Irodalom == | ||
{{reflist}} | {{reflist}} | ||
Revision as of 08:27, 13 March 2019
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Coupled Finite Element Method / Time-Dependent Magnetic Field | |
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Instructor
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Teaching Assistants:
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Contents
Csatolt végeselem-módszer (FEM)
Quasistatic Electromagnetic Field
The most important case for electromagnetic equipment (sensors, actuators, motors, etc.) is the quasistatic case often referred to as the eddy current or magnetodynamic case. For quasistatic electromagnetic field we can neglect the displacement current density term [math]\partial \vec{D}/\partial t[/math], which gives Maxwell's equations to the following form
[math]\nabla\times\vec{H}=\vec{J}[/math]
Ampere's law,
[math]\nabla\times\vec{E}=-\frac{\partial \vec{B}}{\partial t}[/math]
Faraday's law,
[math]\nabla\cdot\vec{B}(\vec{r},t)=0[/math]
Gauss's law (magnetic).
Magnetic Vector Potential and Electric Scalar Potential
According to [math]\nabla\cdot\vec{B}(\vec{r},t)=0[/math] the magnetic flux density is conservative and therefore can be described by the curl of a vector
- [math] \vec{B} = \nabla\times\vec{A}[/math],
where [math]\vec{A}[/math] is the magnetic vector potential [Wb/m]. Substituting this expression into Faraday's law results in
- [math]\nabla\times\vec{E}=-\frac{\partial}{\partial t} \left(\nabla\times\vec{A}\right)=-\nabla\times\left(\frac{\partial\vec{A}}{\partial t}\right) \to \nabla\times\left(\vec{E}+\frac{\partial\vec{A}}{\partial t}\right)=\vec{0}[/math],
because rotation (i.e. derivative by space) and derivation by time can be replaced. The curl-less vector field [math]\vec{E}+\partial\vec{A}/\partial t[/math] can be derived from the so-called electric scalar potential [math]V[/math] ([math]\nabla\times\nabla\varphi\equiv0[/math], for any scalar function [math]\varphi=\varphi(\vec{r})[/math], or [math]\varphi=\varphi(\vec{r},t)[/math]),
- [math]\vec{E}+\frac{\partial\vec{A}}{\partial t}=-\nabla V[/math],
and the [math]\vec{E}[/math] electric field intensity vector can be described by two potentials as
- [math]\vec{E}=-\frac{\partial\vec{A}}{\partial t}-\nabla V[/math].
Substituting expression of [math]\vec{B}[/math] and [math]\vec{E}[/math] into Faraday's law leads to the partial differential equation
- [math]\nabla\times\left(\frac{1}{\mu}\nabla\times\vec{A}\right)=\vec{J}_{S}-\frac{\partial\vec{A}}{\partial t}-\nabla V[/math].
