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authorNaoki Pross <np@0hm.ch>2021-10-22 16:31:09 +0200
committerNaoki Pross <np@0hm.ch>2021-10-22 16:31:09 +0200
commit35e49b9339f23bdc7149a02b06b27239b5c4db1e (patch)
treeea716dff3b29333fba96f4276abf1e2763e56b49
parentAdd PDF (diff)
downloadElMag-35e49b9339f23bdc7149a02b06b27239b5c4db1e.tar.gz
ElMag-35e49b9339f23bdc7149a02b06b27239b5c4db1e.zip
Add boundary problem equations (boundary conditions missing)HEADmaster
Diffstat (limited to '')
-rw-r--r--ElMag.tex44
-rw-r--r--build/ElMag.pdfbin93272 -> 97424 bytes
2 files changed, 42 insertions, 2 deletions
diff --git a/ElMag.tex b/ElMag.tex
index f628fea..646fbb0 100644
--- a/ElMag.tex
+++ b/ElMag.tex
@@ -98,7 +98,7 @@
\partial_{v_i} f(\vec{v})
% = f_{v_i}(\vec{v})
= \frac{\partial f}{\partial v_i}
- = \lim_{h\to 0} \frac{f(\vec{v} + h\vec{e}_i) - f(\vec{v})}{h}
+ = \lim_{h\to 0} \frac{f(\vec{v} + h\uvec{e}_i) - f(\vec{v})}{h}
\]
\end{definition}
@@ -126,7 +126,7 @@ If there is a relation \(x(y)\) or \(y(x)\) the above does not hold.
column vector containing the partial derivatives
in each direction.
\[
- \grad f (\vec{x}) = \sum_{i=1}^m \partial_{x_i} f(\vec{x}) \vec{e}_i
+ \grad f (\vec{x}) = \sum_{i=1}^m \partial_{x_i} f(\vec{x}) \uvec{e}_i
= \begin{pmatrix}
\partial_{x_1} f(\vec{x}) \\
\vdots \\
@@ -375,6 +375,46 @@ Laplacian operator. We will study equations with of form in \S \ref{sec:poisson}
% \subsection{Energy density}
+\section{Boundary value problems}
+
+\subsection{Steady-state flow analysis}
+The equatation for the steady-state analysis is
+\begin{equation}
+ \laplacian \varphi = 0 \quad\text{for}\quad \vec{r} \in \Omega,
+\end{equation}
+with its boundary conditions:
+\(
+ \varphi = 0 \text{ for } \vec{r}\in\Gamma_e,
+ \varphi = U \text{ for } \vec{r}\in\Gamma_b,
+ \nabla_\uvec{n} \varphi = 0 \text{ for } \vec{r}\in\Gamma_s.
+\)
+
+\subsection{Magnetostatic analysis}
+The equation for the magnetostatic analysis is
+\begin{equation}
+ \vlaplacian \vec{A} = -\mu_0 \vec{J} \quad\text{for}\quad \vec{r} \in \Omega.
+\end{equation}
+
+\subsection{Magnetoquasistatic analysis}
+The equation for the magnetoquasistatic analysis is
+\begin{equation}
+ \vlaplacian \vec{A} - \mu_0 \sigma \partial_t \vec{A}
+ = -\mu_0 \vec{J}_q \quad\text{for}\quad \vec{r} \in \Omega.
+\end{equation}
+
+\subsection{Electrodynamic analysis}
+The equationsfor the electrodynamic analysis are
+\begin{subequations}
+ \begin{gather}
+ \vlaplacian \vec{E}
+ - \mu \sigma \partial_t \vec{E}
+ - \mu \epsilon \partial^2_t \vec{E} = \vec{0}, \\
+ \vlaplacian \vec{H}
+ - \mu \sigma \partial_t \vec{H}
+ - \mu \epsilon \partial^2_t \vec{H} = \vec{0}.
+ \end{gather}
+\end{subequations}
+
\section{Laplace and Poisson's equations} \label{sec:poisson}
The so called \emph{Poisson's equation} has the form
diff --git a/build/ElMag.pdf b/build/ElMag.pdf
index ed2ec7c..4c5eb31 100644
--- a/build/ElMag.pdf
+++ b/build/ElMag.pdf
Binary files differ