From 96e8bad4eb9a573a33ad917dbf7c897289e5c386 Mon Sep 17 00:00:00 2001
From: jaseg <git@jaseg.de>
Date: Fri, 13 Sep 2024 17:52:20 +0200
Subject: [PATCH] sim WIP

---
 paper/paper.tex | 23 ++++++++++++++++++++---
 1 file changed, 20 insertions(+), 3 deletions(-)

diff --git a/paper/paper.tex b/paper/paper.tex
index 71257d0..7f29cf7 100644
--- a/paper/paper.tex
+++ b/paper/paper.tex
@@ -179,11 +179,28 @@ values of $n$ are possible, which will rotate the second port around the coordin
 
 \section{FEM Simulation}
 
-\subsection{Ohmic Resistance}
+To validate our analytical approximations, we performed a series of FEM simulations in both Elmer FEM and Simulia CST.
+For a number of inductor layouts, we performed simulations to determine ohmic resistance, inductance, and parasitic
+capacitance. For a subset of these layout variants we additionally performed simulations to determine the coupling
+factor between a pair of identical inductors at a number of different distances and rotations.
 
-\subsection{Inductance}
+\paragraph{Ohmic Resistance}
+Determining ohmic resistance by FEM is reasonably easy. In Elmer FEM, we can use the built-in joint static current and
+joule heating solver to determine the ohmic resistance at a given current.
 
-\subsection{Parasitic Capacitance and Self-Resonant Frequency}
+\paragraph{Inductance}
+We let Elmer determine inductance by first using its coil solver to determine the volumetric current density in our mesh
+given a test current, then applying its magnetodynamics solver to solve the electromagnetic field. Elmer provides
+routines to derive the total magnetic field energy $U_\text{mag}$ from an EM field solution. Since we have only our
+inductor under test inside the simulation volume, with test current $I_\text{test}$, we can then derive the inductor's
+inductance according to the well-known relation\todo{Find decent source}:
+
+\begin{equation}
+    L = \frac{2\cdot U_\text{mag}}{I_\text{test}^2}
+\end{equation}
+
+\paragraph{Parasitic Capacitance and Self-Resonant Frequency}
+Determining parasitic capacitance is more complex.
 
 \subsection{Coupling}