From ed9e1d3e534384a4022cd6ab417898bf5f2caff4 Mon Sep 17 00:00:00 2001
From: jaseg <git@jaseg.de>
Date: Fri, 4 Oct 2024 17:30:12 +0200
Subject: [PATCH] Paper WIP

---
 .../rms_ripple_double_rotation_n10_r4.pdf     | Bin 57345 -> 57345 bytes
 .../rms_ripple_double_rotation_n5_r4.pdf      | Bin 76204 -> 68014 bytes
 paper/paper.tex                               | 161 +++++++++++++-----
 3 files changed, 115 insertions(+), 46 deletions(-)

diff --git a/paper/figures/rms_ripple_double_rotation_n10_r4.pdf b/paper/figures/rms_ripple_double_rotation_n10_r4.pdf
index b473ce5b46b1a9fd33de4b7cc119518477dc0c29..ec7f898417c116de8ebfec6975f0f586c8d28739 100644
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diff --git a/paper/figures/rms_ripple_double_rotation_n5_r4.pdf b/paper/figures/rms_ripple_double_rotation_n5_r4.pdf
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diff --git a/paper/paper.tex b/paper/paper.tex
index 8b8b3d5..13e9318 100644
--- a/paper/paper.tex
+++ b/paper/paper.tex
@@ -87,7 +87,7 @@ in the placement of heavy components will quickly cause a strong vibration.
 \subsection{Twisted inductors}
 
 Applying a principle inspired by rectangular or octagonal RFIC inductor design as well as by the polygonal basket-woven
-air coils used in early radio set, we propose a novel way of laying out circular PCB inductors that twists the
+air coils used in early radio sets, we propose a novel way of laying out circular PCB inductors that twists the
 inductor's windings around one another using a ring of vias each on the inside and outside of the inductor's windings.
 Applying some math, we show that we can layout a twisted inductor for any number of twists that is co-prime to the
 inductor's turn count.
@@ -227,6 +227,21 @@ values of $n$ and $k$.
 
 \subsubsection{Ohmic Resistance}
 
+The arc length $l$ of a spiral can be calculated from its turn count $n$ and the average of its inner and outer diameters
+$\frac{r_1 + r_2}{2}$ as $l = n\pi\frac{r_1 + r_2}{2}$. Since going from a standard inductor to a twisted inductor does
+not change its turn count or dimensions, the combined arc length of all trace pairs of the twisted inductor does not
+change. Twisted inductors require two additional vias per trace pair, which will increase DC resistance slightly, but
+the contribution of these vias will remain small in practical applications since the overall number of vias is still no
+more than a couple per turn, and since each via only bridges the short distance between the inductor's layers.
+
+As a general expression, for a standard or twisted inductor with turn count $n$ and twist count $k$ ($k=0$ for a
+single-layer spiral inductor, and $k=1$ for a standard two-layer spiral inductor), given via resistance $R_\text{via}$
+we derive a first order approximation of the inductor's DC resistance as follows.
+
+\begin{equation}
+    R_L = n\pi\frac{r_1 + r_2}{2} + \left(2k-1\right)R_\text{via}
+\end{equation}
+
 \subsubsection{Inductance}
 
 \subsection{CAD Integration}
@@ -262,7 +277,8 @@ Determining parasitic capacitance is more complex.
 
 To experimentally validate our design with real-world inductors, we produced test coupons with a number of variations of
 twisted inductors with winding count $n$ between $1$ and $25$, and twist count ranging from $k=0$ (simple single-sided
-spiral inductor) to $k=37$.
+spiral inductor) to $k=37$. All test inductors had an inner diameter of \qty{15}{\milli\meter} and an outer diameter of
+\qty{35}{\milli\meter}.
 
 \subsection{Inductance, Q-factor and DC resistance}
 
@@ -321,7 +337,7 @@ performance parameters.
     \qty{35}{\milli\meter} and inner diameter \qty{15}{\milli\meter}.}
 \end{table*}
 
-\subsection{Coupling and Coupling Variation}
+\subsection{Coupling and its Sensitivity to Radial Offset}
 
 The key performance criterion in our application is the voltage ripple that appears on the secondary side of a WPT link
 when one of the inductors is rotating. To experimentally evaluate the magnitude of this ripple in a realistic scenario
@@ -329,31 +345,24 @@ across a large set of rotations and relative displacements, we created a test se
 an old 3D printer, with a fourth rotation axis provided by a small servo that allows us to position two inductor test
 coupons at arbitrary offsets and angles to one another while measuring their coupling.
 
-To evaluate a realistic scenario, we loaded the secondary inductor with a resistive load of \qty{10}{\ohm}, while
-providing a signal at a \qty{300}{\kilo\hertz} carrier frequency to the primary inductor from a Siglent SDG6022X
-function generator. We measured both the input and output voltages of the coupled inductor pair using Keysight 34465A
-multimeters in AC RMS mode. The results of these measurements, with the voltage ratio between the coupled inductors'
-input and output voltages graphed across one revolution in Figure\ \ref{symmetry_3turn_n_twist} for a set of three-turn
-inductors and in Figure\ \ref{symmetry_10turn_n_twist} for a set of 10-turn inductors  with multiple trace pair amounts
-$k$.
-
-From these graphs we observe slightly lower coupling for $k>0$ compared to a single-layer spiral inductor, which is
-in line with our previous inductance measurements. Across one revolution, we find that single-layer spiral inductors
-exhibit the worst voltage ripple, with simple two-layer inductors with $k=1$ already improving ripple by a large margin.
-Increasing $k$ above $1$ does not decrease the amplitude of this ripple further, but it does shift the ripple into
-higher frequencies that are easier to passively filter, as we originally intended.
-
-\todo{new ripple measurements, concrete coupling factor measurements}
-\todo{schematics for illustration of measeurement circuits}
-
 \begin{figure}
     \begin{center}
-        %\includegraphics[width=0.7\linewidth]{figures/symmetry_3turn_n_twist.pdf}
+        \includegraphics[width=.85\linewidth]{figures/test_schematic.pdf}
     \end{center}
-    \caption{Coupling test circuit}
-    \label{symmetry_test_circuit}
+    \caption{The test schematic used in all measurements. For direct coupling factor measurements, the load resistor was
+    disconnected. We measure voltage at the output of the function generator to account for drop in its internal output
+    resistance.}
+    \label{fig_test_schematic}
 \end{figure}
 
+To evaluate a realistic scenario, we loaded the secondary inductor with a resistive load of \qty{10}{\ohm}, while
+providing a signal at a \qty{300}{\kilo\hertz} carrier frequency to the primary inductor from a Siglent SDG6022X
+function generator as shown in Figure\ \ref{fig_test_schematic}. We measured both the input and output voltages
+of the coupled inductor pair using Keysight 34465A multimeters in AC RMS mode. The results of these measurements, with
+the voltage ratio between the coupled inductors' input and output voltages graphed across one revolution in Figure\
+\ref{fig_symmetry_3turn_n_twist} for a set of three-turn inductors with multiple trace pair amounts $k$. A plot for a
+set of 10-turn inductors is shown in Figure\ \ref{fig_symmetry_10turn_n_twist} in the Appendix.
+
 \begin{figure}
     \begin{center}
         \includegraphics[width=\linewidth]{figures/symmetry_3turn_n_twist.pdf}
@@ -362,22 +371,37 @@ higher frequencies that are easier to passively filter, as we originally intende
     inductors with one inductor rotating w.r.t.\ the other. The inductors have $n=3$ turns each and $k=0$, $k=1$, and
     $k=3$, respectively. For each $k$, voltage curves are plotted for a number of different radial offsets
     between the two inductor's centers.}
-    \label{symmetry_3turn_n_twist}
+    \label{fig_symmetry_3turn_n_twist}
 \end{figure}
 
+From these graphs we observe slightly lower coupling for $k>0$ compared to a single-layer spiral inductor, which is
+in line with our previous inductance measurements. Across one revolution, we find that single-layer spiral inductors
+exhibit the worst voltage ripple, with simple two-layer inductors with $k=1$ already improving ripple by a large margin.
+Increasing $k$ above $1$ does not decrease the amplitude of this ripple further, but it does shift the ripple into
+higher frequencies that are easier to passively filter, as we originally intended.
+
+\subsection{Total Coupling Variation}
+
+To further analyze the behavior of our test inductors under offset and rotation, we had our measurement setup sweep
+through the full range of rotation of each of the two inductors when placed at a fixed height of \qty{1}{\milli\meter}
+and radial offset of \qty{4}{\milli\meter}. The resulting plots show the variation in RMS output voltage compared to its
+mean across all rotations as a percentage plotted against both angular dimensions. Figure\ \ref{fig_rms_ripple_n3} shows
+the resulting coupling plot for a set of three-turn inductors, and Figure\ \ref{fig_rms_ripple_n5} for a set of
+five-turn inductors. Measurements for 10- and for 25-turn inductors are shown in Figures \ref{fig_rms_ripple_n10} and
+\ref{fig_rms_ripple_n25} in the Appendix.
+
+From these plots, we can draw a number of conclusions. First, our primary objective of reducing coupling variation
+across rotations works, with twisted inductors ($k>1$) showing a further improvement over simple two-layer inductors,
+which prove to be better than simple single-layer spiral inductors. As one would expect, this gain is greatest for
+inductors with low turn count, as their turns deviate the furthest from a set of ideal, concentric circles. For the
+our test inductor with an inner diameter of \qty{15}{\milli\meter} and an outer diameter of \qty{35}{\milli\meter},
+$k=3$ trace pairs already provided an improvement over standard configurations.
+
+\todo{concrete coupling factor measurements}
+
 \begin{figure}
     \begin{center}
-        \includegraphics[width=\linewidth]{figures/symmetry_10turn_n_twist.pdf}
-    \end{center}
-    \caption{Coupled RMS output voltage of three pairs of matching inductors with $n=10$ turns each and $k=0$, $k=1$,
-    and $k=3$, respectively, shown as in Figure\ \ref{symmetry_3turn_n_twist}}
-    \label{symmetry_10turn_n_twist}
-\end{figure}
-
-
-\begin{figure}
-    \begin{center}
-        \includegraphics[width=\linewidth]{figures/field_plot_3d_n3_k4.pdf}
+        \includegraphics[width=.6\linewidth]{figures/field_plot_3d_n3_k4.pdf}
     \end{center}
     \caption{The coupling between a pair of identical coils (here with $n=3$ and $k=4$) visualized in three dimensions.
     The $x$ and $y$ axis show in-plane displacement, and the $z$ axis shows output amplitude in arbitrary units. Height
@@ -391,25 +415,43 @@ higher frequencies that are easier to passively filter, as we originally intende
 
 \begin{figure}
     \begin{center}
-        \includegraphics[width=\linewidth]{figures/test_schematic.pdf}
+        \includegraphics[width=.75\linewidth]{figures/rms_ripple_double_rotation_n3_r4.pdf}
     \end{center}
-    \caption{The test schematic used in all measurements. For direct coupling factor measurements, the load resistor was
-    disconnected. We measure voltage at the output of the function generator to account for drop in its internal output
-    resistance.}
-    \label{fig_test_schematic}
+    \caption{RMS ripple magnitude as a percentage of mean RMS output voltage, plotted against the rotation of each of
+    the two inductors. The two coils were kept at a constant \qty{4}{\milli\meter} radial offset, and the output coil
+    was loaded with a \qty{10}{\ohm} load. All RMS ripple plots in this paper share the same color scale to allow for
+    visual comparison. This figure shows four variants of 3-turn coils, plots for $n=5$ can be found in Figure\
+    \ref{fig_rms_ripple_n5} and plots for $n=\{10,25\}$ in Figures \ref{fig_rms_ripple_n10} and \ref{fig_rms_ripple_n25}
+    in the Appendix.}
+    \label{fig_rms_ripple_n3}
+\end{figure}
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=.75\linewidth]{figures/rms_ripple_double_rotation_n5_r4.pdf}
+    \end{center}
+    \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 5-turn coils.}
+    \label{fig_rms_ripple_n5}
 \end{figure}
-% rms_ripple_double_rotation_n25_r4.pdf
-% rms_ripple_double_rotation_n5_r4.pdf
-% rms_ripple_double_rotation_n3_r4.pdf
 
 \section{Conclusion}
 
+In this paper, we introduced a novel layout approach for planar, multi-layer inductors inspired by classic basket-wound
+inductors used in the early days of radio. Our \emph{twisted} inductors produce field distributions that have better
+rotational symmetry along the inductor's main axis compared to either simple single-layer spiral inductors or
+counter-wound two-layer spiral inductors. Furthermore, we found that our sample twisted inductors have slightly higher
+self-resonant frequency compared to both traditional layouts. We base this evaluation on laboratory measurements on a
+set of 24 test inductors, which include an automated, four-dimensional mapping of the coupling between a pair of
+identical inductors. We provide both an analytical description of twisted inductor construction as well as a set of
+Open-Source tools for their design.
+
 \section*{Availability}
 This is version \texttt{\input{version.tex}\unskip} of this paper, generated on \today.
 
-% The git repository with the
-% LaTeX source for this paper as well as our data analysis and demo code can be found at:
+The git repository with the LaTeX source for this paper, the data analysis code underlying our measurements as well the
+set of tools for the generation of twisted inductor layouts that we wrote can be found at:
 
+\todo{link here}
 % \center{\url{https://git.jaseg.de/nice-coils.git}}
 
 \printbibliography[heading=bibintoc]
@@ -421,11 +463,38 @@ This is version \texttt{\input{version.tex}\unskip} of this paper, generated on
 
 \begin{figure*}
     \begin{center}
-        \includegraphics[width=\textwidth]{figures/nk_complex_illust.pdf}
+        \includegraphics[width=.75\textwidth]{figures/nk_complex_illust.pdf}
     \end{center}
     \caption{Layout examples for a number of combinations of turn count $n$ and trace pair count $k$. Note that in this
     illustration we chose values for $n$ and $k$ such that all pairs are coprime.}
     \label{fig_nk_complex_illust}
 \end{figure*}
 
+\section{Supplemental plots}
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=\linewidth]{figures/symmetry_10turn_n_twist.pdf}
+    \end{center}
+    \caption{Coupled RMS output voltage of three pairs of matching inductors with $n=10$ turns each and $k=0$, $k=1$,
+    and $k=3$, respectively, shown as in Figure\ \ref{symmetry_3turn_n_twist}}
+    \label{fig_symmetry_10turn_n_twist}
+\end{figure}
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=.75\linewidth]{figures/rms_ripple_double_rotation_n10_r4.pdf}
+    \end{center}
+    \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 10-turn coils.}
+    \label{fig_rms_ripple_n10}
+\end{figure}
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=.75\linewidth]{figures/rms_ripple_double_rotation_n25_r4.pdf}
+    \end{center}
+    \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 25-turn coils.}
+    \label{fig_rms_ripple_n25}
+\end{figure}
+
 \end{document}