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Include reference to torus knots

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paper/paper.tex
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@ -39,9 +39,7 @@
\begin{document}
% TODO
% Put picture on page 1
% Term "twisted"? "interleaved spirals"?
% Early pic / vis of spirals, somewhere in intro
% Put explanation of WPT to front of related work
% One plot instead of big table
% Move measeurements column to the left?
@ -69,6 +67,23 @@ Achieving Rotation-Invariant Coupling using Twisted Multi-Layer PCB Inductors}
\section{Introduction}
\begin{figure}
\begin{center}
\subcaptionbox{\raggedright A classic planar spiral inductor}{
\includegraphics[width=0.3\figurescale]{figures/svg_vis_paper_plain.png}}
\subcaptionbox{\raggedright A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{
\includegraphics[width=0.2\figurescale]{figures/saacke-radiotechnik-3-ledionspule.jpg}}
\subcaptionbox{\raggedright A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{
\includegraphics[width=0.2\figurescale]{figures/klein-spulen-schwingkreise-korbspule.jpg}}
\subcaptionbox{\raggedright Our proposed inductor layout}{
\includegraphics[width=0.3\figurescale]{figures/svg_vis_paper.png}}
\end{center}
\caption{Illustration of our proposed inductor layout compared to contemporary conventional planar inductors and
honeycomb as well as basket-woven coils from the early days of wireless radio.}
\textbf{TODO}: Not final graphics. Get proper scans for camera-ready version
\label{fig_illust_honeycomb_basket}
\end{figure}
Inductive Wireless Power Transfer (WPT) is a widely used technology supported by a large corpus of research literature
\cite{awuahNovelCoilDesign2023, batraEffectFerriteAddition2015, curranModelingCharacterizationPCB2015,
fanSimultaneousWirelessPower2024, leeSimpleWirelessPower2017, liWirelessPowerTransfer2015,
@ -123,12 +138,13 @@ gap between existing literature on low-power planar WPT inductor design and high
\subsection{Twisted inductors}
In this paper, 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. To fit our unique use case,
we applied a principle which the polygonal basket-woven air coils used in early radio sets are based on to an approach
inspired by contemporary planar inductor layouts. We show that we can layout a twisted inductor for any number of twists
that is co-prime to the inductor's turn count, and that in fact, our approach opens up a large design space for inductor
layouts that interpolate between planar spiral inductors on one end, and planar toroidal inductors on the other end. Our
approach thus generalizes a number of previous approaches to the design of planar inductors.
one another using a ring of vias each on the inside and outside of the inductor's windings. Our approach provides better
performance beyond our particular use case, and improves over conventional contemporary planar inductors applying
similar principles to those which inspired the polygonal basket-woven air coils used in early radio sets. We show that
we can layout a twisted inductor for any number of twists that is co-prime to the inductor's turn count, and that in
fact, our approach opens up a large design space for inductor layouts that interpolate between planar spiral inductors
on one end, and planar toroidal inductors on the other end. Our approach thus generalizes a super-set to a number of
previous approaches to the design of planar inductors.
We observe that in high-frequency applications, a moderate number of twists increases the spacing between the beginning
and end of the inductor's conductor, where the majority of the inductor's AC current flows. This decreases the parasitic
@ -185,22 +201,6 @@ surrounded by a minimum-size annular ring.} and increasing equivalent series res
sections that are necessary to accomodate the via structure, as well as the layer pairing limitations when blind vias
are used in multilayer PCBs.
\begin{figure}
\begin{center}
\subcaptionbox{\raggedright A classic planar spiral inductor}{
\includegraphics[width=0.3\figurescale]{figures/svg_vis_paper_plain.png}}
\subcaptionbox{\raggedright A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{
\includegraphics[width=0.2\figurescale]{figures/saacke-radiotechnik-3-ledionspule.jpg}}
\subcaptionbox{\raggedright A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{
\includegraphics[width=0.2\figurescale]{figures/klein-spulen-schwingkreise-korbspule.jpg}}
\subcaptionbox{\raggedright Our proposed inductor layout}{
\includegraphics[width=0.3\figurescale]{figures/svg_vis_paper.png}}
\end{center}
\caption{Illustrations of honeycomb and basket-woven coils from the early days of wireless radio.}
\textbf{TODO}: Not final graphics. Get proper scans for camera-ready version
\label{fig_illust_honeycomb_basket}
\end{figure}
This lack of a way to wind high frequency inductors with a machine led to the creation of a number of related winding
schemes that include honeycomb and basket woven coils
\cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
@ -426,14 +426,16 @@ will call all layouts with $k\ge 2$ \emph{Twisted Inductors}.
\label{fig_nk_combined}
\end{figure}
Figure\ \ref{fig_nk_combined} illustrates how we arrive at the coprimality requirement. \todo{Cleanly handle $k=0$
case.} If we plot the spiral in polar coordinates on a cartesian plot we observe that for a $n$-turn coil with $k$
inversions, the trace crosses the $\varphi$ axis once for each inversion, wrapping around $r$. Likewise, it crosses the
$r$ axis once for each turn of the inductor, wrapping around $\varphi$. Based on this, we can re-label the angular axis
in steps from $0$ to $k$, and re-label the radial axis in steps from $0$ to $n$. Labelling the new angular axis $i$ and
the new radial axis $j$, in the resulting integer lattice, the trace has slope $1$. We can state the trace's trajectory
as a function of a curve parameter $t \in [0, nk]$ as $f(t) = (i, j) = (t \mod n, t \mod k)$. To produce a valid
inductor, the trace must not intersect anywhere. Thus, the system of congruences
Topologically, the shape of our inductors can be described as a $(k, n)$-torus knot. From knot theory, we know that such
a torus knot exists if and only if both $n$ and $k$ are co-prime. Figure\ \ref{fig_nk_combined} illustrates a derivation
of the coprimality requirement. \todo{Cleanly handle $k=0$ case.} If we plot the spiral in polar coordinates on a
cartesian plot we observe that for a $n$-turn coil with $k$ inversions, the trace crosses the $\varphi$ axis once for
each inversion, wrapping around $r$. Likewise, it crosses the $r$ axis once for each turn of the inductor, wrapping
around $\varphi$. Based on this, we can re-label the angular axis in steps from $0$ to $k$, and re-label the radial axis
in steps from $0$ to $n$. Labelling the new angular axis $i$ and the new radial axis $j$, in the resulting integer
lattice, the trace has slope $1$. We can state the trace's trajectory as a function of a curve parameter $t \in [0, nk]$
as $f(t) = (i, j) = (t \mod n, t \mod k)$. To produce a valid inductor, the trace must not intersect anywhere. Thus, the
system of congruences
\begin{align}
t &\equiv i \mod n\\