From 59d1f34d565d2f5048f8ced534db50646b1051bd Mon Sep 17 00:00:00 2001 From: jaseg Date: Thu, 5 Dec 2024 18:01:30 +0100 Subject: [PATCH] Include reference to torus knots --- paper/paper.tex | 66 +++++++++++++++++++++++++------------------------ 1 file changed, 34 insertions(+), 32 deletions(-) diff --git a/paper/paper.tex b/paper/paper.tex index c40c0a4..166804d 100644 --- a/paper/paper.tex +++ b/paper/paper.tex @@ -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\\