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paper/Makefile
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@ -36,7 +36,7 @@ submission.zip:
figures/setup_probe_small.jpg
version.tex: ${main_tex}.tex paper.bib
echo "${VERSION_STRING}" > $@
echo -n "${VERSION_STRING}" > $@
.PHONY: clean
clean:

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paper/paper.tex
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@ -1,4 +1,4 @@
\documentclass[journal,12pt,onecolumn,draftclsnofoot]{IEEEtran}
\documentclass[journal,10pt,a4paper]{IEEEtran}
\usepackage[T1]{fontenc}
\usepackage[
@ -32,9 +32,8 @@
\newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}}
\newcommand{\partnum}[1]{\texttt{#1}}
\newcommand{\todo}[1]{\textbf{TODO}\footnote{#1}}
% Set to 1.0 for final two-column export
\newlength{\figurescale}
\setlength{\figurescale}{0.75\textwidth}
\setlength{\figurescale}{\linewidth}
\begin{document}
@ -43,7 +42,10 @@
University of Darmstadt, 64283 Darmstadt, Germany (e-mail: jan.goette@tu-darmstadt.de).}
\and
\IEEEauthorblockN{Björn Scheuermann}\thanks{Björn Scheuermann is with the Technical University of Darmstadt,
64283 Darmstadt, Germany (e-mail: scheuermann@kom.tu-darmstadt.de).}}
64283 Darmstadt, Germany (e-mail: scheuermann@kom.tu-darmstadt.de).}
\thanks{This work has been funded by the LOEWE initiative (Hesse, Germany) within the emergenCITY center
(LOEWE/1/12/519/03/05.001(0016)/72) as well as by Technical University of Darmstadt.}
}
\title{Wireless Power Transfer with a Twist:
Achieving Rotation-Invariant Coupling using Twisted Multi-Layer PCB Inductors}
\maketitle
@ -61,24 +63,6 @@ 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{Note}: Not final graphics. Higher-quality scans of the middle two graphics will be submitted with the final
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,
@ -125,6 +109,22 @@ the often higher turn count and the tightly packed, circular wires render this e
ripple caused by this oscillation can be filtered through a voltage regulator or by using a large decoupling capacitor
on the secondary side if the application can accomodate such components on the rotating part.
\begin{figure}
\begin{center}
\subcaptionbox{\raggedright A classic planar spiral inductor}{
\includegraphics[width=0.25\figurescale]{figures/svg_vis_paper_plain.png}}
\subcaptionbox{\raggedright Our proposed inductor layout}{
\includegraphics[width=0.25\figurescale]{figures/svg_vis_paper.png}}
\subcaptionbox{\raggedright A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{
\includegraphics[width=0.15\figurescale]{figures/saacke-radiotechnik-3-ledionspule.jpg}}
\subcaptionbox{\raggedright A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{
\includegraphics[width=0.15\figurescale]{figures/klein-spulen-schwingkreise-korbspule.jpg}}
\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.}
\label{fig_illust_honeycomb_basket}
\end{figure}
While there exist a corpus of prior work focusing on efficient power transfer between two coils whose position relative
to one another cannot be precisely controlled as is the case in wireless phone charging systems as well as in proposed
WPT electric vehicle chargers~\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017},
@ -174,7 +174,7 @@ Our contributions in this paper include:
\item We provide detailed instructions for the construction of such layouts, including a mathematical analysis of
the available parameter space.
\item We provide an analytical model of inductance and DC equivalent series resistance of our scheme.
\item We validate our scheme, we provide laboratory measurements of the basic parameters of 39 test specimens
\item We validate our scheme and provide laboratory measurements of the basic parameters of 39 test specimens
comparing our scheme to conventional layouts.
\item We further present the results of Finite Element Method (FEM) simulations to validate our inductance and ESR
approximations.
@ -330,7 +330,7 @@ layer of such windings forms a helix whose pitch is equal to the wire diameter.
helical scheme reversing at the coil ends, but uses a helical pitch larger than the wire diameter to form a structure
similar to a spool of sewing thread.
Other winding techniques include honeycomb and basket woven coils, some historic examples of which are shown in Figure\
Other winding techniques include honeycomb and basket woven coils, some historic examples of which are shown in Fig.\
\ref{fig_illust_honeycomb_basket}. In a honeycomb coil, like in an universal winding, subsequent winding layers are
wound at a criss-cross pattern. The characteristic feature of honeycomb coils is that the winding machine is adjusted to
produce large air gaps between adjacent windings, resulting in a three-dimensional rhomboid pattern that is vaguely
@ -374,7 +374,7 @@ To improve layer utilization, a common technique in PCB inductor design is to us
inductor's spiral trace, instead of only using the bottom layer for a straight jumper trace. Using both layers this way
allows for wider traces, which lowers resistive losses. We can accomodate this optimization in our definition by
re-defining our normalized radius to allow both positive and negative values, defining negative values to designate
traces on the PCB's bottom layer as follows. Figure\ \ref{fig_nk_combined} shows both a simple and a two-layer
traces on the PCB's bottom layer as follows. Fig.\ \ref{fig_nk_combined} shows both a simple and a two-layer
spiral inductor in the first two columns.
Let $n$ be the turn count of our inductor. The resulting parametrization is:
@ -398,7 +398,7 @@ two core observations:
\begin{description}[\IEEEsetlabelwidth{foo}]
\item[Observation 1.]\hfill\\When using an archimedean spiral, multiple such spirals using the same pitch can be
interleaved by spreading out their start and end points at regular angular intervals.
\item[Observation 2.]\hfill\\In a two-layer spiral inductor (Figure\ \ref{fig_nk_combined}), we can adjust the turn
\item[Observation 2.]\hfill\\In a two-layer spiral inductor (Fig.\ \ref{fig_nk_combined}), we can adjust the turn
count of the pair of traces to move the end point of the bottom layer trace anywhere on the inductor's outer
radius.
\end{description}
@ -408,10 +408,10 @@ scheme~\cite{lopeFirstSelfresonantFrequency2021,sproHighVoltageInsulationDesign2
Combining these two observations, we find that by choosing a number $k$ of inversions, i.e. layer jumps, that is coprime
to the number of total turns of the inductor $n$, we achieve a layout where all $k$ pairs of top and bottom-layer traces
naturally connect in series, with the resulting spirals on the top and bottom layers interleaving cleanly. Figure\
naturally connect in series, with the resulting spirals on the top and bottom layers interleaving cleanly. Fig.\
\ref{fig_nk_combined} shows a layout with $n=3$ turns with both a single inversion ($k=1$), which results in a
conventional two-layer inductor, and with $k=2$ inversions, creating two interleaved spirals on both the top and the
bottom layer of the PCB. Figure\ \ref{fig_nk_complex_illust} in Appendix\ \ref{sec_appendix_layout_examples} shows
bottom layer of the PCB. Fig.\ \ref{fig_nk_complex_illust} in the Appendix of this paper shows
additional layout examples for other values of $n$ and $k$. For $k=\frac{1}{2}$, we get a standard single-layer planar
spiral inductor for any turn count $n$, and for $k=1$ we get a standard two-layer planar spiral inductor for any turn
count $n$. In this paper, we will call all layouts with $k\ge 2$ \emph{Twisted Inductors}. The coordinate description of
@ -446,7 +446,7 @@ Equation\ \ref{eqn_twolayer_spiral} thus becomes:
\end{figure}
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
a torus knot exists if and only if both $n$ and $k$ are co-prime. Fig.\ \ref{fig_nk_combined} illustrates a derivation
of the coprimality requirement. 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
@ -558,10 +558,9 @@ In order to minimize ESR and maximize PCB area utilization, we made the tool aut
possible trace width when given a minimum clearance specification.
To handle outputting PCB geometry in a format that can be read from KiCad, we utilized the open source EDA file format
library \emph{gerbonara}~\cite{GerbonaraToolsHandle}. To support the FEM simulations that are described in the next
section below, our tool contains functionality to map gerbonara's geometry representation into that of
gmsh~\cite{geuzaineGmsh3DFinite2009}, the FEM mesher that we chose to interface with Elmer
FEM~\cite{ruokolainenElmerCSCElmerfemElmer2023}.
library \emph{gerbonara}. To support the FEM simulations that are described in the next section below, our tool contains
functionality to map gerbonara's geometry representation into that of gmsh~\cite{geuzaineGmsh3DFinite2009}, the FEM
mesher that we chose to interface with Elmer FEM~\cite{ruokolainenElmerCSCElmerfemElmer2023}.
Our inductor design tool is available in this paper's supplementary material as well as at the git repository linked at
the end of this paper.
@ -650,58 +649,65 @@ inductors allow for lowers resistive losses by approximately a factor of four. I
lead to the choice of a two-layer inductor, twisted inductors provide improved high-frequency performance at no
additional cost and without compromising other performance parameters.
\begin{table*}
\begin{tabular}{cc|cccc|cccc|ccc}
\multicolumn{2}{c|}{\textbf{Parameters}}&
\multicolumn{4}{c|}{\textbf{Design values}}&
\multicolumn{4}{c|}{\textbf{Simulation results}}&
\multicolumn{3}{c}{\textbf{Measurements}}\\
$n$&
$k$&
$L \left[\unit{\micro\henry}\right]$&
Error $\left[\unit{\percent}\right]$&
$R \left[\unit{\ohm}\right]$&
Error $\left[\unit{\percent}\right]$&
$L \left[\unit{\micro\henry}\right]$&
Error $\left[\unit{\percent}\right]$&
$R \left[\unit{\ohm}\right]$&
Error $\left[\unit{\percent}\right]$&
$L \left[\unit{\micro\henry}\right]$&
$f_\text{res} \left[\unit{\mega\hertz}\right]$&
$R \left[\unit{\ohm}\right]$\\\hline
\rowcolor[gray]{0.9}
$1$& $3$& $0.03$& $-93.1$& $0.0095$& $-49.9$& $0.039$& $-43.6$& $0.008$& $-78.8$& $0.056$& $\textbf{465.07}$& $\textbf{0.0143}$\\
$1$& $4$& $0.03$& $-103.4$& $0.0108$& $-38.6$& $0.040$& $-47.5$& $0.008$& $-87.5$& $\textbf{0.059}$& $460.08$& $0.015$\\
$1$& $5$& $0.03$& $-89.7$& $0.0123$& $-35.3$& $0.041$& $-34.1$& $0.009$& $-84.4$& $0.055$& $460.08$& $0.0166$\\
\hline\rowcolor[gray]{0.9}
$2$& $1$& $0.12$& $-28.4$& $0.0253$& $-12.1$& $0.127$& $-17.3$& $0.024$& $-18.3$& $0.149$& $\textbf{245.51}$& $\textbf{0.0284}$\\
$2$& $3$& $0.12$& $-31.0$& $0.0270$& $-7.9$& $0.128$& $-18.8$& $0.025$& $-16.4$& $\textbf{0.152}$& $240.52$& $0.0291$\\
$2$& $5$& $0.12$& $-26.7$& $0.0299$& $-0.2$& $0.130$& $-13.1$& $0.027$& $-11.1$& $0.147$& $225.5$& $0.03$\\
\hline\rowcolor[gray]{0.9}
$3$& $1$& $0.26$& $-10.0$& $0.0454$& $-1.6$& $0.262$& $-9.5$& $0.044$& $-4.8$& $\textbf{0.287}$& $\textbf{145.71}$& $0.0461$\\
$3$& $4$& $0.26$& $-9.6$& $0.0479$& $5.0$& $0.265$& $-7.9$& $0.046$& $1.1$& $\textbf{0.286}$& $\textbf{145.71}$& $\textbf{0.0455}$\\
\hline\rowcolor[gray]{0.9}
$5$& $1$& $0.73$& $4.5$& $0.0755$& $-3.1$& $0.670$& $-3.4$& $0.074$& $-5.1$& $\textbf{0.693}$& $61.345$& $0.0778$\\
$5$& $3$& $0.73$& $4.3$& $0.0763$& $4.7$& $0.671$& $-3.4$& $0.074$& $1.8$& $\textbf{0.694}$& $\textbf{70.285}$& $0.0727$\\
$5$& $7$& $0.73$& $4.4$& $0.0802$& $16.2$& $0.675$& $-2.8$& $0.077$& $12.7$& $\textbf{0.694}$& $68.05$& $\textbf{0.0672}$\\
\hline\rowcolor[gray]{0.9}
$10$& $1$& $2.90$& $6.3$& $0.2513$& $7.6$& $2.700$& $-0.7$& $0.250$& $7.1$& $\textbf{2.718}$& $24.076$& $0.2322$\\
$10$& $3$& $2.90$& $6.4$& $0.2520$& $10.5$& $2.700$& $-0.5$& $0.250$& $9.8$& $2.714$& $\textbf{28.571}$& $0.2255$\\
$10$& $7$& $2.90$& $6.4$& $0.2554$& $16.9$& $2.700$& $-0.5$& $0.252$& $15.8$& $2.713$& $28.072$& $\textbf{0.2122}$\\
\hline\rowcolor[gray]{0.9}
$25$& $1$& $18.15$& $6.7$& $1.8843$& $9.7$& $16.900$& $-0.2$& $1.900$& $10.4$& $16.938$& $8.84$& $1.7024$\\
$25$& $3$& $18.15$& $6.8$& $1.8851$& $13.2$& N/A& N/A& N/A& N/A& $16.919$& $8.595$& $1.636$\\
$25$& $13$& $18.15$& $6.7$& $1.9016$& $18.9$& $16.900$& $-0.2$& $1.900$& $18.8$& $16.931$& $\textbf{10.555}$& $\textbf{1.5429}$\\
$25$& $37$& $18.15$& $6.0$& $2.0197$& $15.9$& $17.100$& $0.2$& $2.000$& $15.1$& $\textbf{17.066}$& $10.31$& $1.698$\\
\end{tabular}
\setlength{\tabcolsep}{4pt}
\begin{table}
\caption{Inductor sample design parameters and measured characteristics. All inductors have outer diameter
\qty{35}{\milli\meter} and inner diameter \qty{15}{\milli\meter}. The missing values in the simulation results
columns result from the solver failing to converge. Bolded values highlight the best performing coil of each
turn count. Shaded rows indicate conventional two-layer planar inductors ($k=1$).}
columns result from the solver failing to converge.}
\begin{tabular}{cc|cc|cc|ccc}
\multicolumn{2}{c|}{}&
\multicolumn{2}{c|}{\textbf{Design}}&
\multicolumn{2}{c|}{\textbf{Simulation}}&
\multicolumn{3}{c}{\textbf{Measured}}\\
$n$&
$k$&
$L$&
$R$&
$L$&
$R$&
$L$&
$f_\text{res}$&
$R$\\
&
&
$\left[\unit{\micro\henry}\right]$&
$\left[\unit{\ohm}\right]$&
$\left[\unit{\micro\henry}\right]$&
$\left[\unit{\ohm}\right]$&
$\left[\unit{\micro\henry}\right]$&
$\left[\unit{\mega\hertz}\right]$&
$\left[\unit{\ohm}\right]$\\\hline
\rowcolor[gray]{0.9}
$1$& $3$& $0.03$& $0.0095$& $0.039$& $0.008$& $0.056$& ${465}$& ${0.0143}$\\
$1$& $4$& $0.03$& $0.0108$& $0.040$& $0.008$& ${0.059}$& $460$& $0.0150$\\
$1$& $5$& $0.03$& $0.0123$& $0.041$& $0.009$& $0.055$& $460$& $0.0166$\\
\hline\rowcolor[gray]{0.9}
$2$& $1$& $0.12$& $0.0253$& $0.127$& $0.024$& $0.149$& ${246}$& ${0.0284}$\\
$2$& $3$& $0.12$& $0.0270$& $0.128$& $0.025$& ${0.152}$& $241$& $0.0291$\\
$2$& $5$& $0.12$& $0.0299$& $0.130$& $0.027$& $0.147$& $226$& $0.0300$\\
\hline\rowcolor[gray]{0.9}
$3$& $1$& $0.26$& $0.0454$& $0.262$& $0.044$& ${0.287}$& ${146}$& $0.0461$\\
$3$& $4$& $0.26$& $0.0479$& $0.265$& $0.046$& ${0.286}$& ${146}$& ${0.0455}$\\
\hline\rowcolor[gray]{0.9}
$5$& $1$& $0.73$& $0.0755$& $0.670$& $0.074$& ${0.693}$& $61.3$& $0.0778$\\
$5$& $3$& $0.73$& $0.0763$& $0.671$& $0.074$& ${0.694}$& ${70.3}$& $0.0727$\\
$5$& $7$& $0.73$& $0.0802$& $0.675$& $0.077$& ${0.694}$& $68.0$& ${0.0672}$\\
\hline\rowcolor[gray]{0.9}
$10$& $1$& $2.90$& $0.2513$& $2.700$& $0.250$& ${2.718}$& $24.1$& $0.2322$\\
$10$& $3$& $2.90$& $0.2520$& $2.700$& $0.250$& $2.714$& ${28.6}$& $0.2255$\\
$10$& $7$& $2.90$& $0.2554$& $2.700$& $0.252$& $2.713$& $28.1$& ${0.2122}$\\
\hline\rowcolor[gray]{0.9}
$25$& $1$& $18.15$& $1.8843$& $16.900$& $1.900$& $16.938$& $8.84$& $1.7024$\\
$25$& $3$& $18.15$& $1.8851$& N/A& N/A& $16.919$& $8.60$& $1.6360$\\
$25$& $13$& $18.15$& $1.9016$& $16.900$& $1.900$& $16.931$& ${10.56}$& ${1.5429}$\\
$25$& $37$& $18.15$& $2.0197$& $17.100$& $2.000$& ${17.066}$& $10.31$& $1.6980$\\
\end{tabular}
\label{tab_coupons}
\end{table*}
\end{table}
\subsection{Inductance and Frequency Behavior of Larger Coils}
@ -726,6 +732,9 @@ see that this effect exceeds what one would reach by a simple series configurati
indicating a contribution from flux linkage.
\begin{table}
\caption{Parameters and measurement results of a set of larger sample inductors. Bold values indicate best
performance at a given size. Shaded rows indicate conventional planar toroidal ($n=1$) or two-layer planar
spiral inductors ($k=1$).}
\begin{tabular}{cc|cc|ccc|c}
$d_1$&
$d_2$&
@ -765,9 +774,6 @@ indicating a contribution from flux linkage.
$75$&$90$&$53$ &$320$& $461$& $76.2$& $8.75$& $0.72$\\
$75$&$90$&$53$ &$480$& $\mathbf{470}$& $92.9$& $8.00$& $0.84$\\
\end{tabular}
\caption{Parameters and measurement results of a set of larger sample inductors. Bold values indicate best
performance at a given size. Shaded rows indicate conventional planar toroidal ($n=1$) or two-layer planar
spiral inductors ($k=1$).}
\label{tab_wide_coils}
\end{table}
@ -777,7 +783,7 @@ indicating a contribution from flux linkage.
To evaluate twisted inductors in our WPT application, we measured the variation of the coupling between a pair of
inductors using an automated measurement setup consisting of a 3D gantry built from 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 (cf.\ Figure\ \ref{fig_setup_probe}).
angles to one another (cf.\ Fig.\ \ref{fig_setup_probe}).
\begin{figure}
\begin{center}
@ -800,21 +806,21 @@ angles to one another (cf.\ Figure\ \ref{fig_setup_probe}).
To approximate our application, we loaded the secondary inductor with a \qty{10}{\ohm} resistor 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
shown in Fig.\ \ref{fig_test_schematic}. We measured both the input and output voltages of the coupled inductor pair
using Keysight 34465A multimeters in AC Root Mean Square (RMS) mode.
\begin{figure}
\begin{center}
\includegraphics[width=\figurescale]{figures/symmetry_3turn_n_twist.pdf}
\end{center}
\caption{RMS output voltage of the test circuit from Figure\ \ref{fig_test_schematic} for three pairs of matching
\caption{RMS output voltage of the test circuit from Fig.\ \ref{fig_test_schematic} for three pairs of matching
inductors with one inductor rotating w.r.t.\ the other. The inductors have $n=3$ turns each and $k=\frac{1}{2}$,
$k=1$, 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{fig_symmetry_3turn_n_twist}
\end{figure}
Figure\ \ref{fig_symmetry_3turn_n_twist} shows the ratio between input and output voltage of our test link for a set of
Fig.\ \ref{fig_symmetry_3turn_n_twist} shows the ratio between input and output voltage of our test link for a set of
three-turn inductors with multiple inversion numbers $k$ when one inductor is rotated. In practical WPT setups, the
transmitter and receiver coils are rarely aligned perfectly, so we show measurements across a range of radial offsets.
In line with our inductance measurements, coupling is lower at $k>0$ compared to a single-layer spiral inductor. Across
@ -824,12 +830,12 @@ into higher frequencies that are easier to passively filter on the WPT link's se
Expanding our measurements in the previous section, we performed a series of measurements rotating both inductors. In
these measurements, the coils' distance is fixed \qty{1}{\milli\meter} and the radial offset is set to a worst-case
value of \qty{4}{\milli\meter}. Figure\ \ref{fig_rms_ripple_n3} shows the normalized output voltage of a WPT link made
value of \qty{4}{\milli\meter}. Fig.\ \ref{fig_rms_ripple_n3} shows the normalized output voltage of a WPT link made
from three-turn inductors with rotation of one inductor shown on the horizontal axis, and the rotation of the other
shown on the vertical axis.
We performed similar measurements on 24 of our test coupons at \qty{1}{\milli\meter} and \qty{4}{\milli\meter} radial
offsets. Figure\ \ref{fig_k_ripple_plot} shows the combined results of these measurements, with worst-case voltage
offsets. Fig.\ \ref{fig_k_ripple_plot} shows the combined results of these measurements, with worst-case voltage
variation plotted across inversion count $k$ for multiple turn counts $n$ and radial offsets $r$. In this graph, we see
that twisted inductors improve ripple compared to conventional designs, even at a low inversion count such as $k=3$.
@ -842,12 +848,11 @@ pitch, as their turns deviate the furthest from a set of ideal, concentric circl
\begin{center}
\includegraphics[width=.85\figurescale]{figures/k_ripple_plot.pdf}
\end{center}
\caption{RMS Voltage ripple in a model rotating WPT setup with $R_L=\qty{10}{\ohm}$ as a percentage of total RMS
output voltage, plotted against inductor inversion count $k$. Measurements were taken with a number of different
coils with turn count $n$ between a single turn and $25$ turns. Measurements were taken at two different radial coil
offsets of $r=\qty{1}{\milli\meter}$ and $\qty{4}{\milli\meter}$. Coil distance was $d=\qty{1}{\milli\meter}$ in all
cases. The shaded area indicates conventional coil layouts, with the remainder of the plot showing twisted
inductors.}
\caption{RMS Voltage ripple as a percentage of total RMS
output voltage in a rotating WPT setup with $R_L=\qty{10}{\ohm}$, coil distance $d=\qty{1}{\milli\meter}$ plotted
w.r.t. inductor inversion count $k$. Measurements were taken at two radial offsets of $r=\qty{1}{\milli\meter}$ and
$\qty{4}{\milli\meter}$. The shaded area indicates conventional coil layouts, with the remainder of the plot showing
twisted inductors.}
\label{fig_k_ripple_plot}
\end{figure}
@ -921,24 +926,8 @@ four-dimensional mapping of the coupling between a pair of identical inductors.
description of twisted inductor construction as well as a set of Open-Source tools for their design, available at the
link at the end of this paper.
\section*{Acknowledgement}
\addcontentsline{toc}{section}{Acknowledgment}
This work has been funded by the LOEWE initiative (Hesse, Germany) within the emergenCITY center
[LOEWE/1/12/519/03/05.001(0016)/72] as well as by Technical University of Darmstadt.
\section*{Availability}
This is version \texttt{\input{version.tex}\unskip} of this paper, generated on \today.
After publication, 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:
\center{\url{https://git.jaseg.de/nice-coils.git}}
\printbibliography[heading=bibintoc]
\FloatBarrier
\appendix
%\section{Supplemental plots}
%\begin{figure}
@ -966,10 +955,12 @@ measurements as well the set of tools for the generation of twisted inductor lay
% \label{fig_rms_ripple_n25}
%\end{figure}
\section{Layout examples}
\label{sec_appendix_layout_examples}
\FloatBarrier
%\section{Layout examples}
%\label{sec_appendix_layout_examples}
\begin{figure*}
\appendix
\begin{center}
\includegraphics[width=.75\textwidth]{figures/nk_complex_illust.pdf}
\end{center}
@ -977,5 +968,4 @@ measurements as well the set of tools for the generation of twisted inductor lay
illustration we chose values for $n$ and $k$ such that all pairs are coprime.}
\label{fig_nk_complex_illust}
\end{figure*}
\end{document}

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v1.0-0-gcfae60e
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