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@ -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}