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@ -610,10 +610,10 @@ inductor, which have the highest voltage differential, are located right on top
between. Making things worse, common PCB substrates have a relative permittivity much larger than air (usually around
$4$).
Interestingly, we observe that this decrease in high-frequency performance is eventually counteracted by increasing
inversion count $k$. While our test samples focused on smaller turn counts, we observe an increase from a self-resonant
frequency of \qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$.
Prompted by this observation, we produced another set of samples focusing on this aspect. We report our results of this
We observe that this decrease in high-frequency performance is eventually counteracted by increasing inversion count
$k$. While our test samples focused on smaller turn counts, we observe a notable increase from a self-resonant frequency
of \qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$. Prompted by
this observation, we produced another set of 15 samples focusing on this aspect. We report our results of this
investigation in the following section.
In conclusion to the above measurement results, we observe that twisted inductors \emph{improve} high-frequency
@ -757,13 +757,10 @@ indicating a contribution from flux linkage.
\subsection{Coupling and its Sensitivity to Radial Offset}
While our accidential findings that twisted inductors improve high-frequency performance are certainly welcome and may
benefit a range of applications, the key performance criterion in our rotating WPT 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 across a large set of rotations and relative displacements, we created
a test 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 while
measuring their coupling.
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.
\todo{pics of 3d printer test setup}
@ -777,17 +774,10 @@ measuring their coupling.
\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 Root Mean Square (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 inversion numbers
$k$. A plot for a set of 10-turn inductors is shown in Figure\ \ref{fig_symmetry_10turn_n_twist} in the Appendix. A key
observation here is that while the asymmetry in the inductor's field is impossible to distinguish visually in field
plots, the ripple induced by rotation is considerable. The sharp dropoff of coupling with radial offset magnifies any
small asymmetry and leads to the ripple voltages we have listed in Table\ \ref{tab_coupons}, in some cases amounting to
several percent of total RMS output voltage.
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
using Keysight 34465A multimeters in AC Root Mean Square (RMS) mode.
\begin{figure}
\begin{center}
@ -800,36 +790,29 @@ several percent of total RMS output voltage.
\label{fig_symmetry_3turn_n_twist}
\end{figure}
From the ripple plots in Figures\ \ref{fig_symmetry_3turn_n_twist} and \ref{fig_symmetry_10turn_n_twist} 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. While increasing $k$
above $1$ does not siginificantly decrease the amplitude of this ripple further, it shifts the ripple into higher
frequencies that are easier to passively filter on the WPT link's secondary side in our application.
Figure\ \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
one revolution, we find that the single-layer spiral inductor exhibits the most voltage ripple, with simple two-layer
inductors with $k=1$ already improving ripple. For $k$ above $1$, ripple amplitude stay sconstant, but energy is shifted
into higher frequencies that are easier to passively filter on the WPT link's secondary side in our application.
\subsection{Total Coupling Variation}
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
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.
In practical WPT setups, the transmitter and receiver coils are rarely aligned perfectly. To 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.
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
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 $k=3$.
Plotting the results of these experiments as well as a series of experiments at a \qty{1}{\milli\meter} radial offset
against inversion count $k$, we arrive at the graph in Figure\ \ref{fig_k_ripple_plot}. In this graph, we see that
twisted inductors improve ripple compared to conventional designs, even at a low inversion count such as $k=3$.
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},
$k=3$ inversions already provided an improvement over standard configurations, with still better performance observed
for $k=7$ inversions.
\todo{concrete coupling factor measurements}
Concluding our measurements, we achieved our primary objective of reducing coupling variation under rotation, with
twisted inductors ($k>1$) improving over conventional two-layer spiral inductors, which perform better than simple
single-layer spiral inductors. This improvement is greatest for inductors with low turn count and consequentially coarse
pitch, as their turns deviate the furthest from a set of ideal, concentric circles.
\begin{figure}
\begin{center}
@ -871,21 +854,21 @@ for $k=7$ inversions.
\label{fig_rms_ripple_n3}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=.75\figurescale]{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}
%\begin{figure}
% \begin{center}
% \includegraphics[width=.75\figurescale]{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}
\section{Future Work}
On the practical side, as part of our inductor design tool, we extended the EDA file format library gerbonara with code
to automatically map gerbonara's geometry description to the gmsh FEM mesher. This code may be of independent interest
since it allows for the extraction of FEM meshes from not just individual planar components, but PCBs in any file format
supported by gerbonara such as KiCad's native file format, as well as the Gerber file format supported by the majority
of EDA tools.
Our derivation of twisted inductors opens up a space for future research. On the practical side, as part of our inductor
design tool, we extended the EDA file format library gerbonara with code to automatically map gerbonara's geometry
description to the gmsh FEM mesher. This code may be of independent interest since it allows for the extraction of FEM
meshes from not just individual planar components, but PCBs in any file format supported by gerbonara such as KiCad's
native file format, as well as the Gerber file format supported by the majority of EDA tools.
On the theoretical side, the fact that our twisted inductor model generalizes both one- or two-layer planar spiral
inductors as well as planar toroidal inductors would make the deduction of key parameters such as inductance and
@ -893,13 +876,12 @@ distributed capacitance by mathematical analysis or by finite element methods in
\section{Conclusion}
In this paper, we introduced a novel layout approach for planar, multi-layer inductors loosely inspired by classic
basket-wound inductors used in the early days of radio. Our \emph{twisted} inductors generalize several types of
conventional planar inductors including conventional single- or two-layer planar spiral inductors as well as planar
toroidal inductors. For inversion count parameter $k\ge 2$, twisted inductors produce magnetic field distributions that
have better rotational symmetry along the inductor's main axis compared to either single- or two-layer planar spiral
inductors, which yields lower output ripple in WPT through rotating joints and enables the use of smaller and lighter
secondary-side circuitry, improving efficiency.
In this paper, we introduced a novel layout approach for planar, multi-layer inductors. Our \emph{twisted} inductors
generalize several types of conventional planar inductors including conventional single- or two-layer planar spiral
inductors as well as planar toroidal inductors. For inversion count parameter $k\ge 2$, twisted inductors produce
magnetic field distributions that have better rotational symmetry along the inductor's main axis compared to either
conventional single- or two-layer planar spiral inductors, which yields lower output ripple in WPT through rotating
joints and enables the use of smaller and lighter secondary-side circuitry, improving efficiency.
Furthermore, besides the advantages twisted inductors show in our particular application, we found that our sample
twisted inductors have up to \qty{50}{\percent} improved self-resonant frequency as well as up to \qty{6.5}{\percent}
@ -923,32 +905,32 @@ set of tools for the generation of twisted inductor layouts that we wrote can be
\FloatBarrier
\appendix
\section{Supplemental plots}
%\section{Supplemental plots}
\begin{figure}
\begin{center}
\includegraphics[width=\figurescale]{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{fig_symmetry_3turn_n_twist}}
\label{fig_symmetry_10turn_n_twist}
\end{figure}
%\begin{figure}
% \begin{center}
% \includegraphics[width=\figurescale]{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{fig_symmetry_3turn_n_twist}}
% \label{fig_symmetry_10turn_n_twist}
%\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=.75\figurescale]{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\figurescale]{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\figurescale]{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}
%\begin{figure}
% \begin{center}
% \includegraphics[width=.75\figurescale]{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}
\section{Layout examples}
\label{sec_appendix_layout_examples}