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@ -358,11 +358,6 @@ inductance according to the well-known relation\todo{Find decent source}:
L = \frac{2\cdot U_\text{mag}}{I_\text{test}^2}
\end{equation}
\paragraph{Parasitic Capacitance and Self-Resonant Frequency}
Determining parasitic capacitance is more complex.
\subsection{Coupling}
\section{Experimental Validation}
\label{sec_experiments}
@ -436,6 +431,8 @@ 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.
\todo{pics of 3d printer test setup}
\begin{figure}
\begin{center}
\includegraphics[width=.85\linewidth]{figures/test_schematic.pdf}
@ -452,7 +449,11 @@ function generator as shown in Figure\ \ref{fig_test_schematic}. We measured bot
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.
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 small, the ripple induced by rotation is considerable. Figure\
\ref{fig_field_plot_3d} shows a 3D plot of an inductor's coupling. While in the plot the field looks perfectly
rotationally symmetric, the sharp dropoff with radial offset, equivalent to a large gradient, ``amplifies'' any small
asymmetry and leads to the ripple voltages we observed, amounting up to several percent of total RMS output voltage.
\begin{figure}
\begin{center}
@ -465,15 +466,16 @@ set of 10-turn inductors is shown in Figure\ \ref{fig_symmetry_10turn_n_twist} i
\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.
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. 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
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
@ -486,22 +488,22 @@ across rotations works, with twisted inductors ($k>1$) showing a further improve
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, with even better performance observed
$k=3$ trace pairs already provided an improvement over standard configurations, with still better performance observed
for $k=7$ trace pairs.
\todo{concrete coupling factor measurements}
\begin{figure}
\begin{center}
\includegraphics[width=.6\linewidth]{figures/field_plot_3d_n3_k4.pdf}
\includegraphics[width=.6\linewidth]{figures/field_plot_3d_n5_k0.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
and rotation are fixed to \qty{1}{\milli\meter} and \qty{15}{\degree}, respectively. The most prominent aspects of
this plot are that coupling falls off steeply with distance, and that the rotation-dependent variation is small in
comparison. The circular valley around the central peak is the region where one inductor is mostly outside the other
inductors, and intersects the field lines returning from the other inductor's back, leading to a negative coupling
coefficient.}
\caption{The coupling between a pair of identical coils (here two simple spiral inductors with $n=5$ and $k=0$)
visualized in three dimensions. The $x$ and $y$ axis show in-plane displacement, and the $z$ axis shows output
amplitude in arbitrary units. Height and rotation are fixed to \qty{1}{\milli\meter} and \qty{0}{\degree},
respectively. The most prominent aspects of this plot are that coupling falls off steeply with distance, and that
the rotation-dependent variation is small in comparison. The circular valley around the central peak is the region
where one inductor is mostly outside the other inductors, and intersects the field lines returning from the other
inductor's back, leading to a negative coupling coefficient.}
\label{fig_field_plot_3d}
\end{figure}