nice-coils/paper/paper.tex

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\begin{document}

\date{}
\title{Wireless Power Transfer with a Twist:
Achieving Rotation-Invariant Coupling using Multi-Layer PCB Inductors}
\maketitle

\begin{abstract}
    % FIXME
\end{abstract}

\section{Introduction}

Inductive wireless power transfer (WPT) is a widely used technology supported by a large corpus of research literature.
% FIXME cite
While working on a novel application of Inductive wireless power transfer in a Inertial Hardware Security Module (IHSM)
as proposed by Götte and Scheuermann, % FIXME cite
we found ourselves presented with an unusual set of constraints around inductive wireless power transfer through a
rotating joint using a PCB inductor that does not yet seem to be addressed adequately in the existing literature on
inductive wireless power transfer.

Our application poses the challenge of transferring power between a stationary and a rotating part. To reduce
manufacturing cost of both parts, and to reduce weight, and thereby inertia as well as susceptibility to vibration in
the rotating part, we decided to use inductors that are directly patterned onto the IHSM's printed circuit boards.
The primary constraint that results from this choice is a highly constrained turn count that is limited by the PCB
manufacturing processes' pattern resolution and by ohmic heating.

We found that the limited turn count of PCB inductors results in a \emph{slightly} asymmetric field, which means that
the coupling coefficient of two such inductors oscillates at one oscillation per revolution when the inductors are
rotated on-axis, even if both inductors are perfectly coaxially aligned.

In other inductive wireless power transfer systems, this oscillation is mitigated by one of several factors: First, for
this effect to matter in the first place, the two coils have to be rotating with respect to one another. In ferrite or
iron-cored inductors, the core shapes the magnetic field and evens out any such imperfection. In wire-wound inductors,
the (much) higher turn count and circular aspect ratio of the wires reduces this effect to almost nothing. Finally, the
output ripple caused by this oscillation can be filtered through a voltage regulator or by using a large decoupling
capacitor on the secondary side.

While there exist a number of prior works 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, it is generally assumed
that the two coils remain (almost) stationary with respect to one another throughout the charging process. % FIXME cite

There exists a small body of work on inductive power transfer through rotating joints, % FIXME cite
but here the focus lies on higher power budgets than our application requires, which often requires ferrite or iron-core
inductors.

Our application is unique in that it requires power transfer through a joint that is constantly rotating at high speed,
while we simultaneously want to avoid heavy components on the (rotating) receiver side. (Liquid) electrolytic capacitors
cannot be used due to the large centrifugal acceleration that the rotating part experiences, and other heavy components
such as large ceramic or polymer electrolytic capacitors or ferrite-core power inductors are inadvisable since they will
exert large stresses onto the assembly due to the same centrifugal acceleration, and any imbalance caused by tolerances
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 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.

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
capacitance of the inductor and raises its self-resonant frequency, raising its maximum possible operating frequency and
improving its efficiency at lower operating frequencies. This is the same effect that is exploited in basket-woven
air core inductors that were commonly used in old radio sets.
% FIXME citation on this, citation on basket weaving -> It's hard to find reliable references on that.

\section{Related Work}

\subsection{Inductive Wireless Power Transfer in Practice}

Inductive WPT has been proposed in a large number of scenarios, each of which comese with a set of
unique constraints. When WPT is used to charge an electric toothbrush, the implementation cost of the system is
critical, while efficiency and total power output are of little concern. Mechanically, in an electric toothbrush's
charging system, the position and spacing of the transmitter and receiver coils can easily be controlled down to
millimeter precision.

In contrast to this, wireless smartphone charging is a much more demanding application. Here, the total cost of the
system is only secondary, but the receiver's form factor is critical, and total power output as well as efficiency
become major objectives. At the same time, in wireless smartphone charging, position tolerances are very coarse, and the
two coils in the charging base and in the phone may be positioned more than a centimeter off-axis, with a gap of several
millimeters and potentially not even in parallel planes.

Power transfer across large distances is even more of a concern in implantable medical devices. Where a wireless phone
charger must be able to bridge distances of a few millimeters, an implantable medical device might be situated
underneath several centimeter of tissue and bones. At the same time, cost is of (almost) no concern in this medical
application, which enables the use of complex manufacturing techniques, customized electronic components and exotic
materials.

While all of the aforementioned applications transfer somewhere between a few hundred milliwatts and several watts of
power, at the other end of the spectrum there is a large body of research suggesting the use of inductive wireless power
transfer for the charging of electric vehicles (EVs). In this application, the wireless power transfer system replaces
the conventional wired charging connector, which improves the systems' user experience given the strong force required
to seat or unseat these rather large connectors, as well as the heft of the required water-cooled cables. In this
application, size is of (almost) no concern, but at several kilowatt up to dozens or even a hundred kilowatt, the
transferred power is enormous and consequentially efficiency becomes of utmost importance. When charging an EV at a
rate of 30 kW, an efficiency improvement of just $0.1\%$ corresponds to a reduction in power dissipation of 30 W.
Besides the monetary cost of the power lost this way, each small improvement enables a reduction in size of heat sinks
and other cooling components, which directly translates to a decrease in cost.

\subsection{Twisted Inductors in RFIC Design}
\subsection{Basket-Woven Air Coils}
\subsection{Air-Core Inductors for Inductive Power Transfer}
\subsection{Ferrite or Iron-Core Inductors for Inductive Power Transfer}

\section{Twisted Inductor Design} 

We can approach twisted inductors by construction. Let us first consider a simple, planar, circular spiral coil with a
fixed pitch. We will ignore trace width for now, and consider the trace a thin wire. We will assume the inductor's ports
are both located on the positive $x$-Axis. We can rotate it so its first port aligns with the $x$-Axis. To
minimize the loop area of the inductor's connections, inductors are usually designed with both ports close to one
another, so we can also assume its second port aligns with the $x$-Axis.

The trace trajectory of a standard planar spiral inductor can be parameterized in polar coordinates $r, \phi$ based on
an Archimedean spiral: \todo{For the lulz, cite Archimedes here}

\begin{equation}
    r = a\cdot\phi
    \label{eqn_arch_spi_basic}
\end{equation}

An Archimedean spiral defined this way always starts at the origin, and it continues to infinity. Let us re-parameterize
this spiral to a curve parameter $t$ with range $\left[0,1\right]$, such that $t=0$ corresponds to the start of the
inductor and $t=1$ corresponds to its end. As is customary in PCB inductors, we place the inductor's start on its outer
radius. To make handling of this easier, we introduce a variable $r' \in \left[0,1\right]$ representing the radius
normalized to the spiral's width. Let $n$ be the turn count of our inductor. The resulting parametrization is:

\begin{align}
    \phi &= 2\pi n t\\
    r' &= 1 - t \\
    r &= r_1 + r' \left(r_2 - r_1\right)
    \label{eqn_simple_spiral_ind}
\end{align}

The resulting spiral trace starts at radius $r_2$ on the positive $x$ axis, and spirals inward until it meets $r_1$. In
its PCB realization, at $r_1$, a via would be placed to connect the end of the spiral trace to a jumper trace on another
layer of the PCB leading back to the start.

To improve layer utilization, a common technique in PCB inductor design is to use both layers of the PCB for the
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_twolayer_spiral} shows both a simple and a two-layer
spiral inductor. 

\begin{align}
    \phi &= 2\pi n t\\
    r' &= 1 - 2 t \\
    r &= r_1 + \left|r'\right| \left(r_2 - r_1\right)
    \label{eqn_twolayer_spiral}
\end{align}

\begin{figure}
    \begin{center}
        \includegraphics[width=0.7\linewidth]{figures/twolayer_spiral.pdf}
    \end{center}
    \caption{A single-layer spiral inductor's layout (left), and a two-layer spiral inductor's layout (right). Traces on
    the PCB top side are shown in red, traces on the bottom side in blue. Both inductors have $n=3$ turns.}
    \label{fig_twolayer_spiral}
\end{figure}

\subsection{From Spiral to Twisted Inductor}

Extending the above parametrization of a spiral inductor's layout, we propose planar \emph{twisted inductors} based on
two core observations:

\begin{itemize}
    \item 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 In a two-layer spiral inductor as shown in Figure\ \ref{fig_twolayer_spiral} \todo{refer to only right side,
        split into (a) and (b) subfigures}, 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{itemize}

Combining these two observations, we find that by choosing a number $k$ of top/bottom-layer spiral trace pairs that is
coprime to the number of total turns of the inductor $n$, we achieve a layout where when we connect all $k$ trace pairs
in series, the resulting spirals on the top and bottom layers interleave cleanly. Figure\ \ref{fig_nk_interleave_illust}
shows a layout with $n=3$ turns with both a single trace pair ($k=1$) as in a conventional two-layer inductor, and with
$k=2$ trace pairs, 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 additional layout examples for larger
values of $n$ and $k$.

\begin{figure}
    \begin{center}
        \includegraphics[width=0.7\linewidth]{figures/nk_interleave_illust.pdf}
    \end{center}
    \caption{A conventional two-layer planar inductor's layout (left), and a twisted inductor with two trace pairs
    (right). In the twisted inductor, each layer contains two archimedean spirals that interleave at a regular spacing.
    The four spirals of the inductor are connected in series such that they form three total turns.}
    \label{fig_nk_interleave_illust}
\end{figure}


\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}

\section{FEM Simulation}

To validate our analytical approximations, we performed a series of FEM simulations in both Elmer FEM and Simulia CST.
For a number of inductor layouts, we performed simulations to determine ohmic resistance, inductance, and parasitic
capacitance. For a subset of these layout variants we additionally performed simulations to determine the coupling
factor between a pair of identical inductors at a number of different distances and rotations.

\paragraph{Ohmic Resistance}
Determining ohmic resistance by FEM is reasonably easy. In Elmer FEM, we can use the built-in joint static current and
joule heating solver to determine the ohmic resistance at a given current.

\paragraph{Inductance}
We let Elmer determine inductance by first using its coil solver to determine the volumetric current density in our mesh
given a test current, then applying its magnetodynamics solver to solve the electromagnetic field. Elmer provides
routines to derive the total magnetic field energy $U_\text{mag}$ from an EM field solution. Since we have only our
inductor under test inside the simulation volume, with test current $I_\text{test}$, we can then derive the inductor's
inductance according to the well-known relation\todo{Find decent source}:

\begin{equation}
    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}

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 $k=37$. 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}

We measured inductance and Q-factor of each test coupon using a Keysight U1733C LCR meter at \qty{100}{\kilo\hertz}. We
measured DC resistance using a Keysight 34465A multimeter in four-wire resistance mode. We further determined the
self-resonant frequency of each inductor using a LiteVNA64 handheld vector network analyzer. The results of our
measurements are shown in Table\ \ref{tab_inductor_params}.

We found our inductance approximation to be accurate within \qty{10}{\percent} and our ESR approximation to be accurate
within \qty{20}{\percent} for inductors with three turns or more. For lower turn-count inductors, inductance
measurements are difficult because the small absolute inductances involved are easily disturbed by stray inductances,
and ESR measurements are affected by contact and trace resistance even when measurements are taken in four-wire mode.

In accordance with our design intuition, we found that for high turn count inductors, the doubled trace width that is
afforded by splitting a simple spiral inductor across two PCB layers in any two-layer configuration improves ESR by
approximately a factor of two. Going from a simple single-layer spiral inductor to a simple two-layer spiral inductor
($k=1$), we observe that the resulting inductance decreases by up to \qty{15}{\percent}. We suspect that the main factor
leading to this decrease is radial magnetic flux leakage through the PCB material between the inductor's layers.
Comparing simple two-layer inductors with $k=1$ to the twisted inductors with larger $k$ values that we propose in this
paper, we observe almost identical performance for $k>1$ with decreases of less than \qty{0.5}{\percent} going from
$k=1$ to $k=3$ irrespective of turn count. From these measurements we can conclude that the flux linkage of twisted
inductors almost perfectly matches that of simple two-layer inductors.

Finally, while not particularly relevant for our application, we decided to evaluate the high-frequency performance of
twisted inductors. We found that going from a single-layer spiral inductor to a two-layer spiral inductor decreases the
self-resonant frequency, this effect being more pronounced with higher turn count. Intuitively, this makes sense if we
consider the mechanics of inductor self-resonance. The primary contributor to self resonance, particularly in higher
turn count inductors, is capacitive coupling between the inductor's windings. In a single-layer spiral inductor, this
effect gets partially mitigated since the strongest coupling exists between adjacent windings.
the SRF have a small voltage differential as only a fraction of the inductor's total voltage appears across each
winding. Compared to this, when the inductor is constructed as a simple two-layer inductor with $k=1$, now the start and
end windings of the inductor, which have the highest voltage differential, are located right on top of each other with
the substrate in between. Making things worse, common PCB substrates have a dielectric constant much larger than air
(usually around $4$).

Interestingly, we observe that this decrease in high-frequency performance is counteracted by larger trace pair count
$k$. While our test samples focused on smaller turn counts, we observe an increase from an SRF of
\qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$.

In conclusion, we observe that twisted inductors \emph{improve} high-frequency performance compared to simple two-layer
inductors while closely matching them in ESR and inductance. While they peform worse than simple single-layer inductors
in high-frequency performance, the increased trace width that two-layer inductors allow for lowers resistive losses by
approximately a factor of four. In applications where resistive losses 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|ccc|}
        Turn Count $n$&
        Trace pair count $k$&
        Inductance $L \left[\mu H\right]$&
        Q-factor&
        DC resistance $R_\text{ESR} \left[\Omega\right]$\\
    \end{tabular}
    \caption{Inductor sample design parameters and measured characteristics. All inductors have outer diameter
    \qty{35}{\milli\meter} and inner diameter \qty{15}{\milli\meter}.}
\end{table*}

\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
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.

\begin{figure}
    \begin{center}
        \includegraphics[width=.85\linewidth]{figures/test_schematic.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}
\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}
    \end{center}
    \caption{RMS output voltage of the test circuit from Figure\ \ref{symmetry_test_circuit} for three pairs of matching
    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{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, with even 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}
    \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.}
    \label{fig_field_plot_3d}
\end{figure}

\begin{figure}
    \begin{center}
        \includegraphics[width=.75\linewidth]{figures/rms_ripple_double_rotation_n3_r4.pdf}
    \end{center}
    \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}

\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, 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]

\clearpage
\appendix
\section{Layout examples}
\label{sec_appendix_layout_examples}

\begin{figure*}
    \begin{center}
        \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}