\chaptertitle{Rotation-Invariant Wireless Power Transfer using Twisted Multi-Layer PCB Inductors}

\begin{abstract}
    We present \emph{twisted inductors}, a generalization of planar single- or two-layer spiral inductors as well as
    planar toroidal inductors. Compared to conventional planar spiral inductors, twisted inductors generate a magnetic
    field with better rotational symmetry, resulting in decreased output ripple in Wireless Power Transfer applications
    with an axially rotating receiver. Additionally, we found that twisted inductors can simultaneously yield a
    significantly improved self-resonant frequency and a higher inductance in the same area as a conventional planar
    spiral inductor, up to \qty{50}{\percent} improved SRF and \qty{6.5}{\percent} increased inductance among our test
    samples. We base our conclusions on several simulations and an extensive set of practical measurements.
\end{abstract}

\section{Introduction}

Inductive Wireless Power Transfer (WPT) is a widely used technology supported by a large corpus of research literature
\cite{awuahNovelCoilDesign2023, batraEffectFerriteAddition2015, curranModelingCharacterizationPCB2015,
fanSimultaneousWirelessPower2024, leeSimpleWirelessPower2017, liWirelessPowerTransfer2015,
maierContributionSystemDesign2019, mooreApplicationsWirelessPower2019, mouEnergyEfficientAdaptiveDesign2017,
mouWirelessPowerTransfer2015, mullenEffectMisalignmentInductive, rezmeritaSelfMutualInductance2017,
zhangWirelessPowerTransfer2019}.
While working on a novel application of Inductive WPT in a Inertial Hardware Security Module (IHSM) as previously
published in\textcite{gotteCantTouchThis2022}, we found ourselves presented with an unusual set of constraints
attempting WPT through a rotating joint using a PCB inductor---a set of constraints that does not yet seem to be
addressed adequately in the existing literature on inductive WPT.

Our application poses the challenge of transferring power between a stationary and a rotating part of an
IHSM\cite{gotteCantTouchThis2022} through a pair of WPT inductors located on the IHSM's axis of rotation. 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 that the PCB manufacturing processes' pattern resolution results in
a strict upper limit to the turn count that can be achieved in an inductor with a given area.

We found that at such small turn counts, a simple spiral PCB inductors exhibits a \emph{slightly} asymmetric field,
which means that the coupling coefficient of two such inductors oscillates at one cycle 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 is the single major factor shaping the magnetic field, and evens out any small effect
asymmetric windings might have. In wire-wound inductors, the often higher turn count and the tightly packed, circular
wires reduce 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 where those components can be
accomodated on the rotating part given the centrifugal forces resulting from a concrete design's rate of rotation.

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 charges,
% TODO cite
it is generally assumed that the two coils remain (almost) stationary with respect to one another.

There exists a small body of work on inductive power transfer through rotating
joints\cite{fanSimultaneousWirelessPower2024},
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 their solder joints and the surrounding assembly due to the same centrifugal acceleration. 
Any imbalance caused by tolerances in the placement of heavy components or the precise shape of their solder fillets
can cause detrimental vibration.

\subsection{Twisted inductors}

To solve this conundrum, we applied 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. In this paper, 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, and that in fact, our approach opens up a large design space
for inductor layouts that interpolate between planar spiral inductors on one end, and planar toroidal inductors on the
other end. Our approach thus generalizes a number of previous approaches to the design of planar inductors.

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. We note that the principle behind this reduction in distributed
capacitance coincides with the intuition that led to the creation of honeycomb or basket-woven inductors in early radio
sets more than a hundred years ago, before the invention of ferrites.

\subsection{Contributions}
In this paper, we introduce twisted inductors, a novel technique of laying out planar inductors that both improves
rotational symmetry in rotating wireless power transfer interface as well as quality factor in other applications. We
provide detailed layout instructions, including a mathematical analysis of the available parameter space and an
analytical model of both inductance and DC equivalent series resistance of our scheme. Validating our scheme, we provide
laboratory measurements of the basic parameters of a number of test specimens comparing our scheme to conventional
techniques. We furhter performed a number of FEM simulations to validate our inductance and ESL approximations. Finally,
to analyze the degree of rotational symmetry in our proposed scheme, we provide the results of a large number of
automated measurements of coupling between pairs of inductors under various rotations, offsets, distances and load
conditions.

\section{Related Work}

% TODO cite fanSimultaneousWirelessPower2024 below (rotating joint)
% TODO cite \cite{mullenEffectMisalignmentInductive} below (misaligned coils)

\subsection{A Short Historical Diversion on Basket-Woven Air Coils}

Since the early days of radio engineering, the parasitic capacitance of inductors has been a point of
concern\cite{nesperHandbuchDrahtlosenTelegraphie1921,flemingPrinciplesElectricWave1910}. Going back to the early days of
wireless telegraphy after the turn of the twentieth century, coils with high inductance were needed for the construction
of both transmitters and receivers, but the ferrites that would later permit their compact construction were still being
developed. The ferromagnetic core material of choice back then was laminated iron, which was only useful at low
frequencies due to eddy current losses. As a result, the inductors in radio circuits of the era were constructed as
air-core coils. While air core inductors are immune to core saturation, the poor magnetic permeability of air
necessitates a large number of wide turns of wire to reach useful inductance values, which for reasons of practicality
or leakage inductance often could not be wound as a single layer cylindrical coil. This could be resolved by winding an
inductor with many turns on multiple layers, which improves compactness and leakage inductance, but this in turn gives
rise to increased distributed capacitance as now turns with a large voltage differential are layered right on top of
each other.

Back then, a number of ways were devised to decrease distributed capacitance in multilayer inductors. These methods can
be divided into two general categories: Optimizing the connecting order of turns to minimize the voltage differential
between adjacent turns---a technique that is still used to this day\cite{lopeFirstSelfResonant2021}, and optimizing the
winding schema to increase the separation between turns. The main technique in the first category concerns winding the
turns of a cylindrical multilayer inductor not layer by layer, but instead layering them diagonally, effectively
connecting adjacent turns in a diagonal zigzag pattern. Then as now, wound inductors applying this technique were not
feasible to manufacture reliably by machine, but the technique can be closely replicated in PCB inductors as shown in
\textcite{leePrintedSpiralWinding2011a}. The main limiting factors in a PCB implementation are the requirement for a
large number of vias inside the inductor's turns limiting the achievable turn count\footnote{In PCBs, as opposed to
ICs, vias limit the achievable turn count when they need to be placed in-line inside the turns as opposed to on the
inside or outside because a PCB's minimum trace/space widths are usually much smaller than the smallest feasible via,
consisting of a minimum-size drill surrounded by a minimum-size annular ring.} and increasing ESR through the thin trace
sections that are necessary to accomodate the via structure, as well as the layer pairing limitations when blind vias
are used in multilayer PCBs.

\begin{figure}
    \begin{center}
        \subcaptionbox{\raggedright A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{
            \includegraphics[width=0.25\linewidth]{saacke-radiotechnik-3-ledionspule.jpg}}
        \subcaptionbox{\raggedright A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{
            \includegraphics[width=0.25\linewidth]{klein-spulen-schwingkreise-korbspule.jpg}}
    \end{center}
    \caption{Illustrations of honeycomb and basket-woven coils from the early days of wireless radio.}
    \textbf{TODO}: Not final graphics. Get proper scans for camera-ready version
    \label{fig_illust_honeycomb_basket}
\end{figure}

This lack of a way to wind high frequency inductors with a machine led to the creation of a number of related winding
schemes that include honeycomb and basket woven coils
\cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
filbigLehrbuchHochfrequenztechnik1942,
kleinSpulenUndSchwingungskreise1941,
meinkeTaschenbuchHochfrequenztechnik1956,
nottebrockSpulen1950,
struttVerstarkerUndEmpfanger1951,
wiggeRundfunktechnischesHandbuch1930,
zicknerSpulen1927}. The simplest such winding technique is the universal winding as described in depth by
\textcite{querfurthCoilWindingDescription1954}. In a simple, cylindrical wire-wound inductor, the windings are laid down
one right next to the other, until the end of the winding area is met, where the winding direction is reversed. One
layer of such windings forms a helix whose pitch is equal to the wire diameter. A universal winding uses the same
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 contemporary examples of which are shown in
Figure\ \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 on the same layer. When multiple layers like this are
stacked, a three-dimensional rhomboid pattern results that is vaguely reminiscent of a honeycomb's structure.

In basket-woven coils, a mandrel consisting of an odd number of sticks pointing either radially or axially is used, and
the wire is woven between adjacent sticks in an alternating direction. While visually similar to honeycomb coils, this
winding technique is more suited to homebrew construction and less amenable to mass production by machine. In axially
basket-woven coils, the mandrel can be pulled out after the coil is finished. Like honeycomb coils, the resulting
structure can be made mechanically stable with some lacquer, with the turns carrying the layers where they cross.

Both construction techniques apply similar principles to those leading to the improved high-frequency behavior of
twisted inductors that we describe in this paper. Interestingly, the winding schemes of both honeycomb and basket-woven
coils are also governed by the same coprimality condition between the number of turns and the number of inversions
within each turn that we describe for our twisted inductors below, although we could not find an example in contemporary
literature where this condition was explicitly stated \cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querfurthCoilWindingDescription1954}.

\subsection{PCB inductor design for wireless power transfer}

For wireless power transfer, air-core inductors with or without ferrite magnetic shielding are the standard solution.
Since in most applications, an air gap of several millimeters between the sending and receiving assemblies is expected,
adding a ferrite core does not result in a large improvement in coupling. Meanwhile, in many WPT applications,
especially for charging portable devices or medical implants, some misalignment between the sending and receiving coils
is expected. Using the available space with an air-core inductor that has a large cross-sectional area reduces the
impact of this misalignment.

Looking at such WPT inductors, they tend to be mostly planar coils with only a few layers, so implementing them in a PCB
process seems natural. Using a PCB for the inductor has the potential to reduce implementation cost since PCBs are
cheap, and they can also serve as structural support.

Implementing inductors in PCBs has a number of disadvantages. First, due to the limited layer count of common PCB
processes, and due to structure size limitations, the number of windings that can be fit into a given volume is much
lower than in wire-wound inductors. Second, due to a PCB's copper layers being thin compared to its dielectric
substrate, PCB inductors tend to have poor DC resistance. A PCBs' thin but wide trace cross-section helps with
skin effect compared to a solid conductor. However, PCBs can still not approach the performance of litz wire used in
high-frequency WPT coils, which commonly use wire diameters in the tens of micrometer
range\cite{zhaoDesignOptimizationLitzWire2023}. \textcite{lopeFrequencyDependentResistancePlanar2014} and
\textcite{nomotoSplittingConductorsCoils2024} propose a mitigation that aims to emulate a litz wire's structure in
large, high-current PCB inductors, but their mitigation is heavily limited by the structure size achievable in common
PCB manufacturing processes\cite{nguyenReviewComparisonSolid2020}.

A further factor that limits the high-frequency performance of PCB inductors is distributed capacitance. Not only do
large air coils exhibit more parasitic capacitance than much smaller ferrite-core inductors simply due to their size,
when implemented in a PCB process a large fraction of the electrical fields responsible for this capacitance pass
through the PCB's substrate, not air. The relative permittivity $\epsilon_r$ of common PCB substrates typically lies in
the range of $4$ to $5$\cite{mumbyDielectricPropertiesFR41989}, which increases the distributed capacitance compared to
a pure air-core inductor by approximately that same factor.

\subsection{Twisted Inductors in RFIC Design}

Planar inductors are commonly used in RF ICs. In RFIC design, the major challenges are area optimization and precisely
predicting the inductor's characteristics during the design phase. Common optimizations include applying a variable
trace pitch to reduce distributed capacitance\cite{lopez-villegasImprovementQualityFactor2000}, and applying variable
trace width to decrease equivalent series resistance while preserving total inductance and quality
factor\cite{hsuAnalyticalDesignAlgorithm2008}.

In RFICs, inductors are commonly designed as \emph{balanced} inductors with a grounded central node. Such designs
interleave two counter-wound planar spiral inductors on the same layer with the help of some jumper connections on a
second layer\cite{daneshDifferentiallyDrivenSymmetric2002,martinMultiturnTwistedInductor2016}. The use of such designs
in RFIC design is mainly focused on their electrical symmetry, so that the two input ports can be fed with a fully
differential signal, with the inductor loading both driver outputs equally across the inductor's frequency range.

Setting the inversion count to $k=1$ in our proposed scheme as shown below yields the counterwound scheme that is
commonly used for two-layer planar
inductors\cite{lopeFirstSelfResonant2021,sproHighVoltageInsulationDesign2021,leePrintedSpiralWinding2011a}, and
which has been used to stack planar coils for more than a century\cite{flemingPrinciplesElectricWave1910}.

% They note that the main point behind the design is electrical symmetry of the two ports to make driving the thing
% differentially cleaner. We should adopt this observation for our inductors, which likewise are electrically symmetric
% when compared to a single-layer spiral inductor.

\subsection{Inductive Wireless Power Transfer in Practice}

Inductive WPT has been proposed in a large number of
scenarios\cite{zhangWirelessPowerTransfer2019,mouWirelessPowerTransfer2015}, each of which comes 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\cite{mooreApplicationsWirelessPower2019}. 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)\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017}. In this application, the wireless power
transfer system usually 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 charging rates up to tens of kilowatt,
efficiency becomes critical. When charging an EV at a rate of 10 kW, an efficiency improvement of just $0.1\%$
corresponds to a reduction in power dissipation of 10 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{Air-Core Inductors for Inductive Power Transfer}

Across application areas, air-core inductors are often used for wireless power transfer since in most applications, an
air gap of several millimeters or more is expected, and adding a ferrite core would not change the system's performance
by much in these circumstances. A common way to use ferrites in WPT applications is by magnetically shielding the
inductor's back side with a ferrite plate such that the field does not extend beyond the coil's back side, thereby
increasing the intended mutual inductance while simultaneously reducing eddy current losses when the WPT coils are
placed near metal
objects\cite{batraEffectFerriteAddition2015,leeSimpleWirelessPower2017,muehlmannMutualCouplingModeling2012}.

\section{Twisted Inductor Design} 

In this section, we will provide a detailed derivation of the layout of twisted inductors. We can approach this layout
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 $r, \varphi$ based
on an Archimedean spiral: \todo{For the lulz, cite Archimedes here}

\begin{equation}
    r = a\cdot\varphi
    \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
circumference. 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}
    \varphi &= 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_nk_interleave_illust} shows both a simple and a two-layer
spiral inductor. 

\begin{align}
    \varphi &= 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}

\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_nk_interleave_illust}, 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 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\
\ref{fig_nk_interleave_illust} shows a layout with $n=3$ turns with both a single inversion ($k=1$) as 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 additional layout
examples for other values of $n$ and $k$. For $k=0$, 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}.

\begin{figure}
    \begin{center}
        \includegraphics[width=\linewidth]{nk_interleave_illust.pdf}
    \end{center}
    \caption{single-layer spiral inductor's layout (left), a conventional two-layer planar inductor's layout (middle),
    and a twisted inductor with two inversions (right). All three inductors have $n=3$ turns. Traces on the PCB top
    side are shown in red, traces on the bottom side in blue. 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}

Figure\ \ref{fig_nk_chinese_remainder_illust} illustrates how we arrive at the coprimality requirement.
\todo{Cleanly handle $k=0$ case.}
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 re-label the angular axis in steps
from $0$ to $k$, and re-label the radial axis in steps from $0$ to $n$. Labelling the new angular axis $i$ and the new
radial axis $j$, in the resulting integer lattice, the trace has slope $1$. We can state the trace's trajectory as a
function of a curve parameter $t \in [0, nk]$ as $f(t) = (i, j) = (t \mod n, t \mod k)$. To produce a valid inductor,
the trace must not intersect anywhere. Thus, the system of congruences

\begin{align}
    t &\equiv i \mod n\\
    t &\equiv j \mod k
\end{align}

must have a unique solution $t \in [0, nk]$ for all $(i, j)$. This statement corresponds exactly to the Chinese
Remainder Theorem, which states that this solution is unique when $k$ and $n$ are coprime.

\begin{figure}
    \begin{center}
        \includegraphics[width=0.8\linewidth]{nk_chinese_remainder_illust.pdf}
    \end{center}
    \caption{Illustration of the winding pattern of two twisted inductors. The upper plots show the inductor's actual
    layout with the traces on each side of the substrate colored in red (top) and blue (bottom), respectively. The lower
    plots show the same traces, but \emph{unwrap} the annulus by plotting the traces' polar coordinates on cartesian
    axes. The left axis labels show the normalized radius, with $0$ being at the inductor's inner diameter, $1$ being at
    its outer diameter on the top layer, and $-1$ being at its outer diameter on the bottom layer. The top and right
    axes labels show the axis scaled to match indices $i\in\left[0, n\right]$ and $j\in\left[0, k\right]$,
    respectively.}
    \label{fig_nk_chinese_remainder_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 diameter
$\frac{2 r_1 + 2 r_2}{2}=r_1+r_2$ as $l = n\pi\left(r_1 + r_2\right)$. Since going from a standard inductor to a twisted
inductor does not change its turn count or dimensions, the combined arc length of all traces of the twisted inductor
does not change. Twisted inductors require two additional vias per inversion, 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.\todo{Does the skin effect affect the influence of vias?}

As a general expression, for a standard or twisted inductor with turn count $n$ and twist count $k$, 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}

Even for geometrically simple inductors, analytically calculating their inductance is a surprisingly hard problem whose
complexity quickly escalates with the inductor's geometric complexity, with realistic wire shapes as opposed to
approximations assuming an infinitely thin wire, or when taking into account differing magnetic permeabilities of
air or dielectrics and core materials. Instead, a number of approximations are commonly used. A commonly referenced
approximation for the inductance of planar spiral inductors is given by \textcite{mohanSimpleAccurateExpressions1999},
whose current-sheet approximation for circular planar spiral inductors we will use here to estimate our inductor's
inductance. The current-sheet approximation from \textcite{mohanSimpleAccurateExpressions1999} reads:

\begin{equation}
    \label{eqn_mohan_approx}
    L = \frac{\mu n^2 d_\text{avg} c_1}{2}\left(\ln\left(c_2/\rho\right)+c_3\rho+c_4\rho^2\right)
\end{equation}

In this equation, $c_{1-4}$ denote four empirically determined coeficcients that together describe the coil's shape. The
values for circular coils are $c_{1-4}=(1.00, 2.46, 0.00, 0.20)$. $\mu$ is the magnetic permeability of air (for an
air-core inductor), $n$ is the number of turns, $d_\text{avg}$ is the \emph{average} turn radius, i.e. $d_\text{avg} =
2\frac{r_1 + r_2}{2} = r_1 + r_2$. $\rho = \frac{r_2-r_1}{r_2+r_1}$ is the planar spiral inductor's \emph{fill ratio}.
The fill ratio encodes the fact that the inductor's turns have less flux linkage the closer to the inductor's center we
get. While turns close to the outside have good flux linkage due to their inner area overlapping well with that of other
turns, turns close to the center not only have a loop area that is only a fraction of that of turns further outwards,
the closer we get to the center, the larger is also the fraction of the field lines returning as leakage flux on the
outside of the inner turn that pass through the inner part of turns further outwards, flipping the sign and contributing
\emph{negative} mutual inductance.

As Equation\ \ref{eqn_mohan_approx} approximates the inductor's whole set of windings as a single, uniform current
sheet, the turn count only appears as a single factor of $n^2$ in the equation, with $\rho$ and $c_{1-4}$ correcting for
the inductor's geometry. To account for twisted inductors, we can separate the inductor into a set of $2k$ simple planar
spiral inductor \emph{branches} that are connected in series by the twisted inductor's vias. Compared to a simple spiral
inductor, for each branch, the inductance according to Equation\ \ref{eqn_mohan_approx} stays the same except that the
factor $n^2$ drops to $\left(\frac{n}{2k}\right)^2$ because the $n$ windings are evenly distributed across the $2k$
branches. Let us now make two assumptions. First, we will assume that the flux linkage between both sides of the
inductor is approximately one. This assumption is grounded in the fact that for practical designs, the substrate
thickness will be small compared to the inductor's diameter. Second, we will for now ignore the spiral inductor's field
asymmetry and assume that the flux linkage between two intertwined branches on the same side of the substrate is
approximately one. In our measurements below we show that for simple spiral inductors this asymmetry, while problematic
in our application, is small in absolute terms, and grows smaller with increasing turn count.

Based on these two assumptions, we can model the twisted inductor as a set of $2k$ series-connected spiral inductors
that are perfectly coupled, with full flux linkage. This results in the total series inductance gaining back the factor
$\frac{1}{2k^2}$ that each branch lost, resulting in identical inductances for a simple planar spiral inductor and a
twisted inductor with the same size and turn count according to Equation\ \ref{eqn_mohan_approx}. This approximation
introduces an error due to the imperfect flux linkage between the two sides of the substrate, and between two spiral
branches located at an angular offset from each other. In our experiments, we found that for our test inductors,
compared to inductances measured with an LCR meter, this error is below \qty{10}{\percent} for $n=5$ turns or more, and
for our test samples matches the performance of Equation\ \ref{eqn_mohan_approx} for the simple planar spiral inductor
case.

\subsection{CAD Integration}

To allow for easy design with twisted inductors and to speed up the laboratory prototyping we performed for this paper,
we created a tool that generates arbitrary twisted inductor layouts, and that is able to output these layouts as PCB
footprint files for the open source KiCad EDA CAD tool. We integrated the ESR and ESL approximations as derived above
with our tool, so that it provides immediate design feedback when generating inductors. In order to minimize ESR and
maximize PCB area utilization, we made the tool automatically calculate the largest 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}\todo{Cite 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\todo{Cite gmsh}, the
FEM mesher that we chose to interface with Elmer FEM\todo{Cite Elmer}.

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.

\section{FEM Simulation}

To validate our analytical approximations, we performed a series of FEM simulations in Elmer FEM. For a number of
inductor layouts, we performed simulations to determine ohmic resistance and inductance. Due to limitations in our
gmsh/Elmer toolchain, we were unable to run simulations for parasitic capacitance and self-resonance, or for coupling
behavior of coil pairs. We found that for these cases which require larger, more complex meshes, gmsh would frequently
crash during meshing, and where we were able to produce meshes, Elmer would only converge for some of them. While these
are problems that can be solved through either a more skillful description of the problem in gmsh and Elmer, or by using
more robust software such as Simulia CST, we decided to instead experimentally measure these quantities instead
(Section\ \ref{sec_experiments}).

\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}

\section{Experimental Validation}
\label{sec_experiments}

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 and DC resistance}

We measured the inductance and DC resistance of each test coupon using a Keysight U1733C LCR meter at
\qty{100}{\kilo\hertz} for inductance and a Keysight 34465A multimeter in four-wire configuration for DC resistance. 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_coupons}.

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, which here have only 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 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 an SRF of
\qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$. Prompted by this
observation, we produced another set of 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
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|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}
        $1$&  $0$&  $0.03$&   $-86.2$&  $0.0076$& $-86.8$&  $0.038$&  $-42.1$&  $0.008$&  $-77.5$&  $0.054$&   $457.585$&$0.0142$\\
        $1$&  $3$&  $0.03$&   $-93.1$&  $0.0095$& $-49.9$&  $0.039$&  $-43.6$&  $0.008$&  $-78.8$&  $0.056$&   $\textbf{465.07}$& $\textbf{0.0143}$\\
        $1$&  $4$&  $0.03$&   $-103.4$& $0.0108$& $-38.6$&  $0.040$&  $-47.5$&  $0.008$&  $-87.5$&  $\textbf{0.059}$&   $460.08$& $0.015$\\
        $1$&  $5$&  $0.03$&   $-89.7$&  $0.0123$& $-35.3$&  $0.041$&  $-34.1$&  $0.009$&  $-84.4$&  $0.055$&   $460.08$& $0.0166$\\\hline
        \rowcolor[gray]{0.9}
        $2$&  $0$&  $0.16$&   $10.0$&   $0.0252$& $-26.7$&  $0.126$&  $-14.3$&  $0.026$&  $-22.7$&  $0.144$&   $266.24$& $0.0319$\\
        \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}
        $3$&  $0$&  $0.26$&   $-19.6$&  $0.0755$& $-5.0$&   $0.285$&  $-9.1$&   $0.077$&  $-2.9$&   $0.311$&   $192.95$& $0.0792$\\
        \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}
        $5$&  $0$&  $0.73$&   $-9.6$&   $0.2357$& $-0.4$&   $0.760$&  $-5.3$&   $0.240$&  $1.4$&    $0.8$&     $125.415$&$0.2366$\\
        \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}
        $10$& $0$&  $2.90$&   $-2.4$&   $0.7539$& $-2.3$&   $2.900$&  $-2.4$&   $0.761$&  $-1.4$&   $2.97$&    $62.835$& $0.7713$\\
        \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}
        $25$& $0$&  $18.15$&  $1.1$&    $3.7693$& $-3.9$&   $18.000$& $0.3$&    $3.800$&  $-3.0$&   $17.955$&  $24.84$&  $3.9156$\\
        \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}
    \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 two-layer coil
        of each turn count. Shaded rows indicate conventional single-layer ($k=0$) or two-layer ($k=1$) planar
        inductors.}
    \label{tab_coupons}
\end{table*}

\subsection{Inductance and Frequency Behavior of Larger Coils}

To investigate the high-frequency behavior of twisted inductors further, we produced and measured 15 additional sample
inductors, this time larger than before, and with more turns. The parameters of these new samples and our measurement
results are shown in Table\ \ref{tab_wide_coils}. In these results, we can identify three clear trends. First, the ESR
of twisted inductors is generally poorer when compared to two-layer spiral inductors. This increase in ESR is due to the
large number of vias used in these sample inductors. It should be noted that while twisted inductors have worse ESR
compared to conventional two-layer inductors, their ESR is still better than that of a single-layer inductor. Our second
observation is that in every set of samples from this second run of physically larger inductors, twisted inductors
outperform conventional inductors in self-resonant frequency by a considerable margin with an increase in SRF of up to
\qty{50}{\percent} in our samples.

Our third observation is that unlike in the smaller inductors from Table\ \ref{tab_coupons}, in these larger instances,
twisted inductors show increased inductance by approximately \qty{3.7}{\percent} for our smallest samples, and
\qty{6.5}{\percent} for our largest samples. This behavior indicates that large twisted inductors indeed behave like a
combination between a conventional planar spiral inductor and a conventional planar toroidal inductor. Comparing the
magnitude of this increase with the measurements listed in Table\ \ref{tab_wide_coils} for planar toroidal inductors, we
see that this effect exceeds what one would reach by a simple series configuration of both styles of inductor,
indicating a contribution from flux linkage.

\begin{table}
    \begin{tabular}{cc|cc|ccc|c}
        $d_1$&
        $d_2$&
        $n$&
        $k$&
        $L$&
        $R_\text{ESR}$&
        $f_\text{Res}$&
        $C_\text{p}$\\
        $\left[\unit{\milli\meter}\right]$&
        $\left[\unit{\milli\meter}\right]$&
        &
        &
        $\left[\unit{\micro\henry}\right]$&
        $\left[\unit{\ohm}\right]$&
        $\left[\unit{\mega\hertz}\right]$&
        $\left[\unit{\pico\farad}\right]$\\\hline
        \rowcolor[gray]{0.9}
        $25$&$40$&$1$   &$150$& $5.00$&         $11.0$&          N/A&              N/A\\
        \rowcolor[gray]{0.9}
        $25$&$40$&$53$  &$1$&   $120$&          $\mathbf{19.6}$& $18.0$&           $0.65$\\
        $25$&$40$&$53$  &$50$&  $121$&          $22.6$&          $\mathbf{27.5}$&  $\mathbf{0.28}$\\
        $25$&$40$&$53$  &$100$& $123$&          $26.9$&          $26.5$&           $0.29$\\
        $25$&$40$&$53$  &$150$& $\mathbf{125}$& $33.2$&          $24.0$&           $0.35$\\\hline
        \rowcolor[gray]{0.9}
        $50$&$65$&$1$   &$300$& $10.2$&         $21.9$&          N/A&              N/A\\
        \rowcolor[gray]{0.9}
        $50$&$65$&$53$  &$1$&   $270$&          $\mathbf{35.7}$& $10.0$&           $0.94$\\
        $50$&$65$&$53$  &$100$& $272$&          $41.9$&          $\mathbf{15.8}$&  $\mathbf{0.37}$\\
        $50$&$65$&$53$  &$200$& $277$&          $50.1$&          $13.3$&           $0.52$\\
        $50$&$65$&$53$  &$300$& $\mathbf{280}$& $65.0$&          $13.8$&           $0.48$\\\hline
        \rowcolor[gray]{0.9}
        $75$&$90$&$1$   &$480$& $17.3$&         $35.5$&          N/A&              N/A\\
        \rowcolor[gray]{0.9}
        $75$&$90$&$53$  &$1$&   $441$&          $\mathbf{50.7}$& $7.00$&           $1.17$\\
        $75$&$90$&$53$  &$160$& $444$&          $60.8$&          $\mathbf{10.0}$&  $\mathbf{0.57}$\\
        $75$&$90$&$53$  &$320$& $461$&          $76.2$&          $8.75$&           $0.72$\\
        $75$&$90$&$53$  &$480$& $\mathbf{470}$& $92.9$&          $8.00$&           $0.84$\\
    \end{tabular}
    \caption{Inductor sample design parameters and measured characteristics for a number of physically larger,
        ring-shaped inductors. $L$ and $R_\text{ESR}$ have been measured with a Keysight U1733C handheld LCR meter.
        $f_\text{Res}$ has been measured with a LiteVNA VNA. $C_p$ has been calculated for the simple parallel LC
        resonator model from $f_\text{Res}$ and $L$. $f_\text{Res}$ was not be measured for the $n=1$ case since these
        are just planar toroidal inductors, which show different resonance characteristics compared to planar spiral or
        multi-turn twisted inductors. Bolded values highlight the best performance among the coils of one size. Shaded
        rows indicate conventional planar toroidal ($n=1$) or two-layer planar spiral inductors ($k=1$).}
    \label{tab_wide_coils}
\end{table}


\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.

\todo{pics of 3d printer test setup}

\begin{figure}
    \begin{center}
        \includegraphics[width=.85\linewidth]{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 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. 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) 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.

\begin{figure}
    \begin{center}
        \includegraphics[width=\linewidth]{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 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.

\subsection{Total Coupling Variation}

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.

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}

\begin{figure}
    \begin{center}
        \includegraphics[width=.85\linewidth]{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.}
    \label{fig_k_ripple_plot}
\end{figure}

\begin{figure}
    \begin{center}
        \includegraphics[width=.6\linewidth]{field_plot_3d_n5_k0.pdf}
    \end{center}
    \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}

\begin{figure}
    \begin{center}
        \includegraphics[width=.75\linewidth]{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]{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.

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
distributed capacitance by mathematical analysis or by finite element methods interesting.

\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 Wireless Power Transfer 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}
increased inductance compared to conventional two-layer planar spiral inductors.

We base our evaluation on laboratory measurements on a set of 39 sample inductors in total, including 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, available at the
link at the end of this paper.

% FIXME make this end up in the thesis appendix
%\appendix
%\section{Supplemental plots}
%
%\begin{figure}
%    \begin{center}
%        \includegraphics[width=\linewidth]{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\linewidth]{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]{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}
%
%\begin{figure*}
%    \begin{center}
%        \includegraphics[width=.75\textwidth]{nk_complex_illust.pdf}
%    \end{center}
%    \caption{Layout examples for a number of combinations of turn count $n$ and inversion 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*}
%
