\chapterquote{Clifford Ashley~\cite{ashleyAshleyBookKnots1993}}{
    A knot is never ``nearly right''; it is either exactly right or it is hopelessly wrong, one or the other; there is
    nothing in between. This is not the impossibly high standard of the idealist, it is a mere fact for the realist to
    face.}
\chaptertitle{Rotation-Invariant Envelope Power Supply}
\label{chapter-nice-coils}
% Twisted Inductor paper

A central engineering challenge in inertial HSMs is transferring power and data between the payload and the rotating
mesh cage (cf.\ Chapter~\ref{chapter-ihsm}). Industrially, power and data transfer through rotating joints is usually
done using slip ring assemblies. A slip ring consists of one or more contacts that wipe on a rotating circular surface.
Industrially, metal spring contacts plated with hard gold or other common surface coatings are used for transferring
small currents and data signals, and carbon brushes are used for higher currents. Slip rings are widely used in motors
and other rotating machinery.

For use in IHSMs, slip rings have several limitations. First, they are complex precision-machined components and thus
are rather expensive. Beyond cost, they also have performance limitations. Generally, slip rings are most well-suited to
slow rotation, as high rotation increases the wear of the contacts. The design target of \qty{1000}{rpm} we use in IHSMs
are at the upper end of what commercial slip rings usually support. A third disadvantage is that they are sensitive, and
any misalignment or contamination by dust can increase wear and cause intermittant contact.

An IHSM's data link can easily be realized using optical communication. Although power transfer using light is also
possible---and we have in fact demonstrated it in our first prototype IHSM---it comes at the disadvantage of a heavy
rotating assembly since large solar cells are needed, and it has poor end-to-end efficiency. For the large-scale meshes
needed in a high-performance IHSM tailored to SMPC applications, we engineered a better solution: A rotation-invariant
inductive Wireless Power Transfer link.

While Wireless Power Transfer (WPT) is widely used and can be implemented in many different ways~\cite{
    awuahNovelCoilDesign2023,
    batraEffectFerriteAddition2015,
    curranModelingCharacterizationPCB2015,
    fanSimultaneousWirelessPower2024,
    leeSimpleWirelessPower2017,
    liWirelessPowerTransfer2015,
    maierContributionSystemDesign2019,
    mooreApplicationsWirelessPower2019,
    mouEnergyEfficientAdaptiveDesign2017,
    mouWirelessPowerTransfer2015,
    zhangWirelessPowerTransfer2019}.
Most WPT variants link the primary and secondary side primarily through the magnetic component of the
electromagnetic field, and coils are used as the transmitting and receiving antenna. Such \emph{inductive} WPT uses low
frequency, which reduces circuit complexity, and it is well-suited for transferring high power across short distances.
The electronic realization of a WPT link is usually similar to that of a DC/DC converter, except that in place of the
inductor or flyback transformer, the pair of transceiver coils is used. Compared to a flyback transformer, the WPT
link's transceiver coil pair has a lower coupling coefficient that varies with distance.

A challenge in WPT links is the strong dependency between link inductor coupling coefficient and distance. In a naïve
implementation that uses the link coils as a simple transformer, link efficiency would drop sharply with distance. To
decrease the impact of this distance dependency, almost all WPT implementations combine the transceiver coils with
capacitors to form a pair of tuned tank circuits that are driven like they would be in a resonant converter. Like in
resonant converters, a variety of topologies such as series, parallel, or series-parallel LC are used for these tuning
circuits.

\section{Construction Approach}

\begin{figure}
    \begin{center}
        \subcaptionbox{\raggedright A classic planar spiral inductor}{
            \includegraphics[width=0.28\textwidth]{svg_vis_paper_plain.png}}
        \subcaptionbox{\raggedright A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{
            \includegraphics[width=0.18\textwidth]{saacke-radiotechnik-3-ledionspule.jpg}}
        \subcaptionbox{\raggedright A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{
            \includegraphics[width=0.18\textwidth]{klein-spulen-schwingkreise-korbspule.jpg}}
        \subcaptionbox{\raggedright Our proposed inductor layout}{
            \includegraphics[width=0.28\textwidth]{svg_vis_paper.png}}
    \end{center}
    \caption[Planar inductor layout comparison]{Illustration of our proposed inductor layout compared to contemporary
        conventional planar inductors and honeycomb as well as basket-woven coils from the early days of wireless
        radio.}
    \label{fig_illust_honeycomb_basket}
\end{figure}

\todo{Not final graphics. Get proper scans for camera-ready version}
In the WPT link powering the rotating mesh of an IHSM presents an unusual set of constraints, which does not seem to be
addressed adequately in the existing literature on inductive WPT yet. To reduce the need for custom-wound inductors, we
settled on using a planar inductor implemented in a Printed Circuit Board (PCB). Such planar PCB inductors are limited
by the structure size limits of the PCB process, resulting in rotational asymmetry due to the trace width. Planar
inductors are usually considered approximately axisymmetric. In our application, we found that the field asymmetry in
feasible PCB inductors is large enough that axial rotation of two such inductors results in an oscillation of their
coupling coefficient that leads to voltage ripple on the secondary side, especially when the coils are
misaligned.

The large centrifugal acceleration on an IHSM mesh prohibits the use of batteries or liquid electrolyte capacitors on
the rotating part, and makes heavy components such as large Multilayer Ceramic Capacitors (MLCCs) and ferrite-core
inductors challenging to balance. As a result, the secondary-side voltage ripple poses a significant issue since the
conventional ways of efficiently filtering such ripple through large bypass capacitors or through a secondary-side
switchmode power supply are difficult to implement due to their mass.

In other inductive WPT systems, this issue 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 core inductors, the core is
the major factor shaping the magnetic field and evens out the small effect of winding asymmetry. In wire-wound
inductors, the often higher turn count and the tightly packed, circular wires render this effect negligible. 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 if the application can accomodate such components on the rotating part.

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 chargers~\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017}, it is generally
assumed that the two coils remain quasi-stationary with respect to one another.

There exists a body of work on inductive power transfer through rotating joints but here the focus often lies on higher
power budgets than our application requires, which in practice requires more space and a ferrite or laminated iron
core~\cite{
    fanSimultaneousWirelessPower2024,
    songRotationLightweightWirelessPower2019,
    wangCoaxialNestedCouplersBased2020,
    }.
Often, these rotating joint WPT systems use coaxial structures, but segmented approaches exist, too~\cite{
    wangNovelRotatingWireless2024,
    yanFreeRotationWirelessPower2023,
    xiaRotaryWirelessPower2024,
    liWirelessPowerTransfer2021,
}.
In lower-power applications, segmented approaches are more common. A key challenge in segmented approaches is the
reduction of secondary-side ripple induced when the segments' alignment changes throught one revolution~\cite{
    zhangWirelessSensorPower2024,
}, which usually requires additional secondary-side circuitry. In this work, we introduce a planar coil topology for WPT
through a rotating joint using a single planar PCB coil on both the transmitting and the receiving side that improves
rotation ripple at low turn counts.

\subsection{Twisted inductors}

To solve these issues, we propose a layout for circular PCB inductors that uses a number of series-connected interleaved
spirals to achieve a topological equivalent to a torus knot from mathematical knot theory. Our layout twists the
inductor's windings around one another by connecting the interleaved spiral segments with a ring of vias each on the
inside and outside of the inductor's windings. Our approach provides better performance beyond our particular use case,
and improves over conventional contemporary planar inductors applying similar principles to those which inspired  the
polygonal basket-woven air coils used in early radio sets. We show that we can layout a twisted inductor for any number
of layer inversions that is co-prime to the inductor's turn count. Our approach opens up a 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 super-set to a number of previous approaches to the design of planar inductors.

We observe that in high-frequency applications, a moderate number of layer inversions 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 increases its Self-Resonant Frequency (SRF), raising its maximum possible
operating frequency and improving its efficiency at lower operating frequencies.

\subsection{Contributions}
Our contributions on this matter include:
\begin{itemize}
    \item We introduce twisted inductors, a planar inductor layout that improves rotational symmetry in WPT through
        rotating joins, and promises improved high-frequency behavior in other applications.
    \item We provide detailed instructions for the construction of such layouts, including a mathematical analysis of
        the available parameter space.
    \item We provide an analytical model of inductance and DC equivalent series resistance of our scheme.
    \item Validating our scheme, we provide laboratory measurements of the basic parameters of 39 test specimens
        comparing our scheme to conventional layouts.
    \item We further present the results of Finite Element Method (FEM) simulations to validate our inductance and ESR
        approximations.
    \item 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.
\end{itemize}

\section{Related Work}

\subsection{Inductive WPT 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 little concern, but at charging rates up to tens of kilowatt,
efficiency becomes critical.
%When charging an EV at a rate of \qty{10}{\kilo\watt}, an efficiency improvement of just
%$0.1\%$ corresponds to a reduction in power dissipation of \qty{10}{\watt}. 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{Core materials in WPT}

Across application areas, air-core inductors are often used for WPT since in most applications, an air gap of several
millimeters or more is expected~\cite{curranModelingCharacterizationPCB2015}. Especially in low-power application such
as mobile device charging, the size and weight of ferrites is an obstacle to their use, and at lower power levels losses
are less of a concern.

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}. Similar to
how the trace layouts of planar WPT coils are optimized to improve power transfer efficiency, the layout of ferrite
components has been proposed for optimization~\cite{batraEffectFerriteAddition2015}.

\subsection{PCB inductor design for wireless power transfer}

Today, air-core inductors are the standard solution in inductive WPT links. Since in most WPT 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. Instead, the impact of this misalignment is reduced by maximizing the area of the
air-core inductors used, or by tiling multiple
inductors~\cite{curranModelingCharacterizationPCB2015,wangNovelRotatingWireless2024,zhangDynamicWirelessPower2025}.

WPT inductors 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. However, implementing inductors in PCBs has several 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\footnote{common values are \qtyrange{15}{30}{\micro\meter} copper thickness and
\qtyrange{600}{1600}{\micro\meter} substrate thickness} PCB inductors tend to have poor DC resistance, albeit the thin
copper layer decreases skin effect losses compared to a solid, round conductors of the same cross-sectional area.
However, PCBs can still not approach the performance of litz wire used in high-frequency WPT coils, which commonly use
wire diameters in the range of tens of micrometer~\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 does a
large air coil exhibit more parasitic capacitance than an equivalent, smaller ferrite-core inductor simply due to its
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{Planar Inductors in RFIC Design}

Beyond WPT, planar inductors are commonly used in radio frequency integrated circuits (RFICs). 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~\cite{lopez-villegasImprovementQualityFactor2000} and variable
trace width~\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.

% Note: 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{A Brief 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 often
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.

Before the invention of ferrites, 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 and
optimizing the winding schema of turns. Both aim at increasing spacing between parts of the coil that have a large
voltage differential.

The connecting order of turns was optimized at the assembly level by stacking coils in a particular
way~\cite{flemingPrinciplesElectricWave1910} and at the component level by winding coils in a particular way to minimize
the voltage differential between adjacent turns---a technique that is still used to this
day~\cite{lopeFirstSelfResonant2021}. The main winding optimization 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
integrated circuits (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
equivalent series resistance (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.

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 historic 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, resulting in a three-dimensional rhomboid pattern 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 chapter.\footnote{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
historic literature where this condition was explicitly stated~\cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querfurthCoilWindingDescription1954}.}

\section{Twisted Inductor Design} 

In this section, we present a detailed derivation of the layout of twisted inductors. We 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 on top of one another on different layers, which also helps to minimize the loop area of the inductor's
connections.

The trace trajectory of a standard planar spiral inductor can be parameterized in polar coordinates $r, \varphi$ based
on an Archimedean spiral:

\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 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_combined} shows both a simple and a two-layer
spiral inductor in the first two columns. 

Let $n$ be the turn count of our inductor. The resulting parametrization is:

\begin{align}
    \varphi &= 2\pi n t\\\nonumber
    r &= r_1 + \left|1 - 2 t\right| \left(r_2 - r_1\right)
    \label{eqn_twolayer_spiral}
\end{align}

The resulting spiral trace starts at radius $r_2$ on the positive $x$ axis, and spirals inward until it meets $r_1$,
where the sign indicates a layer change, and the trace reverses to continue back to $r_2$ on another layer. 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 the other layer
of the PCB leading back to the start.

\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{description}
    \item[Observation 1.]\hfill\\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[Observation 2.]\hfill\\In a two-layer spiral inductor (Figure\ \ref{fig_nk_combined}), 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{description}

Setting the inversion count to $k=1$ in our proposed scheme yields the conventional two-layer counterwound
scheme~\cite{lopeFirstSelfResonant2021,sproHighVoltageInsulationDesign2021,leePrintedSpiralWinding2011a}.

\begin{figure}
    \begin{center}
        \includegraphics[width=\textwidth]{nk_combined.pdf}
    \end{center}
    \caption[Basic twisted planar inductor layouts]{Inductor layouts for several sets of turn count $n$ and inversion
        count $k$. The top row shows the actual trace layout in cartesian coordinates, the bottom row visualizes the
        winding schema.}
    \label{fig_nk_combined}
\end{figure}

\begin{figure}
    \begin{center}
        \includegraphics[width=\textwidth]{nk_complex_illust.pdf}
    \end{center}
    \caption[Complex twisted planar inductor layout variants]{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}

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_combined} shows a layout with $n=3$ turns with both a single inversion ($k=1$), which results 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} shows additional layout examples for other values of $n$
and $k$. For $k=\frac{1}{2}$, 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 chapter, we will call all
layouts with $k\ge 2$ \emph{Twisted Inductors}. The coordinate description of Equation\ \ref{eqn_twolayer_spiral} thus
becomes:

\begin{align}
    \varphi &= 2\pi n t\\\nonumber
    r &= r_1 + \left|1 - \left( 2 k t \mod 2 \right) \right| \left(r_2 - r_1\right)
    \label{eqn_twisted_spiral}
\end{align}

%\begin{figure}
%    \begin{center}
%        \includegraphics[width=\figurescale]{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}

Topologically, the shape of our inductors can be described as a $(k, n)$-torus knot. From knot theory, we know that such
a torus knot exists if and only if both $n$ and $k$ are co-prime. Figure\ \ref{fig_nk_combined} illustrates a derivation
of the coprimality requirement. 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 if and only if $k$ and $n$ are coprime.

In the following paragraphs, we will derive analytical expressions for Ohmic resistance and inductance of inductors
derived under this schema.

%\begin{figure}
%    \begin{center}
%        \includegraphics[width=0.8\figurescale]{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}

\paragraph{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 nor its 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.

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}

\paragraph{Inductance}

Even for geometrically simple inductors, analytically calculating their inductance is a surprisingly hard problem whose
complexity quickly escalates when geometrically complex inductors are analyzed, when realistic wire shapes as opposed to
thin wire or current sheet approximations are used, and when taking into account differing magnetic permeabilities of
air or dielectrics and core materials. Instead of precise analytical models, 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 are specific to 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
chapter, 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~\cite{KiCadEDA}. We integrated the ESR and inductance
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}~\cite{GerbonaraToolsHandle}. 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~\cite{geuzaineGmsh3DFinite2009}, the FEM mesher that we chose to interface with Elmer
FEM~\cite{ruokolainenElmerCSCElmerfemElmer2023}.

\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 (cf.\
Section\ \ref{sec_experiments}). While our measurements only cover a small number of inductor samples, their results are
more reliable than results from FEM and can serve as a baseline for future work on such simulations.

We conducted our FEM simulations as follows:

\paragraph{Ohmic Resistance}
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~\cite{meeekerFiniteElementMethod2015}:

\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 24 test coupons with a number of variations
of twisted inductors with winding count $n$ between $1$ and $25$, and twist count ranging from $k=\frac{1}{2}$ (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} corresponding to the space available in our IHSM implementation.

\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
chapter, 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, we decided to evaluate the high-frequency performance of twisted inductors. It is well-known that self-resonant
frequency decreases when going from a single-layer spiral inductor to a two-layer spiral inductor while keeping
inductance and dimensions constant~\cite{zhangImprovedCompensationMethod2025}. Our measurements show this effect, with
it being more pronounced with higher turn count. Intuitively, this makes sense if we consider the mechanism 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 $4$).

We observe that this decrease in high-frequency performance is eventually counteracted by increasing inversion count
$k$. While our test samples focused on smaller turn counts, we observe a notable increase from a self-resonant frequency
of \qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$. Prompted by
this observation, we produced another set of 15 samples focusing on this aspect. We report our results of this
investigation in the following section.

In conclusion to the above measurement results, we observe that twisted inductors \emph{improve} high-frequency
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{sidewaystable}
    \centering
    \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}
        $1$&  $3$&  $0.03$&   $-93.1$&  $0.0095$& $-49.9$&  $0.039$&  $-43.6$&  $0.008$&  $-78.8$&  $0.056$&   $\textbf{465.07}$& $\textbf{0.0143}$\\
        $1$&  $4$&  $0.03$&   $-103.4$& $0.0108$& $-38.6$&  $0.040$&  $-47.5$&  $0.008$&  $-87.5$&  $\textbf{0.059}$&   $460.08$& $0.015$\\
        $1$&  $5$&  $0.03$&   $-89.7$&  $0.0123$& $-35.3$&  $0.041$&  $-34.1$&  $0.009$&  $-84.4$&  $0.055$&   $460.08$& $0.0166$\\
        \hline\rowcolor[gray]{0.9}
        $2$&  $1$&  $0.12$&   $-28.4$&  $0.0253$& $-12.1$&  $0.127$&  $-17.3$&  $0.024$&  $-18.3$&  $0.149$&   $\textbf{245.51}$& $\textbf{0.0284}$\\
        $2$&  $3$&  $0.12$&   $-31.0$&  $0.0270$& $-7.9$&   $0.128$&  $-18.8$&  $0.025$&  $-16.4$&  $\textbf{0.152}$&   $240.52$& $0.0291$\\
        $2$&  $5$&  $0.12$&   $-26.7$&  $0.0299$& $-0.2$&   $0.130$&  $-13.1$&  $0.027$&  $-11.1$&  $0.147$&   $225.5$&  $0.03$\\
        \hline\rowcolor[gray]{0.9}
        $3$&  $1$&  $0.26$&   $-10.0$&  $0.0454$& $-1.6$&   $0.262$&  $-9.5$&   $0.044$&  $-4.8$&   $\textbf{0.287}$&   $\textbf{145.71}$& $0.0461$\\
        $3$&  $4$&  $0.26$&   $-9.6$&   $0.0479$& $5.0$&    $0.265$&  $-7.9$&   $0.046$&  $1.1$&    $\textbf{0.286}$&   $\textbf{145.71}$& $\textbf{0.0455}$\\
        \hline\rowcolor[gray]{0.9}
        $5$&  $1$&  $0.73$&   $4.5$&    $0.0755$& $-3.1$&   $0.670$&  $-3.4$&   $0.074$&  $-5.1$&   $\textbf{0.693}$&   $61.345$& $0.0778$\\
        $5$&  $3$&  $0.73$&   $4.3$&    $0.0763$& $4.7$&    $0.671$&  $-3.4$&   $0.074$&  $1.8$&    $\textbf{0.694}$&   $\textbf{70.285}$& $0.0727$\\
        $5$&  $7$&  $0.73$&   $4.4$&    $0.0802$& $16.2$&   $0.675$&  $-2.8$&   $0.077$&  $12.7$&   $\textbf{0.694}$&   $68.05$&  $\textbf{0.0672}$\\
        \hline\rowcolor[gray]{0.9}
        $10$& $1$&  $2.90$&   $6.3$&    $0.2513$& $7.6$&    $2.700$&  $-0.7$&   $0.250$&  $7.1$&    $\textbf{2.718}$&   $24.076$& $0.2322$\\
        $10$& $3$&  $2.90$&   $6.4$&    $0.2520$& $10.5$&   $2.700$&  $-0.5$&   $0.250$&  $9.8$&    $2.714$&   $\textbf{28.571}$& $0.2255$\\
        $10$& $7$&  $2.90$&   $6.4$&    $0.2554$& $16.9$&   $2.700$&  $-0.5$&   $0.252$&  $15.8$&   $2.713$&   $28.072$& $\textbf{0.2122}$\\
        \hline\rowcolor[gray]{0.9}
        $25$& $1$&  $18.15$&  $6.7$&    $1.8843$& $9.7$&    $16.900$& $-0.2$&   $1.900$&  $10.4$&   $16.938$&  $8.84$&   $1.7024$\\
        $25$& $3$&  $18.15$&  $6.8$&    $1.8851$& $13.2$&   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.]{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 coil of each turn count. Shaded rows indicate conventional
        two-layer planar inductors ($k=1$).}
    \label{tab_coupons}
\end{sidewaystable}

\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 that were larger (up to \qty{90}{\milli\meter} outer diameter) and that had a higher turn count (up to 53)
compared to our initial set of samples. 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, in our first set of test coupons we saw that their ESR is still better than that of a single-layer
inductor because the traces can be made wider. Our second observation is that in every set of samples from this second
run of physically larger inductors, twisted inductors outperform conventional planar inductors in self-resonant
frequency by a considerable margin with an increase in SRF of up to \qty{58}{\percent} from our
$d_2=\qty{65}{\milli\meter}$ sample going from $k=1$ to $k=100$.

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}
    \centering
    \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[Parameters and measurement results of larger sample inductors.]{Parameters and measurement results of a set
        of larger sample inductors. Bold values indicate best performance at a given 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}

To evaluate twisted inductors in our WPT application, we measured the variation of the coupling between a pair of
inductors using an automated measurement setup consisting of a 3D gantry built from an old 3D printer, with a fourth
rotation axis provided by a small servo that allows us to position two inductor test coupons at arbitrary offsets and
angles to one another.

\todo{pics of 3d printer test setup}

\begin{figure}
    \begin{center}
        \includegraphics[width=.65\textwidth]{test_schematic.pdf}
    \end{center}
    \caption[Planar inductor test schematic]{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 approximate our application, we loaded the secondary inductor with a \qty{10}{\ohm} resistor while providing a signal
at a \qty{300}{\kilo\hertz} carrier frequency to the primary inductor from a Siglent SDG6022X function generator as
shown in Figure\ \ref{fig_test_schematic}. We measured both the input and output voltages of the coupled inductor pair
using Keysight 34465A multimeters in AC Root Mean Square (RMS) mode.

\begin{figure}
    \begin{center}
        \includegraphics[width=0.8\textwidth]{symmetry_3turn_n_twist.pdf}
    \end{center}
    \caption[Planar inductor voltage ripple versus rotation angle]{RMS output voltage of the test circuit from Figure\
        \ref{fig_test_schematic} for three pairs of matching inductors with one inductor rotating w.r.t.\ the other. The
        inductors have $n=3$ turns each and $k=\frac{1}{2}$, $k=1$, 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}

Figure\ \ref{fig_symmetry_3turn_n_twist} shows the ratio between input and output voltage of our test link for a set of
three-turn inductors with multiple inversion numbers $k$ when one inductor is rotated. In practical WPT setups, the
transmitter and receiver coils are rarely aligned perfectly, so we show measurements across a range of radial offsets.
In line with our inductance measurements, coupling is lower at $k>0$ compared to a single-layer spiral inductor. Across
one revolution, we find that the single-layer spiral inductor exhibits the most voltage ripple, with simple two-layer
inductors with $k=1$ already improving ripple. For $k$ above $1$, ripple amplitude stay sconstant, but energy is shifted
into higher frequencies that are easier to passively filter on the WPT link's secondary side in our application.

Expanding our measurements in the previous section, we performed a series of measurements rotating both inductors. In
these measurements, the coils' distance is fixed \qty{1}{\milli\meter} and the radial offset is set to a worst-case
value of \qty{4}{\milli\meter}. Figure\ \ref{fig_rms_ripple_n3} shows the normalized output voltage of a WPT link made
from three-turn inductors with rotation of one inductor shown on the horizontal axis, and the rotation of the other 
shown on the vertical axis.

We performed similar measurements on 24 of our test coupons at \qty{1}{\milli\meter} and \qty{4}{\milli\meter} radial
offsets. Figure\ \ref{fig_k_ripple_plot} shows the combined results of these measurements, with worst-case voltage
variation plotted across inversion count $k$ for multiple turn counts $n$ and radial offsets $r$. In this graph, we see
that twisted inductors improve ripple compared to conventional designs, even at a low inversion count such as $k=3$.

Concluding our measurements, we achieved our primary objective of reducing coupling variation under rotation, with
twisted inductors ($k>1$) improving over conventional two-layer spiral inductors, which perform better than simple
single-layer spiral inductors. This improvement is greatest for inductors with low turn count and consequentially coarse
pitch, as their turns deviate the furthest from a set of ideal, concentric circles.

\begin{figure}
    \begin{center}
        \includegraphics[width=.65\textwidth]{k_ripple_plot.pdf}
    \end{center}
    \caption[Planar inductor voltage ripple versus design parameter]{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\figurescale]{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
%    $k=\frac{1}{2}$)
%    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=.65\textwidth]{rms_ripple_double_rotation_n3_r4.pdf}
    \end{center}
    \caption[Planar inductor voltage ripple versus both angles for $n=3, k=\{0,1,4\}$]{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 chapter 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}.}
    \label{fig_rms_ripple_n3}
\end{figure}

\begin{figure}
    \begin{center}
        \includegraphics[width=.65\textwidth]{rms_ripple_double_rotation_n10_r4.pdf}
    \end{center}
    \caption[Planar inductor voltage ripple versus both angles for $n=10, k=\{0,1,3,7\}$]{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=.65\textwidth]{rms_ripple_double_rotation_n25_r4.pdf}
    \end{center}
    \caption[Planar inductor voltage ripple versus both angles for $n=25, k=\{0,1,3,13\}$]{RMS ripple magnitude as shown
        in Figure\ \ref{fig_rms_ripple_n3} for four different 25-turn coils.}
    \label{fig_rms_ripple_n25}
\end{figure}

\begin{figure}
    \begin{center}
        \includegraphics[width=.65\textwidth]{rms_ripple_double_rotation_n5_r4.pdf}
    \end{center}
    \caption[Planar inductor voltage ripple versus both angles for $n=5, k=\{0,1,3,7\}$]{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}

Our derivation of twisted inductors opens up a space for future research. On the practical side, as part of our inductor
design tool, we extended the EDA file format library gerbonara with code to automatically map gerbonara's geometry
description to the gmsh FEM mesher. This code may be of independent interest since it allows for the extraction of FEM
meshes from not just individual planar components, but PCBs in any file format supported by gerbonara such as KiCad's
native file format, as well as the Gerber file format supported by the majority of EDA tools.

On the theoretical side, the fact that our twisted inductor model generalizes both one- or two-layer planar spiral
inductors as well as planar toroidal inductors would make the deduction of key parameters such as inductance and
distributed capacitance by mathematical analysis or by finite element methods interesting. Furthermore, the precise
contribution of vias to the twisted inductor's parasitics is interesting, especially for layouts with large values of
inversion count $k$. We suspect that via influence will be frequency dependant as vias and traces have distinct DC
resistances, and skin effect will affect both to a differring extent.

\section{Conclusion}

In this chapter, we introduced a novel layout approach for planar, multi-layer inductors. Our \emph{twisted} inductors
generalize several types of conventional planar inductors including conventional single- or two-layer planar spiral
inductors as well as planar toroidal inductors. For inversion count parameter $k\ge 2$, twisted inductors produce
magnetic field distributions that have better rotational symmetry along the inductor's main axis compared to either
conventional single- or two-layer planar spiral inductors, which yields lower output ripple in WPT through rotating
joints and enables the use of smaller and lighter secondary-side circuitry, improving efficiency.

Furthermore, besides the advantages twisted inductors show in our particular application, we found that our sample
twisted inductors have up to \qty{50}{\percent} improved self-resonant frequency as well as up to \qty{6.5}{\percent}
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 in the supplementary
material to this thesis.

Applied to an IHSM design, a wireless power transfer system using twised inductors to power the rotating mesh improves
efficiency by reducing losses due to stray capacitance and reduces secondary-side ripple. The reduced secondary-side
ripple allows the use of smaller filtering components, reducing board mass and mitigating heavy components as a possible
fault location. Additionally, the reduced ripple allows the use of secondary-side voltage regulators with less voltage
headroom, further reducing power transfer losses. By directly embedding twisted inductors into the IHSM's secondary side
mesh monitoring PCB, construction is simplified. The resulting assembly is lighter and smaller, which reduces motor load
and enables the implementation of compact IHSM meshes. 

%\begin{figure}
%    \begin{center}
%        \includegraphics[width=\figurescale]{symmetry_10turn_n_twist.pdf}
%    \end{center}
%    \caption{Coupled RMS output voltage of three pairs of matching inductors with $n=10$ turns each and
%    $k=\frac{1}{2}$, $k=1$, and $k=3$, respectively, shown as in Figure\ \ref{fig_symmetry_3turn_n_twist}}
%    \label{fig_symmetry_10turn_n_twist}
%\end{figure}