\documentclass[journal,12pt,onecolumn,draftclsnofoot]{IEEEtran} \usepackage[T1]{fontenc} \usepackage[ backend=biber, style=numeric, natbib=true, url=false, doi=true, eprint=false ]{biblatex} \addbibresource{paper.bib} \usepackage{amssymb,amsmath} \usepackage{eurosym} \usepackage{wasysym} \usepackage[binary-units]{siunitx} \usepackage{commath} \usepackage{graphicx,color} \usepackage{colortbl} \usepackage{subcaption} \usepackage{placeins} \usepackage{array} \usepackage{censor} \usepackage{hyperref} \usepackage{makecell} \DeclareSIUnit{\baud}{Bd} \DeclareSIUnit{\year}{a} \DeclareSIUnit{\rpm}{rpm} \renewcommand{\floatpagefraction}{.8} \newcommand{\degree}{\ensuremath{^\circ}} \newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}} \newcommand{\partnum}[1]{\texttt{#1}} \newcommand{\todo}[1]{\textbf{TODO}\footnote{#1}} % Set to 1.0 for final two-column export \newlength{\figurescale} \setlength{\figurescale}{0.75\textwidth} \begin{document} % TODO % Define all acronyms % * PCB % Use less acronyms % Term "twisted"? "interleaved spirals"? % Early pic / vis of spirals, somewhere in intro % Put explanation of WPT to front of related work % One plot instead of big table % Move measeurements column to the left? % In experiment schematic / setup schema: what is moving, what is stationary? % Tone down mentioning of inspiration % Go into way more detail on use case \date{November 14 2024} \author{\IEEEauthorblockN{Jan Sebastian Götte}\thanks{Jan Sebastian Götte is with the Technical University of Darmstadt, 64283 Darmstadt, Germany (e-mail: jan.goette@tu-darmstadt.de).}} \title{Wireless Power Transfer with a Twist: Achieving Rotation-Invariant Coupling using Twisted Multi-Layer PCB Inductors} \maketitle \begin{abstract} We present \emph{twisted inductors}, a generalization of planar single- and 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 (WPT) applications with an axially rotating receiver. Additionally, we found that twisted inductors can simultaneously yield a significantly improved Self-Resonant Frequency (SRF) 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 an application of Inductive WPT in a Inertial Hardware Security Module (IHSM) as previously published by \textcite{gotteCantTouchThis2022}, we found ourselves presented with an unusual set of constraints attempting WPT through a rotating joint using a planar inductor implemented in a Printed Circuit Board (PCB)---a set of constraints that does not seem to be addressed adequately in the existing literature on inductive WPT yet. Inertial Hardware Security Modules are a hardware security primitive that discourages tampering with a payload (e.g.\ a single-board computer) by rotating a tamper-sensing enclosure around the payload. The tamper-sensing enclosure continuously monitors itself for tampering using sensors such as tamper-sensing meshes\cite{TamperResistance2020a} and accelerometers. When the tamper-sensing enclosure signals a tamper alarm to the payload, the payload immediately destroys all sensitive data to prevent the attacker from gaining access to it. In principle, an IHSM is similar to an ATM that responds to attempts at opening its vault by dispensing dye over the bank notes within, rendering them unusable. In our IHSM implementation, the tamper-sensing enclosure rotates at \qtyrange{1000}{3000}{\rpm}. The rotating enclosure is powered through a pair of WPT inductors located on the IHSM's axis of rotation. The large centrifugal acceleration prohibits the use of batteries or liquid electrolyte capacitors on the rotating part, and makes heavy components such as large Multilayer Ceramic Capacitors (MLCCs) challenging to balance. 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. Planar inductors are usually considered approximately axisymmetric. In our application, we found that at small turn counts, the asymmetry in a planar spiral inductors's field is large enough that the resulting oscillation of the coupling coefficient of two such inductors with the inductor's revolution leads to voltage ripple on the secondary side. Radial misalignment of the coils further exacerbates this issue. 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 renders 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, % TODO cite it is generally assumed that the two coils remain quasi-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 in practice requires more space and a ferrite or laminated iron core. Therefore, this paper bridges the gap between existing literature on low-power planar WPT inductor design and high-power WPT through rotating joints. \subsection{Twisted inductors} 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. To fit our unique use case, we applied a principle which the polygonal basket-woven air coils used in early radio sets are based on to an approach inspired by contemporary planar inductor layouts. 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 (SRF), 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 further present the results of Finite Element Method (FEM) simulations to validate our inductance and ESR 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. 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 to minimize the voltage differential between adjacent turns---a technique that is still used to this day\cite{lopeFirstSelfresonantFrequency2021}, 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{leePrintedSpiralWinding2011}. 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. \begin{figure} \begin{center} \subcaptionbox{\raggedright A classic planar spiral inductor}{ \includegraphics[width=0.3\figurescale]{figures/svg_vis_paper_plain.png}} \subcaptionbox{\raggedright A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{ \includegraphics[width=0.2\figurescale]{figures/saacke-radiotechnik-3-ledionspule.jpg}} \subcaptionbox{\raggedright A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{ \includegraphics[width=0.2\figurescale]{figures/klein-spulen-schwingkreise-korbspule.jpg}} \subcaptionbox{\raggedright Our proposed inductor layout}{ \includegraphics[width=0.3\figurescale]{figures/svg_vis_paper.png}} \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} Air-core inductors with or without ferrite magnetic shielding are the standard solution in inductive WPT links. 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 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---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 provides some advantage over a solid, round conductors of the same cross-sectional area at higher frequencies due to skin effect. 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 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 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 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{lopeFirstSelfresonantFrequency2021,sproHighVoltageInsulationDesign2021,leePrintedSpiralWinding2011}, 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 \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{Air-Core Inductors 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, 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 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. 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_combined} shows both a simple and a two-layer spiral inductor in the first two columns. \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{description} \item[Observation 1.] 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.] 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} 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 ($k=1$) 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=\figurescale]{figures/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} \begin{figure} \begin{center} \includegraphics[width=\figurescale]{figures/nk_combined.pdf} \end{center} \caption{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} Figure\ \ref{fig_nk_combined} 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\figurescale]{figures/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\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}. 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 (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} 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} 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 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 a self-resonant frequency of \qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$. Prompted by this observation, we produced another set of samples focusing on this aspect. We report our results of this 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{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} 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\figurescale]{figures/test_schematic.pdf} \end{center} \caption{The test schematic used in all measurements. For direct coupling factor measurements, the load resistor was disconnected. We measure voltage at the output of the function generator to account for drop in its internal output resistance.} \label{fig_test_schematic} \end{figure} To evaluate a realistic scenario, we loaded the secondary inductor with a resistive load of \qty{10}{\ohm}, while providing a signal at a \qty{300}{\kilo\hertz} carrier frequency to the primary inductor from a Siglent SDG6022X function generator as shown in Figure\ \ref{fig_test_schematic}. We measured both the input and output voltages of the coupled inductor pair using Keysight 34465A multimeters in AC Root Mean Square (RMS) mode. The results of these measurements, with the voltage ratio between the coupled inductors' input and output voltages graphed across one revolution in Figure\ \ref{fig_symmetry_3turn_n_twist} for a set of three-turn inductors with multiple inversion numbers $k$. A plot for a set of 10-turn inductors is shown in Figure\ \ref{fig_symmetry_10turn_n_twist} in the Appendix. A key observation here is that while the asymmetry in the inductor's field is impossible to distinguish visually in field plots, the ripple induced by rotation is considerable. The sharp dropoff of coupling with radial offset magnifies any small asymmetry and leads to the ripple voltages we have listed in Table\ \ref{tab_coupons}, in some cases amounting to several percent of total RMS output voltage. \begin{figure} \begin{center} \includegraphics[width=\figurescale]{figures/symmetry_3turn_n_twist.pdf} \end{center} \caption{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 $k=0$, $k=1$, 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\figurescale]{figures/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\figurescale]{figures/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\figurescale]{figures/rms_ripple_double_rotation_n3_r4.pdf} \end{center} \caption{RMS ripple magnitude as a percentage of mean RMS output voltage, plotted against the rotation of each of the two inductors. The two coils were kept at a constant \qty{4}{\milli\meter} radial offset, and the output coil was loaded with a \qty{10}{\ohm} load. All RMS ripple plots in this paper share the same color scale to allow for visual comparison. This figure shows four variants of 3-turn coils, plots for $n=5$ can be found in Figure\ \ref{fig_rms_ripple_n5} and plots for $n=\{10,25\}$ in Figures \ref{fig_rms_ripple_n10} and \ref{fig_rms_ripple_n25} in the Appendix.} \label{fig_rms_ripple_n3} \end{figure} \begin{figure} \begin{center} \includegraphics[width=.75\figurescale]{figures/rms_ripple_double_rotation_n5_r4.pdf} \end{center} \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 5-turn coils.} \label{fig_rms_ripple_n5} \end{figure} \section{Future Work} On the practical side, as part of our inductor design tool, we extended the EDA file format library gerbonara with code to automatically map gerbonara's geometry description to the gmsh FEM mesher. This code may be of independent interest since it allows for the extraction of FEM meshes from not just individual planar components, but PCBs in any file format supported by gerbonara such as KiCad's native file format, as well as the Gerber file format supported by the majority of EDA tools. 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 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, available at the link at the end of this paper. \section*{Availability} This is version \texttt{\input{version.tex}\unskip} of this paper, generated on \today. The git repository with the LaTeX source for this paper, the data analysis code underlying our measurements as well the set of tools for the generation of twisted inductor layouts that we wrote can be found at: \todo{link here} % \center{\url{https://git.jaseg.de/nice-coils.git}} \printbibliography[heading=bibintoc] \FloatBarrier \appendix \section{Supplemental plots} \begin{figure} \begin{center} \includegraphics[width=\figurescale]{figures/symmetry_10turn_n_twist.pdf} \end{center} \caption{Coupled RMS output voltage of three pairs of matching inductors with $n=10$ turns each and $k=0$, $k=1$, and $k=3$, respectively, shown as in Figure\ \ref{fig_symmetry_3turn_n_twist}} \label{fig_symmetry_10turn_n_twist} \end{figure} \begin{figure} \begin{center} \includegraphics[width=.75\figurescale]{figures/rms_ripple_double_rotation_n10_r4.pdf} \end{center} \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 10-turn coils.} \label{fig_rms_ripple_n10} \end{figure} \begin{figure} \begin{center} \includegraphics[width=.75\figurescale]{figures/rms_ripple_double_rotation_n25_r4.pdf} \end{center} \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 25-turn coils.} \label{fig_rms_ripple_n25} \end{figure} \section{Layout examples} \label{sec_appendix_layout_examples} \begin{figure*} \begin{center} \includegraphics[width=.75\textwidth]{figures/nk_complex_illust.pdf} \end{center} \caption{Layout examples for a number of combinations of turn count $n$ and 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*} \end{document}