diff --git a/chapter-nice-coils/chapter.tex b/chapter-nice-coils/chapter.tex new file mode 100644 index 0000000..864443d --- /dev/null +++ b/chapter-nice-coils/chapter.tex @@ -0,0 +1,910 @@ +\chapter{Rotation-Invariant Envelope Power Supply} +% Twisted Inductor paper + +A central engineering challenge in inertial HSMs is transferring power and data between the payload and the rotating +mesh cage. 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{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.} + \textbf{TODO}: Not final graphics. Get proper scans for camera-ready version + \label{fig_illust_honeycomb_basket} +\end{figure} + +In the WPT link powering the rotating mesh of an IHSM presentsan 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{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{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. 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 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{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{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{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{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{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{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{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. + + +%\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} + + diff --git a/chapter-nice-coils/figures/divide_by_zero.pdf b/chapter-nice-coils/figures/divide_by_zero.pdf new file mode 100644 index 0000000..fbf0d5a Binary files /dev/null and b/chapter-nice-coils/figures/divide_by_zero.pdf differ diff --git a/chapter-nice-coils/figures/field_plot_3d_n3_k4.pdf b/chapter-nice-coils/figures/field_plot_3d_n3_k4.pdf new file mode 100644 index 0000000..2513222 Binary files /dev/null and b/chapter-nice-coils/figures/field_plot_3d_n3_k4.pdf 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