Incorporate first batch of Leo's comments

paper/paper.tex

@@ -59,13 +59,13 @@ Achieving Rotation-Invariant Coupling using Twisted Multi-Layer PCB Inductors}
 \maketitle
 
 \begin{abstract}
-    We present \emph{twisted inductors}, a generalization of planar single- or two-layer spiral inductors as well as
+    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 applications
-    with an axially rotating receiver. Additionally, we found that twisted inductors can simultaneously yield a
-    significantly improved self-resonant frequency and a higher inductance in the same area as a conventional planar
-    spiral inductor, up to \qty{50}{\percent} improved SRF and \qty{6.5}{\percent} increased inductance among our test
-    samples. We base our conclusions on several simulations and an extensive set of practical measurements.
+    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 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}
@@ -76,31 +76,38 @@ fanSimultaneousWirelessPower2024, leeSimpleWirelessPower2017, liWirelessPowerTra
 maierContributionSystemDesign2019, mooreApplicationsWirelessPower2019, mouEnergyEfficientAdaptiveDesign2017,
 mouWirelessPowerTransfer2015, mullenEffectMisalignmentInductive, rezmeritaSelfMutualInductance2017,
 zhangWirelessPowerTransfer2019}.
-While working on a novel application of Inductive WPT in a Inertial Hardware Security Module (IHSM) as previously
-published in\textcite{gotteCantTouchThis2022}, we found ourselves presented with an unusual set of constraints
-attempting WPT through a rotating joint using a PCB inductor---a set of constraints that does not yet seem to be
-addressed adequately in the existing literature on inductive WPT.
+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.
 
-Our application poses the challenge of transferring power between a stationary part of an
-IHSM\cite{gotteCantTouchThis2022} and part that rotates at high speed (\qtyrange{1000}{3000}{\rpm}) through a pair of
-WPT inductors located on the IHSM's axis of rotation. The large centrifugal acceleration prohibits the use of liquid
-electrolyte capacitors on the rotating part, and makes heavy components such as large 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.
+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.
 
-While planar inductors are usually considered approximately axisymmetric, we found that at the small turn counts in our
-application, 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,
-an issue which is exacerbated by radial misalignment of the coils.
+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.
 
-In other inductive wireless power transfer 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 reduce this effect to almost
-nothing. Finally, the output ripple caused by this oscillation can be filtered through a voltage regulator or by using a
-large decoupling capacitor on the secondary side if the application can accomodate such components on the rotating part.
+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
@@ -110,18 +117,18 @@ it is generally assumed that the two coils remain quasi-stationary with respect
 
 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.
+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. Applying some math, we show that we can layout a twisted inductor for
-any number of twists that is co-prime to the inductor's turn count, and that in fact, our approach opens up a large
-design space for inductor layouts that interpolate between planar spiral inductors on one end, and planar toroidal
-inductors on the other end. Our approach thus generalizes a number of previous approaches to the design of planar
-inductors.
+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
@@ -136,9 +143,9 @@ rotational symmetry in rotating wireless power transfer interface as well as qua
 provide detailed layout instructions, including a mathematical analysis of the available parameter space and an
 analytical model of both inductance and DC equivalent series resistance of our scheme. Validating our scheme, we provide
 laboratory measurements of the basic parameters of a number of test specimens comparing our scheme to conventional
-techniques. We furhter performed a number of FEM simulations to validate our inductance and ESL approximations. Finally,
-to analyze the degree of rotational symmetry in our proposed scheme, we provide the results of a large number of
-automated measurements of coupling between pairs of inductors under various rotations, offsets, distances and load
+techniques. We furhter present the results of FEM simulations to validate our inductance and ESL approximations.
+Finally, to analyze the degree of rotational symmetry in our proposed scheme, we provide the results of a large number
+of automated measurements of coupling between pairs of inductors under various rotations, offsets, distances and load
 conditions.
 
 \section{Related Work}
@@ -161,20 +168,21 @@ inductor with many turns on multiple layers, which improves compactness and leak
 rise to increased distributed capacitance as now turns with a large voltage differential are layered right on top of
 each other.
 
-Back then, a number of ways were devised to decrease distributed capacitance in multilayer inductors. These methods can
-be divided into two general categories: Optimizing the connecting order of turns to minimize the voltage differential
-between adjacent turns---a technique that is still used to this day\cite{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 ICs, vias limit the achievable turn count when they need to be placed in-line inside the turns as opposed
-to on the inside or outside because a PCB's minimum trace/space widths are usually much smaller than the smallest
-feasible via, consisting of a minimum-size drill surrounded by a minimum-size annular ring.} and increasing ESR through
-the thin trace sections that are necessary to accomodate the via structure, as well as the layer pairing limitations
-when blind vias are used in multilayer PCBs.
+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}
@@ -225,12 +233,12 @@ kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querf
 
 \subsection{PCB inductor design for wireless power transfer}
 
-For wireless power transfer, air-core inductors with or without ferrite magnetic shielding are the standard solution.
-Since in most applications, an air gap of several millimeters between the sending and receiving assemblies is expected,
-adding a ferrite core does not result in a large improvement in coupling. Meanwhile, in many WPT applications,
-especially for charging portable devices or medical implants, some misalignment between the sending and receiving coils
-is expected. Using the available space with an air-core inductor that has a large cross-sectional area reduces the
-impact of this misalignment.
+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
@@ -239,28 +247,30 @@ 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, PCB inductors tend to have poor DC resistance. A PCBs' thin but wide trace cross-section helps with
-skin effect compared to a solid conductor. However, PCBs can still not approach the performance of litz wire used in
-high-frequency WPT coils, which commonly use wire diameters in the tens of micrometer
-range\cite{zhaoDesignOptimizationLitzWire2023}. \textcite{lopeFrequencyDependentResistancePlanar2014} and
-\textcite{nomotoSplittingConductorsCoils2024} propose a mitigation that aims to emulate a litz wire's structure in
-large, high-current PCB inductors, but their mitigation is heavily limited by the structure size achievable in common
-PCB manufacturing processes\cite{nguyenReviewComparisonSolid2020}.
+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
+the range of $4$ to $5$ \cite{mumbyDielectricPropertiesFR41989}, which increases the distributed capacitance compared to
 a pure air-core inductor by approximately that same factor.
 
 \subsection{Twisted Inductors in RFIC Design}
 
-Planar inductors are commonly used in RF ICs. In RFIC design, the major challenges are area optimization and precisely
-predicting the inductor's characteristics during the design phase. Common optimizations include applying a variable
-trace pitch to reduce distributed capacitance\cite{lopez-villegasImprovementQualityFactor2000}, and applying variable
-trace width to decrease equivalent series resistance while preserving total inductance and quality
-factor\cite{hsuAnalyticalDesignAlgorithm2008}.
+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
@@ -304,24 +314,23 @@ transfer for the charging of electric vehicles
 transfer system usually replaces the conventional wired charging connector, which improves the systems' user experience
 given the strong force required to seat or unseat these rather large connectors, as well as the heft of the required
 water-cooled cables. In this application, size is of (almost) no concern, but at charging rates up to tens of kilowatt,
-efficiency becomes critical. When charging an EV at a rate of 10 kW, an efficiency improvement of just $0.1\%$
-corresponds to a reduction in power dissipation of 10 W. Besides the monetary cost of the power lost this way, each
-small improvement enables a reduction in size of heat sinks and other cooling components, which directly translates to a
-decrease in cost.
+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 for Inductive Power Transfer}
+\subsection{Air-Core Inductors in WPT}
 
-Across application areas, air-core inductors are often used for wireless power transfer since in most applications, an
-air gap of several millimeters or more is expected, and adding a ferrite core would not change the system's performance
-by much in these circumstances. A common way to use ferrites in WPT applications is by magnetically shielding the
-inductor's back side with a ferrite plate such that the field does not extend beyond the coil's back side, thereby
-increasing the intended mutual inductance while simultaneously reducing eddy current losses when the WPT coils are
-placed near metal
+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 will provide a detailed derivation of the layout of twisted inductors. We can approach this layout
+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
@@ -372,12 +381,12 @@ spiral inductor in the first two columns.
 Extending the above parametrization of a spiral inductor's layout, we propose planar \emph{twisted inductors} based on
 two core observations:
 
-\begin{itemize}
-    \item When using an archimedean spiral, multiple such spirals using the same pitch can be interleaved by spreading
-        out their start and end points at regular angular intervals.
-    \item In a two-layer spiral inductor (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{itemize}
+\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
@@ -452,7 +461,7 @@ slightly, but the contribution of these vias will remain small in practical appl
 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
+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}
@@ -512,15 +521,16 @@ case.
 
 To allow for easy design with twisted inductors and to speed up the laboratory prototyping we performed for this paper,
 we created a tool that generates arbitrary twisted inductor layouts, and that is able to output these layouts as PCB
-footprint files for the open source KiCad EDA CAD tool. We integrated the ESR and ESL approximations as derived above
-with our tool, so that it provides immediate design feedback when generating inductors. In order to minimize ESR and
-maximize PCB area utilization, we made the tool automatically calculate the largest possible trace width when given a
-minimum clearance specification.
+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}\todo{Cite gerbonara}. To support the FEM simulations that are described in the next section
-below, our tool contains functionality to map gerbonara's geometry representation into that of gmsh\todo{Cite gmsh}, the
-FEM mesher that we chose to interface with Elmer FEM\todo{Cite Elmer}.
+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.
@@ -881,8 +891,8 @@ basket-wound inductors used in the early days of radio. Our \emph{twisted} induc
 conventional planar inductors including conventional single- or two-layer planar spiral inductors as well as planar
 toroidal inductors. For inversion count parameter $k\ge 2$, twisted inductors produce magnetic field distributions that
 have better rotational symmetry along the inductor's main axis compared to either single- or two-layer planar spiral
-inductors, which yields lower output ripple in Wireless Power Transfer through rotating joints and enables the use of
-smaller and lighter secondary-side circuitry, improving efficiency.
+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}