Revision work

paper/paper.tex

@@ -55,14 +55,14 @@ Achieving Rotation-Invariant Coupling using Twisted Multi-Layer PCB Inductors}
 \maketitle
 
 \begin{abstract}
-    We present \emph{twisted inductors}, a generalization of planar single- and two-layer spiral inductors as well as
-    planar toroidal inductors. Compared to conventional planar spiral inductors, twisted inductors generate a magnetic
-    field with better rotational symmetry, resulting in decreased output ripple in Wireless Power Transfer (WPT)
-    applications with an axially rotating receiver. Additionally, we found that twisted inductors can simultaneously
-    yield a significantly improved Self-Resonant Frequency (SRF) and a higher inductance in the same area as a
-    conventional planar spiral inductor, up to \qty{50}{\percent} improved SRF and \qty{6.5}{\percent} increased
-    inductance among our test samples. We base our conclusions on several simulations and an extensive set of practical
-    measurements.
+    We present \emph{twisted inductors}, a planar inductor layout that interleaves multiple spiral traces across two
+    layers, increasing Self-Resonant Frequency (SRF), providing higher inductance, and improving rotational field
+    symmetry compared to conventional layouts. Twisted inductors generalize both conventional planar spiral inductors as
+    well as planar toroidal inductors. We experimentally show that in Wireless Power Transfer (WPT) through an axially
+    rotating joint in Inertial Hardware Security Modules (IHSMs), the improved symmetry of twisted inductors results in
+    decreased output ripple. We further provide measurements of 39 test coupons showing that twisted inductors improve
+    SRF by up to \qty{50}{\percent} and increase inductance by up to \qty{6.5}{\percent} compared to conventional planar
+    spiral inductors.
 \end{abstract}
 
 \section{Introduction}
@@ -96,31 +96,31 @@ published by \textcite{gotteCantTouchThis2022}, we found ourselves presented wit
 attempting WPT through a rotating joint using a planar inductor implemented in a Printed Circuit Board (PCB)---a set of
 constraints that does not seem to be addressed adequately in the existing literature on inductive WPT yet.
 
-Inertial Hardware Security Modules are a hardware security primitive that discourages tampering with a payload (e.g.\ a
-single-board computer) by rotating a tamper-sensing enclosure around the payload. The tamper-sensing enclosure
+Inertial Hardware Security Modules are a hardware security primitive that discourages tampering with a payload such as 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.
+ATM that responds to attempts at opening its vault by dispensing dye over the bank notes within, rendering them
+unusable.
 
-In our IHSM implementation, the tamper-sensing enclosure rotates at \qtyrange{1000}{3000}{\rpm}. The rotating enclosure
-is powered through a pair of WPT inductors located on the IHSM's axis of rotation. The large centrifugal acceleration
-prohibits the use of batteries or liquid electrolyte capacitors on the rotating part, and makes heavy components such as
-large Multilayer Ceramic Capacitors (MLCCs) challenging to balance. To reduce manufacturing cost of both parts, and to
-reduce weight and thereby inertia as well as susceptibility to vibration in the rotating part, we decided to use
-inductors that are directly patterned onto the IHSM's printed circuit boards. The primary constraint that results from
-this choice is that the PCB manufacturing processes' pattern resolution results in a strict upper limit to the turn
-count that can be achieved in an inductor with a given area.
+In our IHSM implementation, the tamper-sensing enclosure rotates at \qtyrange{1000}{3000}{\rpm}. The tamper sensing
+circuit on 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. Planar inductors that
+are patterned directly into a PCB provide a cost-effective and lightweight solution to this problem, but the coarse
+pattern resolution of PCBs results in a strict upper limit to the turn count that can be achieved in an inductor with a
+given area.
 
-Planar inductors are usually considered approximately axisymmetric. In our application, we found that at small turn
-counts, the asymmetry in a planar spiral inductors's field is large enough that the resulting oscillation of the
-coupling coefficient of two such inductors with the inductor's revolution leads to voltage ripple on the secondary side.
-Radial misalignment of the coils further exacerbates this issue.
+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.
 
 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,
+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.
 
@@ -137,48 +137,45 @@ gap between existing literature on low-power planar WPT inductor design and high
 
 \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. 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 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 super-set to a number of
-previous approaches to the design of planar inductors.
+In this paper, 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 twists increases the spacing between the beginning
-and end of the inductor's conductor, where the majority of the inductor's AC current flows. This decreases the parasitic
-capacitance of the inductor and raises its Self-Resonant Frequency (SRF), raising its maximum possible operating
-frequency and improving its efficiency at lower operating frequencies. We note that the principle behind this reduction
-in distributed capacitance coincides with the intuition that led to the creation of honeycomb or basket-woven inductors
-in early radio sets more than a hundred years ago, before the invention of ferrites.
+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}
 % TODO itemize this.
-In this paper, we introduce twisted inductors, a novel technique of laying out planar inductors that both improves
-rotational symmetry in rotating wireless power transfer interface as well as quality factor in other applications. We
-provide detailed layout instructions, including a mathematical analysis of the available parameter space and an
-analytical model of both inductance and DC equivalent series resistance of our scheme. Validating our scheme, we provide
-laboratory measurements of the basic parameters of a number of test specimens comparing our scheme to conventional
-techniques. We further present the results of Finite Element Method (FEM) simulations to validate our inductance and ESR
-approximations. Finally, to analyze the degree of rotational symmetry in our proposed scheme, we provide the results of
-a large number of automated measurements of coupling between pairs of inductors under various rotations, offsets,
-distances and load conditions.
+In this paper, we introduce twisted inductors, a planar inductor layout that both improves rotational symmetry in
+rotating wireless power transfer interface as well as quality factor in other applications. We provide detailed layout
+instructions, including a mathematical analysis of the available parameter space and an analytical model of both
+inductance and DC equivalent series resistance of our scheme. Validating our scheme, we provide laboratory measurements
+of the basic parameters of 39 test specimens comparing our scheme to conventional layouts. We further present the
+results of Finite Element Method (FEM) simulations to validate our inductance and ESR approximations. Finally, to
+analyze the degree of rotational symmetry in our proposed scheme, we provide the results of a large number of automated
+measurements of coupling between pairs of inductors under various rotations, offsets, distances and load conditions.
 
 \section{Related Work}
 
-% TODO cite fanSimultaneousWirelessPower2024 below (rotating joint)
 % TODO cite \cite{mullenEffectMisalignmentInductive} below (misaligned coils)
 
-\subsection{A Short Historical Diversion on Basket-Woven Air Coils}
+\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 constructed as
-air-core coils. While air core inductors are immune to core saturation, the poor magnetic permeability of air
+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
@@ -217,11 +214,11 @@ layer of such windings forms a helix whose pitch is equal to the wire diameter.
 helical scheme reversing at the coil ends, but uses a helical pitch larger than the wire diameter to form a structure
 similar to a spool of sewing thread.
 
-Other winding techniques include honeycomb and basket woven coils, some contemporary examples of which are shown in
-Figure\ \ref{fig_illust_honeycomb_basket}. In a honeycomb coil, like in an universal winding, subsequent winding layers
-are wound at a criss-cross pattern. The characteristic feature of honeycomb coils is that the winding machine is
-adjusted to produce large air gaps between adjacent windings on the same layer. When multiple layers like this are
-stacked, a three-dimensional rhomboid pattern results that is vaguely reminiscent of a honeycomb's structure.
+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
@@ -230,31 +227,29 @@ basket-woven coils, the mandrel can be pulled out after the coil is finished. Li
 structure can be made mechanically stable with some lacquer, with the turns carrying the layers where they cross.
 
 Both construction techniques apply similar principles to those leading to the improved high-frequency behavior of
-twisted inductors that we describe in this paper. Interestingly, the winding schemes of both honeycomb and basket-woven
-coils are also governed by the same coprimality condition between the number of turns and the number of inversions
-within each turn that we describe for our twisted inductors below, although we could not find an example in contemporary
-literature where this condition was explicitly stated \cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
-kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querfurthCoilWindingDescription1954}.
+twisted inductors that we describe in this paper.\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}.}
 
 \subsection{PCB inductor design for wireless power transfer}
 
-Air-core inductors with or without ferrite magnetic shielding are the standard solution in inductive WPT links. Since in
-most applications, an air gap of several millimeters between the sending and receiving assemblies is expected, adding a
-ferrite core does not result in a large improvement in coupling. Meanwhile, in many WPT applications, especially for
-charging portable devices or medical implants, some misalignment between the sending and receiving coils is expected.
-Using the available space with an air-core inductor that has a large cross-sectional area reduces the impact of this
-misalignment.
+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.
 
-Looking at such WPT inductors, they tend to be mostly planar coils with only a few layers, so implementing them in a PCB
-process seems natural. Using a PCB for the inductor has the potential to reduce implementation cost since PCBs are
-cheap, and they can also serve as structural support.
+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.
 
 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
+processes and due to structure size limitations, the number of windings that can be fit into a given volume is much
 lower than in wire-wound inductors. Second, due to a PCB's copper layers being thin compared to its dielectric
-substrate---common values are \qtyrange{15}{30}{\micro\meter} copper thickness and \qtyrange{600}{1600}{\micro\meter}
-substrate thickness---PCB inductors tend to have poor DC resistance, albeit the thin copper layer provides some
-advantage over a solid, round conductors of the same cross-sectional area at higher frequencies due to skin effect.
+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
@@ -262,17 +257,17 @@ mitigation that aims to emulate a litz wire's structure in large, high-current P
 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
+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{Twisted Inductors in RFIC Design}
 
-Planar inductors are commonly used in radio frequency integrated circuits (RFICs). In RFIC design, the major challenges
-are area optimization and precisely predicting the inductor's characteristics during the design phase. Common
+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 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}.