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@ -102,7 +102,7 @@ constraints that does not seem to be addressed adequately in the existing litera
Inertial Hardware Security Modules are a hardware security primitive that discourages tampering with a payload such as a 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 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 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 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 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 ATM that responds to attempts at opening its vault by dispensing dye over the bank notes within, rendering them
@ -130,21 +130,28 @@ capacitor on the secondary side if the application can accomodate such component
While there exist a corpus of prior work focusing on efficient power transfer between two coils whose position relative 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 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}, WPT electric vehicle chargers~\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017},
it is generally assumed that the two coils remain quasi-stationary with respect to one another. it is generally assumed that the two coils remain quasi-stationary with respect to one another.
There exists a small body of work on inductive power transfer through rotating There exists a body of work on inductive power transfer through rotating joints but here the focus often lies on higher
joints\cite{ power budgets than our application requires, which in practice requires more space and a ferrite or laminated iron
core~\cite{
fanSimultaneousWirelessPower2024, fanSimultaneousWirelessPower2024,
xiaRotaryWirelessPower2024,
songRotationLightweightWirelessPower2019, songRotationLightweightWirelessPower2019,
wangCoaxialNestedCouplersBased2020,
}.
Often, these rotating joint WPT systems use coaxial structures, but segmented approaches exist, too~\cite{
wangNovelRotatingWireless2024, wangNovelRotatingWireless2024,
yanFreeRotationWirelessPower2023, yanFreeRotationWirelessPower2023,
wangCoaxialNestedCouplersBased2020}, xiaRotaryWirelessPower2024,
but here the focus usually lies on higher power budgets than our application requires, which in practice requires more liWirelessPowerTransfer2021,
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. In lower-power applications, segmented approaches are more common. A key challenge in segmented approaches is the
% FIXME refer to wangNovelRotatingWireless2024,yanFreeRotationWirelessPower2023,liWirelessPowerTransfer2021 as segmented approaches. our system performs better reduction of secondary-side ripple induced when the segments' alignment changes throught one revolution~\cite{
zhangWirelessSensorPower2024,
}, which usually requires additional secondary-side circuitry. This paper introduces 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} \subsection{Twisted inductors}
@ -185,7 +192,7 @@ Our contributions in this paper include:
\subsection{Inductive WPT in Practice} \subsection{Inductive WPT in Practice}
Inductive WPT has been proposed in a large number of Inductive WPT has been proposed in a large number of
scenarios\cite{zhangWirelessPowerTransfer2019,mouWirelessPowerTransfer2015}, each of which comes with a set of unique 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 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 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. position and spacing of the transmitter and receiver coils can easily be controlled down to millimeter precision.
@ -197,7 +204,7 @@ two coils in the charging base and in the phone may be positioned more than a ce
millimeters and potentially not even in parallel planes. millimeters and potentially not even in parallel planes.
Power transfer across large distances is even more of a concern in implantable medical 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 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 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 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. manufacturing techniques, customized electronic components and exotic materials.
@ -205,7 +212,7 @@ 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 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 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 transfer for the charging of electric vehicles
(EVs)\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017}. In this application, the wireless power (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 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 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, water-cooled cables. In this application, size is of little concern, but at charging rates up to tens of kilowatt,
@ -218,16 +225,16 @@ efficiency becomes critical.
\subsection{Core materials in WPT} \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 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 millimeters or more is expected~\cite{curranModelingCharacterizationPCB2015}. Especially in low-power application such
mobile device charging, the size and weight of ferrites is an obstacle to their use, and at lower power levels losses 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. 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 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 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 while simultaneously reducing eddy current losses when the WPT coils are placed near metal
objects\cite{batraEffectFerriteAddition2015,leeSimpleWirelessPower2017,muehlmannMutualCouplingModeling2012}. Similar to 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 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}. components has been proposed for optimization~\cite{batraEffectFerriteAddition2015}.
\subsection{PCB inductor design for wireless power transfer} \subsection{PCB inductor design for wireless power transfer}
@ -235,7 +242,7 @@ Today, air-core inductors are the standard solution in inductive WPT links. Sinc
several millimeters between the sending and receiving assemblies is expected, adding a ferrite core does not result in a 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 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 air-core inductors used, or by tiling multiple
inductors\cite{curranModelingCharacterizationPCB2015,wangNovelRotatingWireless2024,zhangDynamicWirelessPower2025}. 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 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 natural. Using a PCB for the inductor has the potential to reduce implementation cost since PCBs are cheap, and they can
@ -246,36 +253,36 @@ compared to its dielectric substrate\footnote{common values are \qtyrange{15}{30
\qtyrange{600}{1600}{\micro\meter} substrate thickness} PCB inductors tend to have poor DC resistance, albeit the thin \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. 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 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}. wire diameters in the range of tens of micrometer~\cite{zhaoDesignOptimizationLitzWire2023}.
\textcite{lopeFrequencyDependentResistancePlanar2014} and \textcite{nomotoSplittingConductorsCoils2024} propose a \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 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 heavily limited by the structure size achievable in common PCB manufacturing
processes\cite{nguyenReviewComparisonSolid2020}. processes~\cite{nguyenReviewComparisonSolid2020}.
A further factor that limits the high-frequency performance of PCB inductors is distributed capacitance. Not only does a 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 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 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 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. a pure air-core inductor by approximately that same factor.
\subsection{Planar Inductors in RFIC Design} \subsection{Planar Inductors in RFIC Design}
Beyond WPT, planar inductors are commonly used in radio frequency integrated circuits (RFICs). In RFIC design, the major 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 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 optimizations include applying a variable trace pitch~\cite{lopez-villegasImprovementQualityFactor2000} and variable
width\cite{hsuAnalyticalDesignAlgorithm2008}. trace width~\cite{hsuAnalyticalDesignAlgorithm2008}.
In RFICs, inductors are commonly designed as \emph{balanced} inductors with a grounded central node. Such designs 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 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 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 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. differential signal, with the inductor loading both driver outputs equally across the inductor's frequency range.
Setting the inversion count to $k=1$ in our proposed scheme yields the counterwound scheme that is commonly used for Setting the inversion count to $k=1$ in our proposed scheme yields the counterwound scheme that is commonly used for
two-layer planar two-layer planar
inductors\cite{lopeFirstSelfresonantFrequency2021,sproHighVoltageInsulationDesign2021,leePrintedSpiralWinding2011}, and inductors~\cite{lopeFirstSelfresonantFrequency2021,sproHighVoltageInsulationDesign2021,leePrintedSpiralWinding2011}, and
which has been used to stack planar coils for more than a century\cite{flemingPrinciplesElectricWave1910}. which has been used to stack planar coils for more than a century~\cite{flemingPrinciplesElectricWave1910}.
% Note: They note that the main point behind the design is electrical symmetry of the two ports to make driving the % 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 % thing differentially cleaner. We should adopt this observation for our inductors, which likewise are electrically
@ -284,22 +291,22 @@ which has been used to stack planar coils for more than a century\cite{flemingPr
\subsection{A Brief 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 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 concern~\cite{nesperHandbuchDrahtlosenTelegraphie1921,flemingPrinciplesElectricWave1910}. Going back to the early days
wireless telegraphy after the turn of the twentieth century, coils with high inductance were needed for the construction of wireless telegraphy after the turn of the twentieth century, coils with high inductance were needed for the
of both transmitters and receivers, but the ferrites that would later permit their compact construction were still being construction of both transmitters and receivers, but the ferrites that would later permit their compact construction
developed. The ferromagnetic core material of choice back then was laminated iron, which was only useful at low were still being developed. The ferromagnetic core material of choice back then was laminated iron, which was only
frequencies due to eddy current losses. As a result, the inductors in radio circuits of the era were often constructed useful at low frequencies due to eddy current losses. As a result, the inductors in radio circuits of the era were often
as air-core coils. While air-core inductors are immune to core saturation, the poor magnetic permeability of air constructed as air-core coils. While air-core inductors are immune to core saturation, the poor magnetic permeability of
necessitates a large number of wide turns of wire to reach useful inductance values, which for reasons of practicality air necessitates a large number of wide turns of wire to reach useful inductance values, which for reasons of
or leakage inductance often could not be wound as a single layer cylindrical coil. This could be resolved by winding an practicality or leakage inductance often could not be wound as a single layer cylindrical coil. This could be resolved
inductor with many turns on multiple layers, which improves compactness and leakage inductance, but this in turn gives by winding an inductor with many turns on multiple layers, which improves compactness and leakage inductance, but this
rise to increased distributed capacitance as now turns with a large voltage differential are layered right on top of in turn gives rise to increased distributed capacitance as now turns with a large voltage differential are layered right
each other. on top of each other.
Before the invention of ferrites, a number of ways were devised to decrease distributed capacitance in multilayer 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 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 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 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 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 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 pattern. Then as now, wound inductors applying this technique were not feasible to manufacture reliably by machine, but
@ -344,7 +351,7 @@ Both construction techniques apply similar principles to those leading to the im
twisted inductors that we describe in this paper.\footnote{Interestingly, the winding schemes of both honeycomb and 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 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 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, historic literature where this condition was explicitly stated~\cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querfurthCoilWindingDescription1954}.} kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querfurthCoilWindingDescription1954}.}
\section{Twisted Inductor Design} \section{Twisted Inductor Design}
@ -543,16 +550,16 @@ case.
To allow for easy design with twisted inductors and to speed up the laboratory prototyping we performed for this paper, 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 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 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. 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 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. 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 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 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 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 gmsh~\cite{geuzaineGmsh3DFinite2009}, the FEM mesher that we chose to interface with Elmer
FEM\cite{ruokolainenElmerCSCElmerfemElmer2023}. FEM~\cite{ruokolainenElmerCSCElmerfemElmer2023}.
Our inductor design tool is available in this paper's supplementary material as well as at the git repository linked at 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. the end of this paper.
@ -580,7 +587,7 @@ We let Elmer determine inductance by first using its coil solver to determine th
given a test current, then applying its magnetodynamics solver to solve the electromagnetic field. Elmer provides 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 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 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}: inductance according to the well-known relation~\cite{meeekerFiniteElementMethod2015}:
\begin{equation} \begin{equation}
L = \frac{2\cdot U_\text{mag}}{I_\text{test}^2} L = \frac{2\cdot U_\text{mag}}{I_\text{test}^2}
@ -619,7 +626,7 @@ inductors almost perfectly matches that of simple two-layer inductors.
Finally, while not particularly relevant for our application, we decided to evaluate the high-frequency performance of Finally, while not particularly relevant for our application, 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 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 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 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 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 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 inductor's windings. In a single-layer spiral inductor, this effect gets partially mitigated since the strongest