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@ -95,38 +95,38 @@ Inductive Wireless Power Transfer (WPT) is a widely used technology supported by
mouEnergyEfficientAdaptiveDesign2017,
mouWirelessPowerTransfer2015,
zhangWirelessPowerTransfer2019}.
While working on an application of Inductive WPT in a Inertial Hardware Security Module (IHSM) as previously
While working on an application of inductive WPT in a Inertial Hardware Security Module (IHSM) as previously
published by \textcite{gotteCantTouchThis2022}, we found ourselves presented with an unusual set of constraints
attempting WPT through a rotating joint using a planar inductor implemented in a Printed Circuit Board (PCB)---a set of
constraints that does not seem to be addressed adequately in the existing literature on inductive WPT yet.
Inertial Hardware Security Modules are a hardware security primitive that discourages tampering with a payload 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
continuously monitors itself for tampering using sensors such as tamper-sensing meshes~\cite{TamperResistance2020} and
accelerometers. When the tamper-sensing enclosure signals a tamper alarm to the payload, the payload immediately
destroys all sensitive data to prevent the attacker from gaining access to it. In principle, an IHSM is similar to an
ATM that responds to attempts at opening its vault by dispensing dye over the bank notes within, rendering them
unusable.
In our IHSM implementation, the tamper-sensing enclosure rotates at \qtyrange{1000}{3000}{\rpm}. The 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.
In our IHSM implementation, the tamper-sensing enclosure rotates at speeds in the range from
\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 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 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.
In other inductive WPT systems, this issue is mitigated by 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
@ -147,23 +147,22 @@ Often, these rotating joint WPT systems use coaxial structures, but segmented ap
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{
reduction of secondary-side ripple induced when the segments' alignment changes throughout 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.
}, which usually requires additional secondary-side circuitry.
\subsection{Twisted 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.
outside of the inductor's windings. Our approach improves rotation ripple in WPT through a rotating joint. Furthermore,
it provides better performance beyond our particular use case, and improves upon conventional contemporary planar
inductors by 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, gerneralizing 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
@ -178,7 +177,7 @@ Our contributions in this paper include:
\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
\item We validate 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.
@ -240,8 +239,8 @@ components has been proposed for optimization~\cite{batraEffectFerriteAddition20
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
large improvement in coupling. Instead, the impact of this distance as well as 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
@ -253,7 +252,7 @@ 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
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}.
wire diameters in the range of tens of micrometers~\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
@ -361,7 +360,7 @@ width for now, and consider the trace a thin wire. We will assume the inductor's
$x$-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
The trace trajectory of a standard planar spiral inductor can be parametrized in polar coordinates $r, \varphi$ based
on an Archimedean spiral:
\begin{equation}
@ -369,7 +368,7 @@ on an Archimedean spiral:
\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
An Archimedean spiral defined this way always starts at the origin, and it continues to infinity. Let us re-parametrize
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.
@ -495,10 +494,11 @@ vias is still no more than a couple per turn, and since each via only bridges th
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.
resistance $R_\text{via}$ and trace cross-sectional area $A_\text{tr}$, 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}
R_L = n\pi\left(r_1 + r_2\right)\frac{\rho_\text{Cu}}{A_\text{tr}} + \left(2k-1\right)R_\text{via}
\end{equation}
\subsubsection{Inductance}
@ -867,11 +867,12 @@ pitch, as their turns deviate the furthest from a set of ideal, concentric circl
\includegraphics[width=.75\figurescale]{figures/rms_ripple_double_rotation_n3_r4.pdf}
\end{center}
\caption{RMS ripple magnitude as a percentage of mean RMS output voltage, plotted against the rotation of each of
the two inductors. The two coils were kept at a constant \qty{4}{\milli\meter} radial offset, and the output coil
was loaded with a \qty{10}{\ohm} load. All RMS ripple plots in this paper share the same color scale to allow for
visual comparison. This figure shows four variants of 3-turn coils, plots for $n=5$ can be found in Figure\
\ref{fig_rms_ripple_n5} and plots for $n=\{10,25\}$ in Figures \ref{fig_rms_ripple_n10} and \ref{fig_rms_ripple_n25}
in the Appendix.}
the two inductors for three variants of three-turn coils. 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. The plots share the
same color scale to allow for visual comparison.
% , plots for $n=5$ can be found in Figure\ \ref{fig_rms_ripple_n5} and plots for $n=\{10,25\}$ in Figures
% \ref{fig_rms_ripple_n10} and \ref{fig_rms_ripple_n25} in the Appendix.
}
\label{fig_rms_ripple_n3}
\end{figure}