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@ -83,12 +83,18 @@ Achieving Rotation-Invariant Coupling using Twisted Multi-Layer PCB Inductors}
\end{figure}
Inductive Wireless Power Transfer (WPT) is a widely used technology supported by a large corpus of research literature
\cite{awuahNovelCoilDesign2023, batraEffectFerriteAddition2015, curranModelingCharacterizationPCB2015,
fanSimultaneousWirelessPower2024, leeSimpleWirelessPower2017, liWirelessPowerTransfer2015,
maierContributionSystemDesign2019, mooreApplicationsWirelessPower2019, mouEnergyEfficientAdaptiveDesign2017,
mouWirelessPowerTransfer2015, mullenEffectMisalignmentInductive, rezmeritaSelfMutualInductance2017,
zhangWirelessPowerTransfer2019}.
% FIXME todo too many refs, weed out ones that don't appear elsewhere.
\cite{
awuahNovelCoilDesign2023,
batraEffectFerriteAddition2015,
curranModelingCharacterizationPCB2015,
fanSimultaneousWirelessPower2024,
leeSimpleWirelessPower2017,
liWirelessPowerTransfer2015,
maierContributionSystemDesign2019,
mooreApplicationsWirelessPower2019,
mouEnergyEfficientAdaptiveDesign2017,
mouWirelessPowerTransfer2015,
zhangWirelessPowerTransfer2019}.
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
@ -160,8 +166,8 @@ operating frequency and improving its efficiency at lower operating frequencies.
\subsection{Contributions}
Our contributions in this paper include:
\begin{itemize}
\item 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.
\item We introduce twisted inductors, a planar inductor layout that improves rotational symmetry in WPT through
rotating joins, and promises improved high-frequency behavior in other applications.
\item We provide detailed instructions for the construction of such layouts, including a mathematical analysis of
the available parameter space.
\item We provide an analytical model of inductance and DC equivalent series resistance of our scheme.
@ -176,8 +182,6 @@ Our contributions in this paper include:
\section{Related Work}
% TODO cite \cite{mullenEffectMisalignmentInductive} below (misaligned coils)
\subsection{Inductive WPT in Practice}
Inductive WPT has been proposed in a large number of
@ -204,36 +208,41 @@ transfer for the charging of electric vehicles
(EVs)\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017}. In this application, the wireless power
transfer system usually replaces the conventional wired charging connector, which improves the systems' user experience
given the strong force required to seat or unseat these rather large connectors, as well as the heft of the required
water-cooled cables. In this application, size is of (almost) no concern, but at charging rates up to tens of kilowatt,
efficiency becomes critical. When charging an EV at a rate of \qty{10}{\kilo\watt}, an efficiency improvement of just
$0.1\%$ corresponds to a reduction in power dissipation of \qty{10}{\watt}. Besides the monetary cost of the power lost
this way, each small improvement enables a reduction in size of heat sinks and other cooling components, which directly
translates to a decrease in cost.
water-cooled cables. In this application, size is of little concern, but at charging rates up to tens of kilowatt,
efficiency becomes critical.
%When charging an EV at a rate of \qty{10}{\kilo\watt}, an efficiency improvement of just
%$0.1\%$ corresponds to a reduction in power dissipation of \qty{10}{\watt}. Besides the monetary cost of the power lost
%this way, each small improvement enables a reduction in size of heat sinks and other cooling components, which directly
%translates to a decrease in cost.
\subsection{Air-Core Inductors 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
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}.
millimeters or more is expected\cite{curranModelingCharacterizationPCB2015}. Especially in low-power application such as
mobile device charging, the size and weight of ferrites is an obstacle to their use, and at lower power levels losses
are less of a concern.
A common way to use ferrites in WPT applications is by magnetically shielding the inductor's back side with a ferrite
plate such that the field does not extend beyond the coil's back side, thereby increasing the intended mutual inductance
while simultaneously reducing eddy current losses when the WPT coils are placed near metal
objects\cite{batraEffectFerriteAddition2015,leeSimpleWirelessPower2017,muehlmannMutualCouplingModeling2012}. Similar to
how the trace layouts of planar WPT coils are optimized to improve power transfer efficiency, the layout of ferrite
components has been proposed for optimization\cite{batraEffectFerriteAddition2015}.
\subsection{PCB inductor design for wireless power transfer}
Today, air-core inductors are the standard solution in inductive WPT links. Since in most WPT applications an air gap of
several millimeters between the sending and receiving assemblies is expected, adding a ferrite core does not result in a
large improvement in coupling. Instead, the impact of this misalignment is reduced by maximizing the area of the
air-core inductors used.
air-core inductors used, or by tiling multiple
inductors\cite{curranModelingCharacterizationPCB2015,wangNovelRotatingWireless2024,zhangDynamicWirelessPower2025}.
WPT inductors tend to be mostly planar coils with only a few layers, so implementing them in a PCB process seems
natural. Using a PCB for the inductor has the potential to reduce implementation cost since PCBs are cheap, and they can
also serve as structural support.
Implementing inductors in PCBs has several disadvantages. First, due to the limited layer count of common PCB
processes and due to structure size limitations, the number of windings that can be fit into a given volume is much
lower than in wire-wound inductors. Second, due to a PCB's copper layers being thin compared to its dielectric
substrate\footnote{common values are \qtyrange{15}{30}{\micro\meter} copper thickness and
also serve as structural support. However, implementing inductors in PCBs has several disadvantages. First, due to the
limited layer count of common PCB processes and due to structure size limitations, the number of windings that can be
fit into a given volume is much lower than in wire-wound inductors. Second, due to a PCB's copper layers being thin
compared to its dielectric substrate\footnote{common values are \qtyrange{15}{30}{\micro\meter} copper thickness and
\qtyrange{600}{1600}{\micro\meter} substrate thickness} PCB inductors tend to have poor DC resistance, albeit the thin
copper layer decreases skin effect losses compared to a solid, round conductors of the same cross-sectional area.
However, PCBs can still not approach the performance of litz wire used in high-frequency WPT coils, which commonly use
@ -250,13 +259,12 @@ through the PCB's substrate, not air. The relative permittivity $\epsilon_r$ of
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}
\subsection{Planar Inductors in RFIC Design}
Beyond WPT, planar inductors are commonly used in radio frequency integrated circuits (RFICs). In RFIC design, the major
challenges are area optimization and precisely predicting the inductor's characteristics during the design phase. Common
optimizations include applying a variable trace pitch 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}.
optimizations include applying a variable trace pitch\cite{lopez-villegasImprovementQualityFactor2000} and variable trace
width\cite{hsuAnalyticalDesignAlgorithm2008}.
In RFICs, inductors are commonly designed as \emph{balanced} inductors with a grounded central node. Such designs
interleave two counter-wound planar spiral inductors on the same layer with the help of some jumper connections on a
@ -609,16 +617,17 @@ $k=1$ to $k=3$ irrespective of turn count. From these measurements we can conclu
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
twisted inductors. We found that going from a single-layer spiral inductor to a two-layer spiral inductor decreases the
self-resonant frequency, this effect being more pronounced with higher turn count. Intuitively, this makes sense if we
consider the mechanics of inductor self-resonance. The primary contributor to self resonance, particularly in higher
turn count inductors, is capacitive coupling between the inductor's windings. In a single-layer spiral inductor, this
effect gets partially mitigated since the strongest coupling exists between adjacent windings, which here have only a
small voltage differential as only a fraction of the inductor's total voltage appears across each winding. Compared to
this, when the inductor is constructed as a simple two-layer inductor with $k=1$, now the start and end windings of the
inductor, which have the highest voltage differential, are located right on top of each other with the substrate in
between. Making things worse, common PCB substrates have a relative permittivity much larger than air (usually around
$4$).
twisted inductors. It is well-known that self-resonant frequency decreases when going from a single-layer spiral
inductor to a two-layer spiral inductor while keeping inductance and dimensions
constant\cite{zhangImprovedCompensationMethod2025}. Our measurements show this effect, with it being more pronounced
with higher turn count. Intuitively, this makes sense if we consider the mechanism of inductor self-resonance. The
primary contributor to self resonance, particularly in higher turn count inductors, is capacitive coupling between the
inductor's windings. In a single-layer spiral inductor, this effect gets partially mitigated since the strongest
coupling exists between adjacent windings, which here have only a small voltage differential as only a fraction of the
inductor's total voltage appears across each winding. Compared to this, when the inductor is constructed as a simple
two-layer inductor with $k=1$, now the start and end windings of the inductor, which have the highest voltage
differential, are located right on top of each other with the substrate in between. Making things worse, common PCB
substrates have a relative permittivity much larger than air (usually around $4$).
We observe that this decrease in high-frequency performance is eventually counteracted by increasing inversion count
$k$. While our test samples focused on smaller turn counts, we observe a notable increase from a self-resonant frequency