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\begin{document}

\date{}
\title{Wireless Power Transfer with a Twist:
Achieving Rotation-Invariant Coupling using Multi-Layer PCB Inductors}
\maketitle

\begin{abstract}
    % FIXME
\end{abstract}

\section{Introduction}

Inductive wireless power transfer (WPT) is a widely used technology supported by a large corpus of research literature.
% FIXME cite
While working on a novel application of Inductive wireless power transfer in a Inertial Hardware Security Module (IHSM)
as proposed by Götte and Scheuermann, % FIXME cite
we found ourselves presented with an unusual set of constraints around inductive wireless power transfer through a
rotating joint using a PCB inductor that does not yet seem to be addressed adequately in the existing literature on
inductive wireless power transfer.

Our application poses the challenge of transferring power between a stationary and a rotating part. 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 a highly constrained turn count that is limited by the PCB
manufacturing processes' pattern resolution and by ohmic heating.

We found that the limited turn count of PCB inductors results in a \emph{slightly} asymmetric field, which means that
the coupling coefficient of two such inductors oscillates at one oscillation per revolution when the inductors are
rotated on-axis, even if both inductors are perfectly coaxially aligned.

In other inductive wireless power transfer systems, this oscillation 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 or
iron-cored inductors, the core shapes the magnetic field and evens out any such imperfection. In wire-wound inductors,
the (much) higher turn count and circular aspect ratio of the wires reduces 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.

While there exist a number of prior works 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, it is generally assumed
that the two coils remain (almost) stationary with respect to one another throughout the charging process. % FIXME cite

There exists a small body of work on inductive power transfer through rotating joints, % FIXME cite
but here the focus lies on higher power budgets than our application requires, which often requires ferrite or iron-core
inductors.

Our application is unique in that it requires power transfer through a joint that is constantly rotating at high speed,
while we simultaneously want to avoid heavy components on the (rotating) receiver side. (Liquid) electrolytic capacitors
cannot be used due to the large centrifugal acceleration that the rotating part experiences, and other heavy components
such as large ceramic or polymer electrolytic capacitors or ferrite-core power inductors are inadvisable since they will
exert large stresses onto the assembly due to the same centrifugal acceleration, and any imbalance caused by tolerances
in the placement of heavy components will quickly cause a strong vibration.

\subsection{Twisted inductors}

Applying a principle inspired by rectangular or octagonal RFIC inductor design as well as by the polygonal basket-woven
air coils used in early radio set, 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.
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.

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, raising its maximum possible operating frequency and
improving its efficiency at lower operating frequencies. This is the same effect that is exploited in basket-woven
air core inductors that were commonly used in old radio sets.
% FIXME citation on this, citation on basket weaving -> It's hard to find reliable references on that.

\section{Related Work}

\subsection{Inductive Wireless Power Transfer in Practice}

Inductive WPT has been proposed in a large number of scenarios, each of which comese with a set of
unique 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 position and spacing of the transmitter and receiver coils can easily be controlled down to
millimeter precision.

In contrast to this, wireless smartphone charging is a much more demanding application. Here, the total cost of the
system is only secondary, but the receiver's form factor is critical, and total power output as well as efficiency
become major objectives. At the same time, in wireless smartphone charging, position tolerances are very coarse, and the
two coils in the charging base and in the phone may be positioned more than a centimeter off-axis, with a gap of several
millimeters and potentially not even in parallel planes.

Power transfer across large distances is even more of a concern in implantable medical devices. 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 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.

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
transfer for the charging of electric vehicles (EVs). In this application, the wireless power transfer system 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 several kilowatt up to dozens or even a hundred kilowatt, the
transferred power is enormous and consequentially efficiency becomes of utmost importance. When charging an EV at a
rate of 30 kW, an efficiency improvement of just $0.1\%$ corresponds to a reduction in power dissipation of 30 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.

\subsection{Twisted Inductors in RFIC Design}
\subsection{Basket-Woven Air Coils}
\subsection{Air-Core Inductors for Inductive Power Transfer}
\subsection{Ferrite or Iron-Core Inductors for Inductive Power Transfer}

\section{Twisted Inductor Design} 

We can approach twisted inductors 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 connections, inductors are usually designed with both ports close to one
another, so we can also assume its second port aligns with the $x$-Axis.

The trace trajectory of a standard planar spiral inductor can be parameterized in polar coordinates $r, \phi$ based on
an Archimedean spiral: \todo{For the lulz, cite Archimedes here}

\begin{equation}
    r = a\cdot\phi
    \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
this spiral to a curve parameter $t$ with range $\left[0,1\right]$.

inductor has a defined inner radius $r_0$ and outer radius $r_1$, and a
fixed turn count $n$. Let us further re-parameterize the spiral to a curve parameter $t$ with range $\left[0,
1\right]$. Taking into account that the input ports of a spiral inductor are usually placed on the outside of the
spiral, we define the inductor's first port to lie at $\left(\phi, r\right)=\left(0, r_1\right)$, and we define that
this corresponds to $t=0$. The resulting parametrization is:

\begin{align}
       r &= r_1 - \frac{t}{n} \cdot \left(r_1 - r_0\right) \\
    \phi &= 2\pi \cdot n \cdot t
    \label{eqn_simple_spiral_ind}
\end{align}

For integer $n$, the spiral's second port will lie at $\left(\phi, r\right)=\left(0, r_0\right)$, however, other
values of $n$ are possible, which will rotate the second port around the coordinate origin.

%Let us further flip the radial coordinate axis such that the spiral's outer end is at $\phi=0$
%because spiral inductors usually have their input ports at the outside. By normalizing the coordinate axes substituting
%$\phi' = \frac{1}{2\pi}\phi$ and $r' = \left\(r - r_0\right\) \cdot \frac{1}{r_1 - r_0}$:

\subsection{From Spiral to Twisted Inductor}

\begin{figure*}
    \begin{center}
        \includegraphics[width=\textwidth]{figures/nk_simple_illust.pdf}
    \end{center}
\end{figure*}

\begin{figure*}
    \begin{center}
        \includegraphics[width=\textwidth]{figures/nk_complex_illust.pdf}
    \end{center}
\end{figure*}

\subsubsection{Ohmic Resistance}

\subsubsection{Inductance}

\subsection{CAD Integration}

\section{FEM Simulation}

To validate our analytical approximations, we performed a series of FEM simulations in both Elmer FEM and Simulia CST.
For a number of inductor layouts, we performed simulations to determine ohmic resistance, inductance, and parasitic
capacitance. For a subset of these layout variants we additionally performed simulations to determine the coupling
factor between a pair of identical inductors at a number of different distances and rotations.

\paragraph{Ohmic Resistance}
Determining ohmic resistance by FEM is reasonably easy. In Elmer FEM, we can use the built-in joint static current and
joule heating solver to determine the ohmic resistance at a given current.

\paragraph{Inductance}
We let Elmer determine inductance by first using its coil solver to determine the volumetric current density in our mesh
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
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\todo{Find decent source}:

\begin{equation}
    L = \frac{2\cdot U_\text{mag}}{I_\text{test}^2}
\end{equation}

\paragraph{Parasitic Capacitance and Self-Resonant Frequency}
Determining parasitic capacitance is more complex.

\subsection{Coupling}

\section{Experimental Validation}

\subsection{Inductance and Parasitic Capacitance}

\subsection{Self-Resonant Frequency}

\subsection{Coupling}

\section{Conclusion}

\section*{Availability}
This is version \texttt{\input{version.tex}\unskip} of this paper, generated on \today.

% The git repository with the
% LaTeX source for this paper as well as our data analysis and demo code can be found at:

% \center{\url{https://git.jaseg.de/nice-coils.git}}

\printbibliography[heading=bibintoc]

\end{document}
