Compact paper a bit

paper/paper.tex

@@ -13,10 +13,7 @@
 \usepackage{amssymb,amsmath}
 \usepackage{eurosym}
 \usepackage{wasysym}
-
 \usepackage[binary-units]{siunitx}
-\DeclareSIUnit{\baud}{Bd}
-\DeclareSIUnit{\year}{a}
 \usepackage{commath}
 \usepackage{graphicx,color}
 \usepackage{colortbl}
@@ -27,6 +24,9 @@
 \usepackage{hyperref}
 \usepackage{makecell}
 
+\DeclareSIUnit{\baud}{Bd}
+\DeclareSIUnit{\year}{a}
+\DeclareSIUnit{\rpm}{rpm}
 \renewcommand{\floatpagefraction}{.8}
 \newcommand{\degree}{\ensuremath{^\circ}}
 \newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}}
@@ -68,53 +68,47 @@ published in\textcite{gotteCantTouchThis2022}, we found ourselves presented with
 attempting WPT through a rotating joint using a PCB inductor---a set of constraints that does not yet seem to be
 addressed adequately in the existing literature on inductive WPT.
 
-Our application poses the challenge of transferring power between a stationary and a rotating part of an
-IHSM\cite{gotteCantTouchThis2022} through a pair of WPT inductors located on the IHSM's axis of rotation. 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.
+Our application poses the challenge of transferring power between a stationary part of an
+IHSM\cite{gotteCantTouchThis2022} and part that rotates at high speed (\qtyrange{1000}{3000}{\rpm}) through a pair of
+WPT inductors located on the IHSM's axis of rotation. The large centrifugal acceleration prohibits the use of liquid
+electrolyte capacitors on the rotating part, and makes heavy components such as large 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.
 
-We found that at such small turn counts, a simple spiral PCB inductors exhibits a \emph{slightly} asymmetric field,
-which means that the coupling coefficient of two such inductors oscillates at one cycle per revolution when the
-inductors are rotated on-axis, even if both inductors are perfectly coaxially aligned.
+While planar inductors are usually considered approximately axisymmetric, we found that at the small turn counts in our
+application, 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,
+an issue which is exacerbated by radial misalignment of the coils.
 
-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 is the single major factor shaping the magnetic field, and evens out any small effect
-asymmetric windings might have. In wire-wound inductors, the often higher turn count and the tightly packed, circular
-wires reduce 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 where those components can be
-accomodated on the rotating part given the centrifugal forces resulting from a concrete design's rate of rotation.
+In other inductive wireless power transfer 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 reduce 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 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
-WPT electric vehicle charges,
+WPT electric vehicle chargers,
 % TODO cite
-it is generally assumed that the two coils remain (almost) 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
-joints\cite{fanSimultaneousWirelessPower2024},
-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 their solder joints and the surrounding assembly due to the same centrifugal acceleration. 
-Any imbalance caused by tolerances in the placement of heavy components or the precise shape of their solder fillets
-can cause detrimental vibration.
+joints\cite{fanSimultaneousWirelessPower2024}, but here the focus lies on higher power budgets than our application
+requires, which in practice requires more space and a ferrite or laminated iron core.
 
 \subsection{Twisted inductors}
 
-To solve this conundrum, we applied a principle inspired by rectangular or octagonal RFIC inductor design as well as by
-the polygonal basket-woven air coils used in early radio sets. 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. 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, 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 number of previous approaches to the design of planar 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. To fit our unique use case,
+we applied a principle which the polygonal basket-woven air coils used in early radio sets are based on to an approach
+inspired by contemporary planar inductor layouts. 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, 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 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
@@ -229,7 +223,7 @@ Looking at such WPT inductors, they tend to be mostly planar coils with only a f
 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 a number of disadvantages. First, due to the limited layer count of common PCB
+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, PCB inductors tend to have poor DC resistance. A PCBs' thin but wide trace cross-section helps with
@@ -350,8 +344,8 @@ To improve layer utilization, a common technique in PCB inductor design is to us
 inductor's spiral trace, instead of only using the bottom layer for a straight jumper trace. Using both layers this way
 allows for wider traces, which lowers resistive losses. We can accomodate this optimization in our definition by
 re-defining our normalized radius to allow both positive and negative values, defining negative values to designate
-traces on the PCB's bottom layer as follows. Figure\ \ref{fig_nk_interleave_illust} shows both a simple and a two-layer
-spiral inductor. 
+traces on the PCB's bottom layer as follows. Figure\ \ref{fig_nk_combined} shows both a simple and a two-layer
+spiral inductor in the first two columns. 
 
 \begin{align}
     \varphi &= 2\pi n t\\
@@ -368,42 +362,50 @@ two core observations:
 \begin{itemize}
     \item When using an archimedean spiral, multiple such spirals using the same pitch can be interleaved by spreading
         out their start and end points at regular angular intervals.
-    \item In a two-layer spiral inductor as shown in Figure\ \ref{fig_nk_interleave_illust}, we can adjust the turn
-        count of the pair of traces to move the end point of the bottom layer trace anywhere on the inductor's outer
-        radius.
+    \item In a two-layer spiral inductor (Figure\ \ref{fig_nk_combined}), we can adjust the turn count of the
+        pair of traces to move the end point of the bottom layer trace anywhere on the inductor's outer radius.
 \end{itemize}
 
 Combining these two observations, we find that by choosing a number $k$ of inversions, i.e. layer jumps, that is coprime
 to the number of total turns of the inductor $n$, we achieve a layout where all $k$ pairs of top and bottom-layer traces
 naturally connect in series, with the resulting spirals on the top and bottom layers interleaving cleanly. Figure\
-\ref{fig_nk_interleave_illust} shows a layout with $n=3$ turns with both a single inversion ($k=1$) as in a conventional
+\ref{fig_nk_combined} shows a layout with $n=3$ turns with both a single inversion ($k=1$) as in a conventional
 two-layer inductor, and with $k=2$ inversions, creating two interleaved spirals on both the top and the bottom layer of
 the PCB. Figure\ \ref{fig_nk_complex_illust} in Appendix\ \ref{sec_appendix_layout_examples} shows additional layout
 examples for other values of $n$ and $k$. For $k=0$, we get a standard single-layer planar spiral inductor for any turn
 count $n$, and for $k=1$ we get a standard two-layer planar spiral inductor for any turn count $n$. In this paper, we
 will call all layouts with $k\ge 2$ \emph{Twisted Inductors}.
 
+%\begin{figure}
+%    \begin{center}
+%        \includegraphics[width=\figurescale]{figures/nk_interleave_illust.pdf}
+%    \end{center}
+%    \caption{single-layer spiral inductor's layout (left), a conventional two-layer planar inductor's layout (middle),
+%    and a twisted inductor with two inversions (right). All three inductors have $n=3$ turns. Traces on the PCB top
+%    side are shown in red, traces on the bottom side in blue. In the twisted inductor, each layer contains two
+%    archimedean spirals that interleave at a regular spacing. The four spirals of the inductor are connected in series
+%    such that they form three total turns.}
+%    \label{fig_nk_interleave_illust}
+%\end{figure}
+
 \begin{figure}
     \begin{center}
-        \includegraphics[width=\figurescale]{figures/nk_interleave_illust.pdf}
+        \includegraphics[width=\figurescale]{figures/nk_combined.pdf}
     \end{center}
-    \caption{single-layer spiral inductor's layout (left), a conventional two-layer planar inductor's layout (middle),
-    and a twisted inductor with two inversions (right). All three inductors have $n=3$ turns. Traces on the PCB top
-    side are shown in red, traces on the bottom side in blue. In the twisted inductor, each layer contains two
-    archimedean spirals that interleave at a regular spacing. The four spirals of the inductor are connected in series
-    such that they form three total turns.}
-    \label{fig_nk_interleave_illust}
+    \caption{Inductor layouts for several sets of turn count $n$ and inversion count $k$. The top row shows the actual
+    trace layout in cartesian coordinates, the bottom row visualizes the winding schema.
+    }
+    \label{fig_nk_combined}
 \end{figure}
 
-Figure\ \ref{fig_nk_chinese_remainder_illust} illustrates how we arrive at the coprimality requirement.
-\todo{Cleanly handle $k=0$ case.}
-If we plot the spiral in polar coordinates on a cartesian plot we observe that for a $n$-turn coil with $k$ inversions,
-the trace crosses the $\varphi$ axis once for each inversion, wrapping around $r$. Likewise, it crosses the $r$ axis
-once for each turn of the inductor, wrapping around $\varphi$. Based on this, we can re-label the angular axis in steps
-from $0$ to $k$, and re-label the radial axis in steps from $0$ to $n$. Labelling the new angular axis $i$ and the new
-radial axis $j$, in the resulting integer lattice, the trace has slope $1$. We can state the trace's trajectory as a
-function of a curve parameter $t \in [0, nk]$ as $f(t) = (i, j) = (t \mod n, t \mod k)$. To produce a valid inductor,
-the trace must not intersect anywhere. Thus, the system of congruences
+Figure\ \ref{fig_nk_combined} illustrates how we arrive at the coprimality requirement. \todo{Cleanly handle $k=0$
+case.} If we plot the spiral in polar coordinates on a cartesian plot we observe that for a $n$-turn coil with $k$
+inversions, the trace crosses the $\varphi$ axis once for each inversion, wrapping around $r$. Likewise, it crosses the
+$r$ axis once for each turn of the inductor, wrapping around $\varphi$. Based on this, we can re-label the angular axis
+in steps from $0$ to $k$, and re-label the radial axis in steps from $0$ to $n$. Labelling the new angular axis $i$ and
+the new radial axis $j$, in the resulting integer lattice, the trace has slope $1$. We can state the trace's trajectory
+as a function of a curve parameter $t \in [0, nk]$ as $f(t) = (i, j) = (t \mod n, t \mod k)$. To produce a valid
+inductor, the trace must not intersect anywhere. Thus, the system of congruences
 
 \begin{align}
     t &\equiv i \mod n\\
@@ -413,19 +415,19 @@ the trace must not intersect anywhere. Thus, the system of congruences
 must have a unique solution $t \in [0, nk]$ for all $(i, j)$. This statement corresponds exactly to the Chinese
 Remainder Theorem, which states that this solution is unique when $k$ and $n$ are coprime.
 
-\begin{figure}
-    \begin{center}
-        \includegraphics[width=0.8\figurescale]{figures/nk_chinese_remainder_illust.pdf}
-    \end{center}
-    \caption{Illustration of the winding pattern of two twisted inductors. The upper plots show the inductor's actual
-    layout with the traces on each side of the substrate colored in red (top) and blue (bottom), respectively. The lower
-    plots show the same traces, but \emph{unwrap} the annulus by plotting the traces' polar coordinates on cartesian
-    axes. The left axis labels show the normalized radius, with $0$ being at the inductor's inner diameter, $1$ being at
-    its outer diameter on the top layer, and $-1$ being at its outer diameter on the bottom layer. The top and right
-    axes labels show the axis scaled to match indices $i\in\left[0, n\right]$ and $j\in\left[0, k\right]$,
-    respectively.}
-    \label{fig_nk_chinese_remainder_illust}
-\end{figure}
+%\begin{figure}
+%    \begin{center}
+%        \includegraphics[width=0.8\figurescale]{figures/nk_chinese_remainder_illust.pdf}
+%    \end{center}
+%    \caption{Illustration of the winding pattern of two twisted inductors. The upper plots show the inductor's actual
+%    layout with the traces on each side of the substrate colored in red (top) and blue (bottom), respectively. The lower
+%    plots show the same traces, but \emph{unwrap} the annulus by plotting the traces' polar coordinates on cartesian
+%    axes. The left axis labels show the normalized radius, with $0$ being at the inductor's inner diameter, $1$ being at
+%    its outer diameter on the top layer, and $-1$ being at its outer diameter on the bottom layer. The top and right
+%    axes labels show the axis scaled to match indices $i\in\left[0, n\right]$ and $j\in\left[0, k\right]$,
+%    respectively.}
+%    \label{fig_nk_chinese_remainder_illust}
+%\end{figure}
 
 \subsubsection{Ohmic Resistance}
 
@@ -716,13 +718,9 @@ indicating a contribution from flux linkage.
         $75$&$90$&$53$  &$320$& $461$&          $76.2$&          $8.75$&           $0.72$\\
         $75$&$90$&$53$  &$480$& $\mathbf{470}$& $92.9$&          $8.00$&           $0.84$\\
     \end{tabular}
-    \caption{Inductor sample design parameters and measured characteristics for a number of physically larger,
-        ring-shaped inductors. $L$ and $R_\text{ESR}$ have been measured with a Keysight U1733C handheld LCR meter.
-        $f_\text{Res}$ has been measured with a LiteVNA VNA. $C_p$ has been calculated for the simple parallel LC
-        resonator model from $f_\text{Res}$ and $L$. $f_\text{Res}$ was not be measured for the $n=1$ case since these
-        are just planar toroidal inductors, which show different resonance characteristics compared to planar spiral or
-        multi-turn twisted inductors. Bolded values highlight the best performance among the coils of one size. Shaded
-        rows indicate conventional planar toroidal ($n=1$) or two-layer planar spiral inductors ($k=1$).}
+    \caption{Parameters and measurement results of a set of larger sample inductors. Bold values indicate best
+        performance at a given size. Shaded rows indicate conventional planar toroidal ($n=1$) or two-layer planar
+        spiral inductors ($k=1$).}
     \label{tab_wide_coils}
 \end{table}
 
@@ -757,10 +755,9 @@ the voltage ratio between the coupled inductors' input and output voltages graph
 \ref{fig_symmetry_3turn_n_twist} for a set of three-turn inductors with multiple inversion numbers $k$. A plot for a set
 of 10-turn inductors is shown in Figure\ \ref{fig_symmetry_10turn_n_twist} in the Appendix. A key observation here is
 that while the asymmetry in the inductor's field is impossible to distinguish visually in field plots, the ripple
-induced by rotation is considerable. Figure\ \ref{fig_field_plot_3d} shows a 3D plot of an inductor's coupling. While in
-the plot the field looks perfectly rotationally symmetric, the sharp dropoff with radial offset (equivalent to a large
-gradient) magnifies any small asymmetry and leads to the ripple voltages we have listed in Table\ \ref{tab_coupons},
-in some cases amounting to several percent of total RMS output voltage.
+induced by rotation is considerable. The sharp dropoff of coupling with radial offset magnifies any small asymmetry and
+leads to the ripple voltages we have listed in Table\ \ref{tab_coupons}, in some cases amounting to several percent of
+total RMS output voltage.
 
 \begin{figure}
     \begin{center}
@@ -817,19 +814,19 @@ for $k=7$ inversions.
     \label{fig_k_ripple_plot}
 \end{figure}
 
-\begin{figure}
-    \begin{center}
-        \includegraphics[width=.6\figurescale]{figures/field_plot_3d_n5_k0.pdf}
-    \end{center}
-    \caption{The coupling between a pair of identical coils (here two simple spiral inductors with $n=5$ and $k=0$)
-    visualized in three dimensions. The $x$ and $y$ axis show in-plane displacement, and the $z$ axis shows output
-    amplitude in arbitrary units. Height and rotation are fixed to \qty{1}{\milli\meter} and \qty{0}{\degree},
-    respectively. The most prominent aspects of this plot are that coupling falls off steeply with distance, and that
-    the rotation-dependent variation is small in comparison. The circular valley around the central peak is the region
-    where one inductor is mostly outside the other inductors, and intersects the field lines returning from the other
-    inductor's back, leading to a negative coupling coefficient.}
-    \label{fig_field_plot_3d}
-\end{figure}
+%\begin{figure}
+%    \begin{center}
+%        \includegraphics[width=.6\figurescale]{figures/field_plot_3d_n5_k0.pdf}
+%    \end{center}
+%    \caption{The coupling between a pair of identical coils (here two simple spiral inductors with $n=5$ and $k=0$)
+%    visualized in three dimensions. The $x$ and $y$ axis show in-plane displacement, and the $z$ axis shows output
+%    amplitude in arbitrary units. Height and rotation are fixed to \qty{1}{\milli\meter} and \qty{0}{\degree},
+%    respectively. The most prominent aspects of this plot are that coupling falls off steeply with distance, and that
+%    the rotation-dependent variation is small in comparison. The circular valley around the central peak is the region
+%    where one inductor is mostly outside the other inductors, and intersects the field lines returning from the other
+%    inductor's back, leading to a negative coupling coefficient.}
+%    \label{fig_field_plot_3d}
+%\end{figure}
 
 \begin{figure}
     \begin{center}