Add related work on old coils, rename trace pair to inversion

paper/paper.tex

@@ -101,6 +101,58 @@ air core inductors that were commonly used in old radio sets.
 
 \section{Related Work}
 
+\subsection{A Short Historical Diversion on Basket-Woven Air Coils}
+
+Since the early days of radio engineering, the parasitic capacitance of inductors has been a point of concern. Going
+back to the early days of wireless telegraphy after the turn of the twentieth century, coils with high inductance were
+needed for the construction of both transmitters and receivers, but the ferrites that would later permit their compact
+construction were still being developed. The ferromagnetic core material of choice back then was laminated iron, which
+was only useful at low frequencies due to eddy current losses. As a result, the inductors in radio circuits of the era
+were constructed as air-core coils. While air core inductors are immune to core saturation, the poor magnetic
+permeability of air leads necessitates many large turns of wire for practical inductance values, which for reasons of
+practicality or leakage inductance often could not be wound in a single layer. Winding an inductor with many turns on
+multiple layers improves compactness and leakage inductance, but in turn gives rise to increased distributed
+capacitance.
+
+Back then, 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 minimize the voltage differential
+between adjacent turns, 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 layer by layer, but instead
+layering them diagonally, effectively connecting adjacent turns in a diagonal zigzag pattern. Then as now, this
+technique was only feasible for winding by hand, and could not be executed reliably by machine.
+
+\begin{figure}
+    \begin{center}
+        \subcaptionbox{A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{
+            \includegraphics[width=\linewidth]{figures/saacke-radiotechnik-3-ledionspule.jpg}}
+        \subcaptionbox{A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{
+            \includegraphics[width=\linewidth]{figures/klein-spulen-schwingkreise-korbspule.jpg}}
+    \end{center}
+    \caption{Illustrations of honeycomb and basket-woven coils from the early days of wireless radio.}
+    \label{fig_illust_honeycomb_basket}
+\end{figure}
+
+This lack of a way to wind high frequency inductors with a machine led to the creation of a number of related winding
+schemes that including honeycomb and basket woven coil. Examples of both from contemporary literature are shown in
+Figure\ \ref{fig_illust_honeycomb_basket}. In a honeycomb coil, subsequent winding layers are wound at a criss-cross
+pattern similar to how in sewing, a spool of thread is wound. The characteristic feature of honeycomb coils is that the
+winding machine is adjusted to produce large air gaps between adjacent windings on the same layer. When multiple layers
+like this are stacked, a rhomboid pattern results that is vaguely reminiscent of a honeycomb's structure.
+
+In basket-woven coils, a mandrel consisting of an odd number of sticks pointing either radially or axially is used, and
+the wire is fed woven between adjacent sticks in an alternating direction. While visually similar to honeycomb coils,
+this winding technique is more suited to homebrew construction and less amenable to mass production by machine. In
+axially basket-woven coils, the mandrel can be pulled out after the coil is finished. Like honeycomb coils, the
+resulting structure can be made mechanically stable with some lacquer, with the turns carrying the layers where they
+cross.
+
+Both construction techniques apply similar principles to those leading to the improved high-frequency behavior of
+twisted inductors. Interestingly, both honeycomb and 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 in contemporary literature, this condition is never explicitly stated.
+
+\subsection{Twisted Inductors in RFIC Design}
+
 \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
@@ -132,8 +184,6 @@ rate of 30 kW, an efficiency improvement of just $0.1\%$ corresponds to a reduct
 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}
 
@@ -197,11 +247,11 @@ two core observations:
         radius.
 \end{itemize}
 
-Combining these two observations, we find that by choosing a number $k$ of top/bottom-layer spiral trace pairs that is
-coprime to the number of total turns of the inductor $n$, we achieve a layout where when we connect all $k$ trace pairs
+Combining these two observations, we find that by choosing a number $k$ of inversions that is coprime to the number of
+total turns of the inductor $n$, we achieve a layout where when we connect all $k$ pairs of top and bottom-layer traces
 in series, the resulting spirals on the top and bottom layers interleave cleanly. Figure\ \ref{fig_nk_interleave_illust}
-shows a layout with $n=3$ turns with both a single trace pair ($k=1$) as in a conventional two-layer inductor, and with
-$k=2$ trace pairs, creating two interleaved spirals on both the top and the bottom layer of the PCB. Figure\
+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 larger
 values of $n$ and $k$.
 
@@ -210,7 +260,7 @@ values of $n$ and $k$.
         \includegraphics[width=\linewidth]{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 trace pairs (right). All three inductors have $n=3$ turns. Traces on the PCB top
+    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.}
@@ -218,8 +268,8 @@ values of $n$ and $k$.
 \end{figure}
 
 Figure\ \ref{fig_nk_chinese_remainder_illust} illustrates how we arrive at the coprimality requirement. If we plot the
-spiral in polar coordinates on a cartesian plot we observe that for a $n$-turn coil with $k$ trace pairs, the trace
-crosses the $\varphi$ axis once for each trace pair, wrapping around $r$. Likewise, it crosses the $r$ axis once for
+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
@@ -252,11 +302,11 @@ Remainder Theorem, which states that this solution is unique when $k$ and $n$ ar
 
 The arc length $l$ of a spiral can be calculated from its turn count $n$ and the average of its inner and outer diameter
 $\frac{2 r_1 + 2 r_2}{2}=r_1+r_2$ as $l = n\pi\left(r_1 + r_2\right)$. Since going from a standard inductor to a twisted
-inductor does not change its turn count or dimensions, the combined arc length of all trace pairs of the twisted
-inductor does not change. Twisted inductors require two additional vias per trace pair, which will increase DC
-resistance slightly, but the contribution of these vias will remain small in practical applications since the overall
-number of vias is still no more than a couple per turn, and since each via only bridges the short distance between the
-inductor's layers.
+inductor does not change its turn count or dimensions, the combined arc length of all traces of the twisted inductor
+does not change. Twisted inductors require two additional vias per inversion, which will increase DC resistance
+slightly, but the contribution of these vias will remain small in practical applications since the overall number of
+vias is still no more than a couple per turn, and since each via only bridges the short distance between the inductor's
+layers.
 
 As a general expression, for a standard or twisted inductor with turn count $n$ and twist count $k$ ($k=0$ for a
 single-layer spiral inductor, and $k=1$ for a standard two-layer spiral inductor), given via resistance $R_\text{via}$
@@ -400,7 +450,7 @@ end windings of the inductor, which have the highest voltage differential, are l
 the substrate in between. Making things worse, common PCB substrates have a dielectric constant much larger than air
 (usually around $4$).
 
-Interestingly, we observe that this decrease in high-frequency performance is counteracted by larger trace pair count
+Interestingly, we observe that this decrease in high-frequency performance is counteracted by larger inversion count
 $k$. While our test samples focused on smaller turn counts, we observe an increase from an SRF of
 \qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$.
 
@@ -414,7 +464,7 @@ performance parameters.
 \begin{table*}
     \begin{tabular}{cc|ccc|}
         Turn Count $n$&
-        Trace pair count $k$&
+        Inversion count $k$&
         Inductance $L \left[\mu H\right]$&
         Q-factor&
         DC resistance $R_\text{ESR} \left[\Omega\right]$\\
@@ -448,9 +498,9 @@ providing a signal at a \qty{300}{\kilo\hertz} carrier frequency to the primary
 function generator as shown in Figure\ \ref{fig_test_schematic}. We measured both the input and output voltages
 of the coupled inductor pair using Keysight 34465A multimeters in AC RMS mode. The results of these measurements, with
 the voltage ratio between the coupled inductors' input and output voltages graphed across one revolution in Figure\
-\ref{fig_symmetry_3turn_n_twist} for a set of three-turn inductors with multiple trace pair amounts $k$. 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 small, the ripple induced by rotation is considerable. Figure\
+\ref{fig_symmetry_3turn_n_twist} for a set of three-turn inductors with multiple inversion amounts $k$. 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 small, 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, ``amplifies'' any small
 asymmetry and leads to the ripple voltages we observed, amounting up to several percent of total RMS output voltage.
@@ -488,8 +538,8 @@ across rotations works, with twisted inductors ($k>1$) showing a further improve
 which prove to be better than simple single-layer spiral inductors. As one would expect, this gain is greatest for
 inductors with low turn count, as their turns deviate the furthest from a set of ideal, concentric circles. For the
 our test inductor with an inner diameter of \qty{15}{\milli\meter} and an outer diameter of \qty{35}{\milli\meter},
-$k=3$ trace pairs already provided an improvement over standard configurations, with still better performance observed
-for $k=7$ trace pairs.
+$k=3$ inversions pairs already provided an improvement over standard configurations, with still better performance observed
+for $k=7$ inversions.
 
 \todo{concrete coupling factor measurements}
 
@@ -537,7 +587,7 @@ well as the Gerber file format supported by the majority of EDA tools.
 
 In the measurements we performed on our set of test inductors, we observed that while at the dimensions we chose, a
 twisted inductor has slightly lower inductance by \qty{2.0}{\percent} for $n=2$, or \qty{0.11}{\percent} for $n=25$,
-when compared to a simple two-layer planar spiral inductor, its inductance \emph{increases} with increasing trace pair
+when compared to a simple two-layer planar spiral inductor, its inductance \emph{increases} with increasing inversion
 count $k$. In one of our test coupons with $(n, k)=(25, 37)$, we even measured \emph{higher} inductance compared to a
 simple two-layer planar spiral inductor. We suspect that this increase in inductance is due to the twists of our twisted
 inductor effectively forming the structure of a planar toroidal inductor, with twisted inductors with $k\gg n$
@@ -579,7 +629,7 @@ set of tools for the generation of twisted inductor layouts that we wrote can be
     \begin{center}
         \includegraphics[width=.75\textwidth]{figures/nk_complex_illust.pdf}
     \end{center}
-    \caption{Layout examples for a number of combinations of turn count $n$ and trace pair count $k$. Note that in this
+    \caption{Layout examples for a number of combinations of turn count $n$ and inversion count $k$. Note that in this
     illustration we chose values for $n$ and $k$ such that all pairs are coprime.}
     \label{fig_nk_complex_illust}
 \end{figure*}