paper: Work on spiral parametrization

paper/paper.tex

@@ -154,40 +154,75 @@ an Archimedean spiral: \todo{For the lulz, cite Archimedes here}
 \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:
+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
+radius. To make handling of this easier, we introduce a variable $r' \in \left[0,1\right]$ representing the radius
+normalized to the spiral's width. Let $n$ be the turn count of our inductor. 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
+    \phi &= 2\pi n t\\
+    r' &= 1 - t \\
+    r &= r_1 + r' \left(r_2 - r_1\right)
     \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.
+The resulting spiral trace starts at radius $r_2$ on the positive $x$ axis, and spirals inward until it meets $r_1$. In
+its PCB realization, at $r_1$, a via would be placed to connect the end of the spiral trace to a jumper trace on another
+layer of the PCB leading back to the start.
 
-%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}$:
+To improve layer utilization, a common technique in PCB inductor design is to use both layers of the PCB for the
+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_twolayer_spiral} shows both a simple and a two-layer
+spiral inductor. 
+
+\begin{align}
+    \phi &= 2\pi n t\\
+    r' &= 1 - 2 t \\
+    r &= r_1 + \left|r'\right| \left(r_2 - r_1\right)
+    \label{eqn_twolayer_spiral}
+\end{align}
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=0.7\linewidth]{figures/twolayer_spiral.pdf}
+    \end{center}
+    \caption{A single-layer spiral inductor's layout (left), and a two-layer spiral inductor's layout (right). Traces on
+    the PCB top side are shown in red, traces on the bottom side in blue. Both inductors have $n=3$ turns.}
+    \label{fig_twolayer_spiral}
+\end{figure}
 
 \subsection{From Spiral to Twisted Inductor}
 
-\begin{figure*}
-    \begin{center}
-        \includegraphics[width=\textwidth]{figures/nk_simple_illust.pdf}
-    \end{center}
-\end{figure*}
+Extending the above parametrization of a spiral inductor's layout, we propose planar \emph{twisted inductors} based on
+two core observations:
 
-\begin{figure*}
+\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_twolayer_spiral} \todo{refer to only right side,
+        split into (a) and (b) subfigures}, 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 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
+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\
+\ref{fig_nk_complex_illust} in Appendix\ \ref{sec_appendix_layout_examples} shows additional layout examples for larger
+values of $n$ and $k$.
+
+\begin{figure}
     \begin{center}
-        \includegraphics[width=\textwidth]{figures/nk_complex_illust.pdf}
+        \includegraphics[width=0.7\linewidth]{figures/nk_interleave_illust.pdf}
     \end{center}
-\end{figure*}
+    \caption{A conventional two-layer planar inductor's layout (left), and a twisted inductor with two trace pairs
+    (right). 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}
 
 \subsubsection{Ohmic Resistance}
 
@@ -242,4 +277,18 @@ This is version \texttt{\input{version.tex}\unskip} of this paper, generated on
 
 \printbibliography[heading=bibintoc]
 
+\clearpage
+\appendix
+\section{Layout examples}
+\label{sec_appendix_layout_examples}
+
+\begin{figure*}
+    \begin{center}
+        \includegraphics[width=\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
+    illustration we chose values for $n$ and $k$ such that all pairs are coprime.}
+    \label{fig_nk_complex_illust}
+\end{figure*}
+
 \end{document}