spirals WIP

paper/paper.tex

@@ -29,6 +29,7 @@
 \newcommand{\degree}{\ensuremath{^\circ}}
 \newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}}
 \newcommand{\partnum}[1]{\texttt{#1}}
+\newcommand{\todo}[1]{\textbf{TODO}\footnote{#1}}
 
 \begin{document}
 
@@ -138,6 +139,36 @@ and other cooling components, which directly translates to a decrease in cost.
 
 \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 also ignore the placement
+of the two electrical ports of the inductor for now. 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 taking into account that our spiral 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}
 
 \subsubsection{Ohmic Resistance}