Add nice plots

paper/paper.tex

@@ -140,9 +140,13 @@ 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}
+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
@@ -150,7 +154,9 @@ parameterized in polar coordinates $r, \phi$ based on an Archimedean spiral: \to
 \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
+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