Paper WIP

paper/paper.tex

@@ -145,11 +145,11 @@ are both located on the positive $x$-Axis. We can rotate it so its first port al
 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}
+The trace trajectory of a standard planar spiral inductor can be parameterized in polar coordinates $r, \varphi$ based
+on an Archimedean spiral: \todo{For the lulz, cite Archimedes here}
 
 \begin{equation}
-    r = a\cdot\phi
+    r = a\cdot\varphi
     \label{eqn_arch_spi_basic}
 \end{equation}
 
@@ -160,7 +160,7 @@ radius. To make handling of this easier, we introduce a variable $r' \in \left[0
 normalized to the spiral's width. Let $n$ be the turn count of our inductor. The resulting parametrization is:
 
 \begin{align}
-    \phi &= 2\pi n t\\
+    \varphi &= 2\pi n t\\
     r' &= 1 - t \\
     r &= r_1 + r' \left(r_2 - r_1\right)
     \label{eqn_simple_spiral_ind}
@@ -178,7 +178,7 @@ traces on the PCB's bottom layer as follows. Figure\ \ref{fig_twolayer_spiral} s
 spiral inductor. 
 
 \begin{align}
-    \phi &= 2\pi n t\\
+    \varphi &= 2\pi n t\\
     r' &= 1 - 2 t \\
     r &= r_1 + \left|r'\right| \left(r_2 - r_1\right)
     \label{eqn_twolayer_spiral}
@@ -224,15 +224,40 @@ values of $n$ and $k$.
     \label{fig_nk_interleave_illust}
 \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
+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
+    t &\equiv j \mod k
+\end{align}
+
+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=0.7\linewidth]{figures/nk_chinese_remainder_illust.pdf}
+    \end{center}
+    \caption{TODO}
+    \label{fig_nk_chinese_remainder_illust}
+\end{figure}
 
 \subsubsection{Ohmic Resistance}
 
-The arc length $l$ of a spiral can be calculated from its turn count $n$ and the average of its inner and outer diameters
-$\frac{r_1 + r_2}{2}$ as $l = n\pi\frac{r_1 + r_2}{2}$. 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.
+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.
 
 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 $R_\text{via}$
@@ -244,14 +269,46 @@ we derive a first order approximation of the inductor's DC resistance as follows
 
 \subsubsection{Inductance}
 
+Even for geometrically simple inductors, analytically calculating their inductance is a surprisingly hard problem whose
+complexity quickly escalates with the inductor's geometric complexity, with realistic wire shapes as opposed to
+approximations assuming an infinitely thin wire, or when taking into account differing magnetic permeabilities of
+air or dielectrics and core materials. Instead, a number of approximations are commonly used. A commonly referenced
+approximation for the inductance of planar spiral inductors is given by \textcite{mohanSimpleAccurateExpressions1999},
+whose current-sheet approximation for circular planar spiral inductors we will use here to estimate our inductor's
+inductance. The current-sheet approximation from \textcite{mohanSimpleAccurateExpressions1999} reads:
+
+\begin{equation}
+    \label{eqn_mohan_approx}
+    L = \frac{\mu n^2 d_\text{avg} c_1}{2}\left(\ln\left(c_2/\rho\right)+c_3\rho+c_4\rho^2\right)
+\end{equation}
+
+In this equation, $c_{1-4}$ denote four empirically determined coeficcients that together describe the coil's shape. The
+values for circular coils are $c_{1-4}=(1.00, 2.46, 0.00, 0.20)$. $\mu$ is the magnetic permeability of air (for an
+air-core inductor), $n$ is the number of turns, $d_\text{avg}$ is the \emph{average} turn radius, i.e. $d_\text{avg} =
+2\frac{r_1 + r_2}{2} = r_1 + r_2$. $\rho = \frac{r_2-r_1}{r_2+r_1}$ is the planar spiral inductor's \emph{fill ratio}.
+The fill ratio encodes the fact that the inductor's turns have less flux linkage the closer to the inductor's center we
+get. While turns close to the outside have good flux linkage due to their inner area overlapping well with that of other
+turns, turns close to the center not only have a loop area that is only a fraction of that of turns further outwards,
+the closer we get to the center, the larger is also the fraction of the field lines returning on the outside of the
+inner turn that pass through the inner part of turns further outwards, flipping the sign and contributing
+\emph{negative} mutual inductance.
+
+As Equation\ \ref{eqn_mohan_approx} approximates the inductor's whole set of windings as a single, uniform current
+sheet, the turn count only appears as a single factor of $n^2$ in the equation, with $\rho$ and $c_{1-4}$ correcting for
+the inductor's geometry.
+
 \subsection{CAD Integration}
 
 \section{FEM Simulation}
 
-To validate our analytical approximations, we performed a series of FEM simulations in both Elmer FEM and Simulia CST.
-For a number of inductor layouts, we performed simulations to determine ohmic resistance, inductance, and parasitic
-capacitance. For a subset of these layout variants we additionally performed simulations to determine the coupling
-factor between a pair of identical inductors at a number of different distances and rotations.
+To validate our analytical approximations, we performed a series of FEM simulations in Elmer FEM. For a number of
+inductor layouts, we performed simulations to determine ohmic resistance and inductance. Due to limitations in our
+gmsh/Elmer toolchain, we were unable to run simulations for parasitic capacitance and self-resonance, or for coupling
+behavior of coil pairs. We found that for these cases which require larger, more complex meshes, gmsh would frequently
+crash during meshing, and where we were able to produce meshes, Elmer would only converge for some of them. While these
+are problems that can be solved through either a more skillful description of the problem in gmsh and Elmer, or by using
+more robust software such as Simulia CST, we decided to instead experimentally measure these quantities instead
+(Section\ \ref{sec_experiments}).
 
 \paragraph{Ohmic Resistance}
 Determining ohmic resistance by FEM is reasonably easy. In Elmer FEM, we can use the built-in joint static current and
@@ -274,6 +331,7 @@ Determining parasitic capacitance is more complex.
 \subsection{Coupling}
 
 \section{Experimental Validation}
+\label{sec_experiments}
 
 To experimentally validate our design with real-world inductors, we produced test coupons with a number of variations of
 twisted inductors with winding count $n$ between $1$ and $25$, and twist count ranging from $k=0$ (simple single-sided