Paper text WIP

paper/paper.tex

@@ -174,7 +174,7 @@ To improve layer utilization, a common technique in PCB inductor design is to us
 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
+traces on the PCB's bottom layer as follows. Figure\ \ref{fig_nk_interleave_illust} shows both a simple and a two-layer
 spiral inductor. 
 
 \begin{align}
@@ -184,15 +184,6 @@ spiral inductor.
     \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}
 
 Extending the above parametrization of a spiral inductor's layout, we propose planar \emph{twisted inductors} based on
@@ -201,9 +192,9 @@ two core observations:
 \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.
+    \item In a two-layer spiral inductor as shown in Figure\ \ref{fig_nk_interleave_illust}, 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
@@ -216,11 +207,13 @@ values of $n$ and $k$.
 
 \begin{figure}
     \begin{center}
-        \includegraphics[width=0.7\linewidth]{figures/nk_interleave_illust.pdf}
+        \includegraphics[width=\linewidth]{figures/nk_interleave_illust.pdf}
     \end{center}
-    \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.}
+    \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
+    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.}
     \label{fig_nk_interleave_illust}
 \end{figure}
 
@@ -234,7 +227,7 @@ of a curve parameter $t \in [0, nk]$ as $f(t) = (i, j) = (t \mod n, t \mod k)$.
 must not intersect anywhere. Thus, the system of congruences
 
 \begin{align}
-    t &\equiv i \mod n
+    t &\equiv i \mod n\\
     t &\equiv j \mod k
 \end{align}
 
@@ -245,7 +238,13 @@ Remainder Theorem, which states that this solution is unique when $k$ and $n$ ar
     \begin{center}
         \includegraphics[width=0.7\linewidth]{figures/nk_chinese_remainder_illust.pdf}
     \end{center}
-    \caption{TODO}
+    \caption{Illustration of the winding pattern of two twisted inductors. The upper plots show the inductor's actual
+    layout with the traces on each side of the substrate colored in red (top) and blue (bottom), respectively. The lower
+    plots show the same traces, but \emph{unwrap} the annulus by plotting the traces' polar coordinates on cartesian
+    axes. The left axis labels show the normalized radius, with $0$ being at the inductor's inner diameter, $1$ being at
+    its outer diameter on the top layer, and $-1$ being at its outer diameter on the bottom layer. The top and right
+    axes labels show the axis scaled to match indices $i\in\left[0, n\right]$ and $j\in\left[0, k\right]$,
+    respectively.}
     \label{fig_nk_chinese_remainder_illust}
 \end{figure}
 
@@ -289,16 +288,50 @@ air-core inductor), $n$ is the number of turns, $d_\text{avg}$ is the \emph{aver
 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
+the closer we get to the center, the larger is also the fraction of the field lines returning as leakage flux 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.
+the inductor's geometry. To account for twisted inductors, we can separate the inductor into a set of $2k$ simple planar
+spiral inductor \emph{branches} that are connected in series by the twisted inductor's vias. Compared to a simple spiral
+inductor, for each branch, the inductance according to Equation\ \ref{eqn_mohan_approx} stays the same except that the
+factor $n^2$ drops to $\left(\frac{n}{2k}\right)^2$ because the $n$ windings are evenly distributed across the $2k$
+branches. Let us now make two assumptions. First, we will assume that the flux linkage between both sides of the
+inductor is approximately one. This assumption is grounded in the fact that for practical designs, the substrate
+thickness will be small compared to the inductor's diameter. Second, we will for now ignore the spiral inductor's field
+asymmetry and assume that the flux linkage between two intertwined branches on the same side of the substrate is
+approximately one. In our measurements below we show that for simple spiral inductors this asymmetry, while problematic
+in our application, is small in absolute terms, and grows smaller with increasing turn count.
+
+Based on these two assumptions, we can model the twisted inductor as a set of $2k$ series-connected spiral inductors
+that are perfectly coupled, with full flux linkage. This results in the total series inductance gaining back the factor
+$\frac{1}{2k^2}$ that each branch lost, resulting in identical inductances for a simple planar spiral inductor and a
+twisted inductor with the same size and turn count according to Equation\ \ref{eqn_mohan_approx}. This approximation
+introduces an error due to the imperfect flux linkage between the two sides of the substrate, and between two spiral
+branches located at an angular offset from each other. In our experiments, we found that for our test inductors,
+compared to inductances measured with an LCR meter, this error is below \qty{10}{\percent} for $n=5$ turns or more, and
+for our test samples matches the performance of Equation\ \ref{eqn_mohan_approx} for the simple planar spiral inductor
+case.
 
 \subsection{CAD Integration}
 
+To allow for easy design with twisted inductors, and to speed up the laboratory prototyping we performed for this paper,
+we created a tool that generates arbitrary twisted inductor layouts, and that is able to output these layouts as PCB
+footprint files for the open source KiCad EDA CAD tool. We integrated the ESR and ESL approximations as derived above
+with our tool, so that it provides immediate design feedback when generating inductors. In order to minimize ESR and
+maximize PCB area utilization, we made the tool automatically calculate the largest possible trace width when given a
+minimum clearance specification.
+
+To handle outputting PCB geometry in a format that can be read from KiCad, we utilized the open source EDA file format
+library \emph{gerbonara}\todo{Cite gerbonara}. To support the FEM simulations that are described in the next section
+below, our tool contains functionality to map gerbonara's geometry representation into that of gmsh\todo{Cite gmsh}, the
+FEM mesher that we chose to interface with Elmer FEM\todo{Cite Elmer}.
+
+Our inductor design tool is available in this paper's supplementary material as well as at the git repository linked at
+the end of this paper.
+
 \section{FEM Simulation}
 
 To validate our analytical approximations, we performed a series of FEM simulations in Elmer FEM. For a number of
@@ -493,6 +526,26 @@ for $k=7$ trace pairs.
     \label{fig_rms_ripple_n5}
 \end{figure}
 
+\section{Future Work}
+
+As part of our inductor design tool, we extended the EDA file format library gerbonara with code to automatically map
+gerbonara's geometry description to the gmsh FEM mesher. This code may be of independent interest since it allows for
+the extraction of FEM meshes from PCBs in any file format supported by gerbonara such as KiCad's native file format, as
+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
+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 $k\gg n$
+approximating planar toroidal inductors. In particular, except for the slight curvature of our twisted inductor's
+traces, a twisted inductor with $(n, k)=(1, n')$ \emph{is} effectively a planar toroidal inductor with turn count $n'$.
+
+We suspect that for some choices of parameters, this effect might lead to an appreciable increase in useful inductance
+as well as potentially interesting high-frequency behavior, and we aim at producing additional simulations and new
+measurements for some of these choices of parameters in a future paper.
+
 \section{Conclusion}
 
 In this paper, we introduced a novel layout approach for planar, multi-layer inductors inspired by classic basket-wound
@@ -536,7 +589,7 @@ set of tools for the generation of twisted inductor layouts that we wrote can be
         \includegraphics[width=\linewidth]{figures/symmetry_10turn_n_twist.pdf}
     \end{center}
     \caption{Coupled RMS output voltage of three pairs of matching inductors with $n=10$ turns each and $k=0$, $k=1$,
-    and $k=3$, respectively, shown as in Figure\ \ref{symmetry_3turn_n_twist}}
+    and $k=3$, respectively, shown as in Figure\ \ref{fig_symmetry_3turn_n_twist}}
     \label{fig_symmetry_10turn_n_twist}
 \end{figure}