diff --git a/paper/figures/rms_ripple_double_rotation_n10_r4.pdf b/paper/figures/rms_ripple_double_rotation_n10_r4.pdf index b473ce5..ec7f898 100644 Binary files a/paper/figures/rms_ripple_double_rotation_n10_r4.pdf and b/paper/figures/rms_ripple_double_rotation_n10_r4.pdf differ diff --git a/paper/figures/rms_ripple_double_rotation_n5_r4.pdf b/paper/figures/rms_ripple_double_rotation_n5_r4.pdf index 108299b..520d179 100644 Binary files a/paper/figures/rms_ripple_double_rotation_n5_r4.pdf and b/paper/figures/rms_ripple_double_rotation_n5_r4.pdf differ diff --git a/paper/paper.tex b/paper/paper.tex index 8b8b3d5..13e9318 100644 --- a/paper/paper.tex +++ b/paper/paper.tex @@ -87,7 +87,7 @@ in the placement of heavy components will quickly cause a strong vibration. \subsection{Twisted inductors} Applying a principle inspired by rectangular or octagonal RFIC inductor design as well as by the polygonal basket-woven -air coils used in early radio set, we propose a novel way of laying out circular PCB inductors that twists the +air coils used in early radio sets, we propose a novel way of laying out circular PCB inductors that twists the inductor's windings around one another using a ring of vias each on the inside and outside of the inductor's windings. Applying some math, we show that we can layout a twisted inductor for any number of twists that is co-prime to the inductor's turn count. @@ -227,6 +227,21 @@ values of $n$ and $k$. \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. + +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 $R_\text{via}$ +we derive a first order approximation of the inductor's DC resistance as follows. + +\begin{equation} + R_L = n\pi\frac{r_1 + r_2}{2} + \left(2k-1\right)R_\text{via} +\end{equation} + \subsubsection{Inductance} \subsection{CAD Integration} @@ -262,7 +277,8 @@ Determining parasitic capacitance is more complex. 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 -spiral inductor) to $k=37$. +spiral inductor) to $k=37$. All test inductors had an inner diameter of \qty{15}{\milli\meter} and an outer diameter of +\qty{35}{\milli\meter}. \subsection{Inductance, Q-factor and DC resistance} @@ -321,7 +337,7 @@ performance parameters. \qty{35}{\milli\meter} and inner diameter \qty{15}{\milli\meter}.} \end{table*} -\subsection{Coupling and Coupling Variation} +\subsection{Coupling and its Sensitivity to Radial Offset} The key performance criterion in our application is the voltage ripple that appears on the secondary side of a WPT link when one of the inductors is rotating. To experimentally evaluate the magnitude of this ripple in a realistic scenario @@ -329,31 +345,24 @@ across a large set of rotations and relative displacements, we created a test se an old 3D printer, with a fourth rotation axis provided by a small servo that allows us to position two inductor test coupons at arbitrary offsets and angles to one another while measuring their coupling. -To evaluate a realistic scenario, we loaded the secondary inductor with a resistive load of \qty{10}{\ohm}, while -providing a signal at a \qty{300}{\kilo\hertz} carrier frequency to the primary inductor from a Siglent SDG6022X -function generator. We measured both the input and output voltages of the coupled inductor pair using Keysight 34465A -multimeters in AC RMS mode. The results of these measurements, with the voltage ratio between the coupled inductors' -input and output voltages graphed across one revolution in Figure\ \ref{symmetry_3turn_n_twist} for a set of three-turn -inductors and in Figure\ \ref{symmetry_10turn_n_twist} for a set of 10-turn inductors with multiple trace pair amounts -$k$. - -From these graphs we observe slightly lower coupling for $k>0$ compared to a single-layer spiral inductor, which is -in line with our previous inductance measurements. Across one revolution, we find that single-layer spiral inductors -exhibit the worst voltage ripple, with simple two-layer inductors with $k=1$ already improving ripple by a large margin. -Increasing $k$ above $1$ does not decrease the amplitude of this ripple further, but it does shift the ripple into -higher frequencies that are easier to passively filter, as we originally intended. - -\todo{new ripple measurements, concrete coupling factor measurements} -\todo{schematics for illustration of measeurement circuits} - \begin{figure} \begin{center} - %\includegraphics[width=0.7\linewidth]{figures/symmetry_3turn_n_twist.pdf} + \includegraphics[width=.85\linewidth]{figures/test_schematic.pdf} \end{center} - \caption{Coupling test circuit} - \label{symmetry_test_circuit} + \caption{The test schematic used in all measurements. For direct coupling factor measurements, the load resistor was + disconnected. We measure voltage at the output of the function generator to account for drop in its internal output + resistance.} + \label{fig_test_schematic} \end{figure} +To evaluate a realistic scenario, we loaded the secondary inductor with a resistive load of \qty{10}{\ohm}, while +providing a signal at a \qty{300}{\kilo\hertz} carrier frequency to the primary inductor from a Siglent SDG6022X +function generator as shown in Figure\ \ref{fig_test_schematic}. We measured both the input and output voltages +of the coupled inductor pair using Keysight 34465A multimeters in AC RMS mode. The results of these measurements, with +the voltage ratio between the coupled inductors' input and output voltages graphed across one revolution in Figure\ +\ref{fig_symmetry_3turn_n_twist} for a set of three-turn inductors with multiple trace pair amounts $k$. A plot for a +set of 10-turn inductors is shown in Figure\ \ref{fig_symmetry_10turn_n_twist} in the Appendix. + \begin{figure} \begin{center} \includegraphics[width=\linewidth]{figures/symmetry_3turn_n_twist.pdf} @@ -362,22 +371,37 @@ higher frequencies that are easier to passively filter, as we originally intende inductors with one inductor rotating w.r.t.\ the other. The inductors have $n=3$ turns each and $k=0$, $k=1$, and $k=3$, respectively. For each $k$, voltage curves are plotted for a number of different radial offsets between the two inductor's centers.} - \label{symmetry_3turn_n_twist} + \label{fig_symmetry_3turn_n_twist} \end{figure} +From these graphs we observe slightly lower coupling for $k>0$ compared to a single-layer spiral inductor, which is +in line with our previous inductance measurements. Across one revolution, we find that single-layer spiral inductors +exhibit the worst voltage ripple, with simple two-layer inductors with $k=1$ already improving ripple by a large margin. +Increasing $k$ above $1$ does not decrease the amplitude of this ripple further, but it does shift the ripple into +higher frequencies that are easier to passively filter, as we originally intended. + +\subsection{Total Coupling Variation} + +To further analyze the behavior of our test inductors under offset and rotation, we had our measurement setup sweep +through the full range of rotation of each of the two inductors when placed at a fixed height of \qty{1}{\milli\meter} +and radial offset of \qty{4}{\milli\meter}. The resulting plots show the variation in RMS output voltage compared to its +mean across all rotations as a percentage plotted against both angular dimensions. Figure\ \ref{fig_rms_ripple_n3} shows +the resulting coupling plot for a set of three-turn inductors, and Figure\ \ref{fig_rms_ripple_n5} for a set of +five-turn inductors. Measurements for 10- and for 25-turn inductors are shown in Figures \ref{fig_rms_ripple_n10} and +\ref{fig_rms_ripple_n25} in the Appendix. + +From these plots, we can draw a number of conclusions. First, our primary objective of reducing coupling variation +across rotations works, with twisted inductors ($k>1$) showing a further improvement over simple two-layer inductors, +which prove to be better than simple single-layer spiral inductors. As one would expect, this gain is greatest for +inductors with low turn count, as their turns deviate the furthest from a set of ideal, concentric circles. For the +our test inductor with an inner diameter of \qty{15}{\milli\meter} and an outer diameter of \qty{35}{\milli\meter}, +$k=3$ trace pairs already provided an improvement over standard configurations. + +\todo{concrete coupling factor measurements} + \begin{figure} \begin{center} - \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}} - \label{symmetry_10turn_n_twist} -\end{figure} - - -\begin{figure} - \begin{center} - \includegraphics[width=\linewidth]{figures/field_plot_3d_n3_k4.pdf} + \includegraphics[width=.6\linewidth]{figures/field_plot_3d_n3_k4.pdf} \end{center} \caption{The coupling between a pair of identical coils (here with $n=3$ and $k=4$) visualized in three dimensions. The $x$ and $y$ axis show in-plane displacement, and the $z$ axis shows output amplitude in arbitrary units. Height @@ -391,25 +415,43 @@ higher frequencies that are easier to passively filter, as we originally intende \begin{figure} \begin{center} - \includegraphics[width=\linewidth]{figures/test_schematic.pdf} + \includegraphics[width=.75\linewidth]{figures/rms_ripple_double_rotation_n3_r4.pdf} \end{center} - \caption{The test schematic used in all measurements. For direct coupling factor measurements, the load resistor was - disconnected. We measure voltage at the output of the function generator to account for drop in its internal output - resistance.} - \label{fig_test_schematic} + \caption{RMS ripple magnitude as a percentage of mean RMS output voltage, plotted against the rotation of each of + the two inductors. The two coils were kept at a constant \qty{4}{\milli\meter} radial offset, and the output coil + was loaded with a \qty{10}{\ohm} load. All RMS ripple plots in this paper share the same color scale to allow for + visual comparison. This figure shows four variants of 3-turn coils, plots for $n=5$ can be found in Figure\ + \ref{fig_rms_ripple_n5} and plots for $n=\{10,25\}$ in Figures \ref{fig_rms_ripple_n10} and \ref{fig_rms_ripple_n25} + in the Appendix.} + \label{fig_rms_ripple_n3} +\end{figure} + +\begin{figure} + \begin{center} + \includegraphics[width=.75\linewidth]{figures/rms_ripple_double_rotation_n5_r4.pdf} + \end{center} + \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 5-turn coils.} + \label{fig_rms_ripple_n5} \end{figure} -% rms_ripple_double_rotation_n25_r4.pdf -% rms_ripple_double_rotation_n5_r4.pdf -% rms_ripple_double_rotation_n3_r4.pdf \section{Conclusion} +In this paper, we introduced a novel layout approach for planar, multi-layer inductors inspired by classic basket-wound +inductors used in the early days of radio. Our \emph{twisted} inductors produce field distributions that have better +rotational symmetry along the inductor's main axis compared to either simple single-layer spiral inductors or +counter-wound two-layer spiral inductors. Furthermore, we found that our sample twisted inductors have slightly higher +self-resonant frequency compared to both traditional layouts. We base this evaluation on laboratory measurements on a +set of 24 test inductors, which include an automated, four-dimensional mapping of the coupling between a pair of +identical inductors. We provide both an analytical description of twisted inductor construction as well as a set of +Open-Source tools for their design. + \section*{Availability} This is version \texttt{\input{version.tex}\unskip} of this paper, generated on \today. -% The git repository with the -% LaTeX source for this paper as well as our data analysis and demo code can be found at: +The git repository with the LaTeX source for this paper, the data analysis code underlying our measurements as well the +set of tools for the generation of twisted inductor layouts that we wrote can be found at: +\todo{link here} % \center{\url{https://git.jaseg.de/nice-coils.git}} \printbibliography[heading=bibintoc] @@ -421,11 +463,38 @@ This is version \texttt{\input{version.tex}\unskip} of this paper, generated on \begin{figure*} \begin{center} - \includegraphics[width=\textwidth]{figures/nk_complex_illust.pdf} + \includegraphics[width=.75\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*} +\section{Supplemental plots} + +\begin{figure} + \begin{center} + \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}} + \label{fig_symmetry_10turn_n_twist} +\end{figure} + +\begin{figure} + \begin{center} + \includegraphics[width=.75\linewidth]{figures/rms_ripple_double_rotation_n10_r4.pdf} + \end{center} + \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 10-turn coils.} + \label{fig_rms_ripple_n10} +\end{figure} + +\begin{figure} + \begin{center} + \includegraphics[width=.75\linewidth]{figures/rms_ripple_double_rotation_n25_r4.pdf} + \end{center} + \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 25-turn coils.} + \label{fig_rms_ripple_n25} +\end{figure} + \end{document}