From ac9239486684a84652a57afb4622f71960c3202f Mon Sep 17 00:00:00 2001 From: jaseg Date: Wed, 11 Dec 2024 17:11:29 +0100 Subject: [PATCH] Last updates --- paper/paper.tex | 138 +++++++++++++++++++++++++----------------------- 1 file changed, 72 insertions(+), 66 deletions(-) diff --git a/paper/paper.tex b/paper/paper.tex index 569033d..eab2127 100644 --- a/paper/paper.tex +++ b/paper/paper.tex @@ -59,7 +59,7 @@ Achieving Rotation-Invariant Coupling using Twisted Multi-Layer PCB Inductors} well as planar toroidal inductors. We experimentally show that in Wireless Power Transfer (WPT) through an axially rotating joint in Inertial Hardware Security Modules (IHSMs), the improved symmetry of twisted inductors results in decreased output ripple. We further provide measurements of 39 test coupons showing that twisted inductors improve - SRF by up to \qty{50}{\percent} and increase inductance by up to \qty{6.5}{\percent} compared to conventional planar + SRF by up to \qty{58}{\percent} and increase inductance by up to \qty{6.5}{\percent} compared to conventional planar spiral inductors. \end{abstract} @@ -279,11 +279,6 @@ second layer~\cite{daneshDifferentiallyDrivenSymmetric2002,martinMultiturnTwiste in RFIC design is mainly focused on their electrical symmetry, so that the two input ports can be fed with a fully differential signal, with the inductor loading both driver outputs equally across the inductor's frequency range. -Setting the inversion count to $k=1$ in our proposed scheme yields the counterwound scheme that is commonly used for -two-layer planar -inductors~\cite{lopeFirstSelfresonantFrequency2021,sproHighVoltageInsulationDesign2021,leePrintedSpiralWinding2011}, and -which has been used to stack planar coils for more than a century~\cite{flemingPrinciplesElectricWave1910}. - % Note: They note that the main point behind the design is electrical symmetry of the two ports to make driving the % thing differentially cleaner. We should adopt this observation for our inductors, which likewise are electrically % symmetric when compared to a single-layer spiral inductor. @@ -304,20 +299,24 @@ in turn gives rise to increased distributed capacitance as now turns with a larg on top of each other. Before the invention of ferrites, a number of ways were devised to decrease distributed capacitance in multilayer -inductors. These methods can be divided into two general categories: Optimizing the connecting order of turns to -minimize the voltage differential between adjacent turns---a technique that is still used to this -day~\cite{lopeFirstSelfresonantFrequency2021}, and optimizing the winding schema to increase the separation between -turns. The main technique in the first category concerns winding the turns of a cylindrical multilayer inductor not -layer by layer, but instead layering them diagonally, effectively connecting adjacent turns in a diagonal zigzag -pattern. Then as now, wound inductors applying this technique were not feasible to manufacture reliably by machine, but -the technique can be closely replicated in PCB inductors as shown in \textcite{leePrintedSpiralWinding2011}. The main -limiting factors in a PCB implementation are the requirement for a large number of vias inside the inductor's turns -limiting the achievable turn count\footnote{In PCBs, as opposed to integrated circuits (ICs), vias limit the achievable -turn count when they need to be placed in-line inside the turns as opposed to on the inside or outside because a PCB's -minimum trace/space widths are usually much smaller than the smallest feasible via, consisting of a minimum-size drill -surrounded by a minimum-size annular ring.} and increasing equivalent series resistance (ESR) through the thin trace -sections that are necessary to accomodate the via structure, as well as the layer pairing limitations when blind vias -are used in multilayer PCBs. +inductors. These methods can be divided into two general categories: Optimizing the connecting order of turns and +optimizing the winding schema of turns. Both aim at increasing spacing between parts of the coil that have a large +voltage differential. + +The connecting order of turns was optimized at the assembly level by stacking coils in a particular +way~\cite{flemingPrinciplesElectricWave1910} and at the component level by winding coils in a particular way to minimize +the voltage differential between adjacent turns---a technique that is still used to this +day~\cite{lopeFirstSelfresonantFrequency2021}. The main winding optimization in the first category concerns winding the +turns of a cylindrical multilayer inductor not layer by layer, but instead layering them diagonally, effectively +connecting adjacent turns in a diagonal zigzag pattern. Then as now, wound inductors applying this technique were not +feasible to manufacture reliably by machine, but the technique can be closely replicated in PCB inductors as shown in +\textcite{leePrintedSpiralWinding2011}. The main limiting factors in a PCB implementation are the requirement for a +large number of vias inside the inductor's turns limiting the achievable turn count\footnote{In PCBs, as opposed to +integrated circuits (ICs), vias limit the achievable turn count when they need to be placed in-line inside the turns as +opposed to on the inside or outside because a PCB's minimum trace/space widths are usually much smaller than the +smallest feasible via, consisting of a minimum-size drill surrounded by a minimum-size annular ring.} and increasing +equivalent series resistance (ESR) through the thin trace sections that are necessary to accomodate the via structure, +as well as the layer pairing limitations when blind vias are used in multilayer PCBs. This lack of a way to wind high frequency inductors with a machine led to the creation of a number of related winding schemes that include honeycomb and basket woven coils @@ -356,12 +355,11 @@ kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querf \section{Twisted Inductor Design} -In this section, we present a detailed derivation of the layout of twisted inductors. We approach this layout -by construction. Let us first consider a simple, planar, circular spiral coil with a fixed pitch. We will ignore trace +In this section, we present a detailed derivation of the layout of twisted inductors. We approach this layout 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 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. +$x$-Axis on top of one another on different layers, which also helps to minimize the loop area of the inductor's +connections. The trace trajectory of a standard planar spiral inductor can be parameterized in polar coordinates $r, \varphi$ based on an Archimedean spiral: @@ -409,16 +407,19 @@ two core observations: radius. \end{description} +Setting the inversion count to $k=1$ in our proposed scheme yields the conventional two-layer counterwound +scheme~\cite{lopeFirstSelfresonantFrequency2021,sproHighVoltageInsulationDesign2021,leePrintedSpiralWinding2011}. + Combining these two observations, we find that by choosing a number $k$ of inversions, i.e. layer jumps, that is coprime to the number of total turns of the inductor $n$, we achieve a layout where all $k$ pairs of top and bottom-layer traces naturally connect in series, with the resulting spirals on the top and bottom layers interleaving cleanly. Figure\ -\ref{fig_nk_combined} shows a layout with $n=3$ turns with both a single inversion ($k=1$) as in a conventional -two-layer inductor, and with $k=2$ inversions, creating two interleaved spirals on both the top and the bottom layer of -the PCB. Figure\ \ref{fig_nk_complex_illust} in Appendix\ \ref{sec_appendix_layout_examples} shows additional layout -examples for other values of $n$ and $k$. For $k=\frac{1}{2}$, we get a standard single-layer planar spiral inductor for -any turn count $n$, and for $k=1$ we get a standard two-layer planar spiral inductor for any turn count $n$. In this -paper, we will call all layouts with $k\ge 2$ \emph{Twisted Inductors}. The coordinate description of Equation\ -\ref{eqn_twolayer_spiral} thus becomes: +\ref{fig_nk_combined} shows a layout with $n=3$ turns with both a single inversion ($k=1$), which results in a +conventional two-layer inductor, and with $k=2$ inversions, creating two interleaved spirals on both the top and the +bottom layer of the PCB. Figure\ \ref{fig_nk_complex_illust} in Appendix\ \ref{sec_appendix_layout_examples} shows +additional layout examples for other values of $n$ and $k$. For $k=\frac{1}{2}$, we get a standard single-layer planar +spiral inductor for any turn count $n$, and for $k=1$ we get a standard two-layer planar spiral inductor for any turn +count $n$. In this paper, we will call all layouts with $k\ge 2$ \emph{Twisted Inductors}. The coordinate description of +Equation\ \ref{eqn_twolayer_spiral} thus becomes: \begin{align} \varphi &= 2\pi n t\\\nonumber @@ -464,7 +465,10 @@ k)$. To produce a valid inductor, the trace must not intersect anywhere. Thus, t \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. +Remainder Theorem, which states that this solution is unique if and only if $k$ and $n$ are coprime. + +In the following paragraphs, we will derive analytical expressions for Ohmic resistance and inductance of inductors +derived under this schema. %\begin{figure} % \begin{center} @@ -484,8 +488,8 @@ Remainder Theorem, which states that this solution is unique when $k$ and $n$ ar 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 traces of the twisted inductor -does not change. Twisted inductors require two additional vias per inversion, which will increase DC resistance +inductor does not change its turn count nor its dimensions, the combined arc length of all traces of the twisted +inductor does not change. Twisted inductors require two additional vias per inversion, 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. @@ -500,19 +504,20 @@ resistance $R_\text{via}$ we derive a first order approximation of the inductor' \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: +complexity quickly escalates when geometrically complex inductors are analyzed, when realistic wire shapes as opposed to +thin wire or current sheet approximations are used, and when taking into account differing magnetic permeabilities of +air or dielectrics and core materials. Instead of precise analytical models, 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 +In this equation, $c_{1-4}$ denote four empirically determined coeficcients that are specific to 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}. @@ -579,8 +584,8 @@ more reliable than results from FEM and can serve as a baseline for future work We conducted our FEM simulations as follows: \paragraph{Ohmic Resistance} -Determining ohmic resistance by FEM is reasonably easy. In Elmer FEM, we can use the built-in joint static current and -joule heating solver to determine the ohmic resistance at a given current. +In Elmer FEM, we can use the built-in joint static current and joule heating solver to determine the ohmic resistance at +a given current. \paragraph{Inductance} We let Elmer determine inductance by first using its coil solver to determine the volumetric current density in our mesh @@ -596,8 +601,8 @@ inductance according to the well-known relation~\cite{meeekerFiniteElementMethod \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 $k=\frac{1}{2}$ (simple +To experimentally validate our design with real-world inductors, we produced 24 test coupons with a number of variations +of twisted inductors with winding count $n$ between $1$ and $25$, and twist count ranging from $k=\frac{1}{2}$ (simple single-sided 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} corresponding to the space available in our IHSM implementation. @@ -623,18 +628,17 @@ paper, we observe almost identical performance for $k>1$ with decreases of less $k=1$ to $k=3$ irrespective of turn count. From these measurements we can conclude that the flux linkage of twisted inductors almost perfectly matches that of simple two-layer inductors. -Finally, while not particularly relevant for our application, we decided to evaluate the high-frequency performance of -twisted inductors. It is well-known that self-resonant frequency decreases when going from a single-layer spiral -inductor to a two-layer spiral inductor while keeping inductance and dimensions -constant~\cite{zhangImprovedCompensationMethod2025}. Our measurements show this effect, with it being more pronounced -with higher turn count. Intuitively, this makes sense if we consider the mechanism of inductor self-resonance. The -primary contributor to self resonance, particularly in higher turn count inductors, is capacitive coupling between the -inductor's windings. In a single-layer spiral inductor, this effect gets partially mitigated since the strongest -coupling exists between adjacent windings, which here have only a small voltage differential as only a fraction of the -inductor's total voltage appears across each winding. Compared to this, when the inductor is constructed as a simple -two-layer inductor with $k=1$, now the start and end windings of the inductor, which have the highest voltage -differential, are located right on top of each other with the substrate in between. Making things worse, common PCB -substrates have a relative permittivity much larger than air (usually around $4$). +Finally, we decided to evaluate the high-frequency performance of twisted inductors. It is well-known that self-resonant +frequency decreases when going from a single-layer spiral inductor to a two-layer spiral inductor while keeping +inductance and dimensions constant~\cite{zhangImprovedCompensationMethod2025}. Our measurements show this effect, with +it being more pronounced with higher turn count. Intuitively, this makes sense if we consider the mechanism of inductor +self-resonance. The primary contributor to self resonance, particularly in higher turn count inductors, is capacitive +coupling between the inductor's windings. In a single-layer spiral inductor, this effect gets partially mitigated since +the strongest coupling exists between adjacent windings, which here have only a small voltage differential as only a +fraction of the inductor's total voltage appears across each winding. Compared to this, when the inductor is constructed +as a simple two-layer inductor with $k=1$, now the start and end windings of the inductor, which have the highest +voltage differential, are located right on top of each other with the substrate in between. Making things worse, common +PCB substrates have a relative permittivity much larger than air (usually around $4$). We observe that this decrease in high-frequency performance is eventually counteracted by increasing inversion count $k$. While our test samples focused on smaller turn counts, we observe a notable increase from a self-resonant frequency @@ -705,14 +709,16 @@ additional cost and without compromising other performance parameters. \subsection{Inductance and Frequency Behavior of Larger Coils} To investigate the high-frequency behavior of twisted inductors further, we produced and measured 15 additional sample -inductors, this time larger than before, and with more turns. The parameters of these new samples and our measurement -results are shown in Table\ \ref{tab_wide_coils}. In these results, we can identify three clear trends. First, the ESR -of twisted inductors is generally poorer when compared to two-layer spiral inductors. This increase in ESR is due to the -large number of vias used in these sample inductors. It should be noted that while twisted inductors have worse ESR -compared to conventional two-layer inductors, their ESR is still better than that of a single-layer inductor. Our second -observation is that in every set of samples from this second run of physically larger inductors, twisted inductors -outperform conventional inductors in self-resonant frequency by a considerable margin with an increase in SRF of up to -\qty{50}{\percent} in our samples. +inductors that were larger (up to \qty{90}{\milli\meter} outer diameter) and that had a higher turn count (up to 53) +compared to our initial set of samples. The parameters of these new samples and our measurement results are shown in +Table\ \ref{tab_wide_coils}. In these results, we can identify three clear trends. First, the ESR of twisted inductors +is generally poorer when compared to two-layer spiral inductors. This increase in ESR is due to the large number of vias +used in these sample inductors. It should be noted that while twisted inductors have worse ESR compared to conventional +two-layer inductors, in our first set of test coupons we saw that their ESR is still better than that of a single-layer +inductor because the traces can be made wider. Our second observation is that in every set of samples from this second +run of physically larger inductors, twisted inductors outperform conventional planar inductors in self-resonant +frequency by a considerable margin with an increase in SRF of up to \qty{58}{\percent} from our +$d_2=\qty{65}{\milli\meter}$ sample going from $k=1$ to $k=100$. Our third observation is that unlike in the smaller inductors from Table\ \ref{tab_coupons}, in these larger instances, twisted inductors show increased inductance by approximately \qty{3.7}{\percent} for our smallest samples, and