jaseg/nice-coils
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

Last updates

Browse source
This commit is contained in:
jaseg 2024-12-11 17:11:29 +01:00
parent 7a54792439
commit ac92394866
1 changed files with 72 additions and 66 deletions
Download patch file Download diff file Expand all files Collapse all files

138
paper/paper.tex
View file

@ -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