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

Finish first proof of paper

Browse source
This commit is contained in:
jaseg 2024-11-13 21:49:17 +01:00
parent 180547dc7a
commit aacb58c567
1 changed files with 102 additions and 88 deletions
Download patch file Download diff file Expand all files Collapse all files

190
paper/paper.tex
View file

@ -320,8 +320,8 @@ on an Archimedean spiral: \todo{For the lulz, cite Archimedes here}
An Archimedean spiral defined this way always starts at the origin, and it continues to infinity. Let us re-parameterize
this spiral to a curve parameter $t$ with range $\left[0,1\right]$, such that $t=0$ corresponds to the start of the
inductor and $t=1$ corresponds to its end. As is customary in PCB inductors, we place the inductor's start on its outer
radius. To make handling of this easier, we introduce a variable $r' \in \left[0,1\right]$ representing the radius
normalized to the spiral's width. Let $n$ be the turn count of our inductor. The resulting parametrization is:
circumference. To make handling of this easier, we introduce a variable $r' \in \left[0,1\right]$ representing the
radius normalized to the spiral's width. Let $n$ be the turn count of our inductor. The resulting parametrization is:
\begin{align}
\varphi &= 2\pi n t\\
@ -361,13 +361,15 @@ two core observations:
radius.
\end{itemize}
Combining these two observations, we find that by choosing a number $k$ of inversions that is coprime to the number of
total turns of the inductor $n$, we achieve a layout where when we connect all $k$ pairs of top and bottom-layer traces
in series, the resulting spirals on the top and bottom layers interleave cleanly. Figure\ \ref{fig_nk_interleave_illust}
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 larger
values of $n$ and $k$.
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_interleave_illust} 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 $k=0$, 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}.
\begin{figure}
\begin{center}
@ -381,14 +383,15 @@ 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 $k$ inversions, the trace
crosses the $\varphi$ axis once for each inversion, 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
Figure\ \ref{fig_nk_chinese_remainder_illust} illustrates how we arrive at the coprimality requirement.
\todo{Cleanly handle $k=0$ case.}
If we plot the spiral in polar coordinates on a cartesian plot we observe that for a $n$-turn coil with $k$ inversions,
the trace crosses the $\varphi$ axis once for each inversion, 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\\
@ -420,11 +423,10 @@ inductor does not change its turn count or dimensions, the combined arc length o
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.
layers.\todo{Does the skin effect affect the influence of vias?}
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.
As a general expression, for a standard or twisted inductor with turn count $n$ and twist count $k$, 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}
@ -481,7 +483,7 @@ case.
\subsection{CAD Integration}
To allow for easy design with twisted inductors, and to speed up the laboratory prototyping we performed for this paper,
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
@ -535,7 +537,7 @@ spiral inductor) to $k=37$. All test inductors had an inner diameter of \qty{15}
We measured the inductance and DC resistance of each test coupon using a Keysight U1733C LCR meter at
\qty{100}{\kilo\hertz} for inductance and a Keysight 34465A multimeter in four-wire configuration for DC resistance. We
further determined the self-resonant frequency of each inductor using a LiteVNA64 handheld vector network analyzer. The
results of our measurements are shown in Table\ \ref{tab_inductor_params}.
results of our measurements are shown in Table\ \ref{tab_coupons}.
We found our inductance approximation to be accurate within \qty{10}{\percent} and our ESR approximation to be accurate
within \qty{20}{\percent} for inductors with three turns or more. For lower turn-count inductors, inductance
@ -557,23 +559,25 @@ twisted inductors. We found that going from a single-layer spiral inductor to a
self-resonant frequency, this effect being more pronounced with higher turn count. Intuitively, this makes sense if we
consider the mechanics 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.
the SRF have 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 $4$).
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
$4$).
Interestingly, we observe that this decrease in high-frequency performance is counteracted by larger inversion count
$k$. While our test samples focused on smaller turn counts, we observe an increase from an SRF of
\qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$.
Interestingly, 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 an increase from an SRF of
\qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$. Prompted by this
observation, we produced another set of samples focusing on this aspect. We report our results of this investigation in
the following section.
In conclusion, we observe that twisted inductors \emph{improve} high-frequency performance compared to simple two-layer
inductors while closely matching them in ESR and inductance. While they peform worse than simple single-layer inductors
in high-frequency performance, the increased trace width that two-layer inductors allow for lowers resistive losses by
approximately a factor of four. In applications where resistive losses lead to the choice of a two-layer inductor,
twisted inductors provide improved high-frequency performance at no additional cost and without compromising other
performance parameters.
In conclusion to the above measurement results, we observe that twisted inductors \emph{improve} high-frequency
performance compared to simple two-layer inductors while closely matching them in ESR and inductance. While they peform
worse than simple single-layer inductors in high-frequency performance, the increased trace width that two-layer
inductors allow for lowers resistive losses by approximately a factor of four. In applications where resistive losses
lead to the choice of a two-layer inductor, twisted inductors provide improved high-frequency performance at no
additional cost and without compromising other performance parameters.
\begin{table*}
\begin{tabular}{cc|cccc|cccc|ccc}
@ -583,17 +587,17 @@ performance parameters.
\multicolumn{3}{c}{\textbf{Measurements}}\\
$n$&
$k$&
$L_\text{design} \left[\unit{\micro\henry}\right]$&
$L \left[\unit{\micro\henry}\right]$&
Error $\left[\unit{\percent}\right]$&
$R_\text{design} \left[\unit{\ohm}\right]$&
$R \left[\unit{\ohm}\right]$&
Error $\left[\unit{\percent}\right]$&
$L_\text{sim} \left[\unit{\micro\henry}\right]$&
$L \left[\unit{\micro\henry}\right]$&
Error $\left[\unit{\percent}\right]$&
$R_\text{sim} \left[\unit{\ohm}\right]$&
$R \left[\unit{\ohm}\right]$&
Error $\left[\unit{\percent}\right]$&
$L_\text{meas} \left[\unit{\micro\henry}\right]$&
$L \left[\unit{\micro\henry}\right]$&
$f_\text{res} \left[\unit{\mega\hertz}\right]$&
$R_\text{meas} \left[\unit{\ohm}\right]$\\\hline
$R \left[\unit{\ohm}\right]$\\\hline
\rowcolor[gray]{0.9}
$1$& $0$& $0.03$& $-86.2$& $0.0076$& $-86.8$& $0.038$& $-42.1$& $0.008$& $-77.5$& $0.054$& $457.585$&$0.0142$\\
@ -642,15 +646,15 @@ performance parameters.
\subsection{Inductance and Frequency Behavior of Larger Coils}
To investigate the high-frequency behavior of twisted inductors further, we produced and measured several additional
sample inductors, this time larger than before, and with more turns. The results of these measurements 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 all cases we tested, 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.
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.
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
@ -714,12 +718,12 @@ indicating a contribution from flux linkage.
\subsection{Coupling and its Sensitivity to Radial Offset}
While our accidential findings that twisted inductors improve high-frequency performance are certainly welcome and may
benefit many applications, 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 across a large set of rotations and relative displacements, we created a test setup
consisting of a 3D gantry built from 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.
benefit a range of applications, the key performance criterion in our rotating WPT 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 across a large set of rotations and relative displacements, we created
a test setup consisting of a 3D gantry built from 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.
\todo{pics of 3d printer test setup}
@ -738,12 +742,13 @@ providing a signal at a \qty{300}{\kilo\hertz} carrier frequency to the primary
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 inversion amounts $k$. A plot for a set
\ref{fig_symmetry_3turn_n_twist} for a set of three-turn inductors with multiple inversion numbers $k$. A plot for a set
of 10-turn inductors is shown in Figure\ \ref{fig_symmetry_10turn_n_twist} in the Appendix. A key observation here is
that while the asymmetry in the inductor's field is small, the ripple induced by rotation is considerable. Figure\
\ref{fig_field_plot_3d} shows a 3D plot of an inductor's coupling. While in the plot the field looks perfectly
rotationally symmetric, the sharp dropoff with radial offset, equivalent to a large gradient, ``amplifies'' any small
asymmetry and leads to the ripple voltages we observed, amounting up to several percent of total RMS output voltage.
that while the asymmetry in the inductor's field is impossible to distinguish visually in field plots, the ripple
induced by rotation is considerable. Figure\ \ref{fig_field_plot_3d} shows a 3D plot of an inductor's coupling. While in
the plot the field looks perfectly rotationally symmetric, the sharp dropoff with radial offset (equivalent to a large
gradient) magnifies any small asymmetry and leads to the ripple voltages we have listed in Table\ \ref{tab_coupons},
in some cases amounting to several percent of total RMS output voltage.
\begin{figure}
\begin{center}
@ -759,19 +764,19 @@ asymmetry and leads to the ripple voltages we observed, amounting up to several
From the ripple plots in Figures\ \ref{fig_symmetry_3turn_n_twist} and \ref{fig_symmetry_10turn_n_twist} 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.
ripple, with simple two-layer inductors with $k=1$ already improving ripple by a large margin. While increasing $k$
above $1$ does not siginificantly decrease the amplitude of this ripple further, it shifts the ripple into higher
frequencies that are easier to passively filter on the WPT link's secondary side in our application.
\subsection{Total Coupling Variation}
To 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.
In practical WPT setups, the transmitter and receiver coils are rarely aligned perfectly. To 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.
Plotting the results of these experiments as well as a series of experiments at a \qty{1}{\milli\meter} radial offset
against inversion count $k$, we arrive at the graph in Figure\ \ref{fig_k_ripple_plot}. In this graph, we see that
@ -782,7 +787,7 @@ across rotations works, with twisted inductors ($k>1$) showing a further improve
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},
$k=3$ inversions pairs already provided an improvement over standard configurations, with still better performance observed
$k=3$ inversions already provided an improvement over standard configurations, with still better performance observed
for $k=7$ inversions.
\todo{concrete coupling factor measurements}
@ -837,25 +842,34 @@ for $k=7$ inversions.
\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 not just individual planar components, but 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.
On the practical side, 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 not just individual planar components, but 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.
On the theoretical side, the fact that our twisted inductor model generalizes both one- or two-layer planar spiral
inductors as well as planar toroidal inductors would make the deduction of key parameters such as inductance and
distributed capacitance by mathematical analysis or by finite element methods interesting.
\section{Conclusion}
In this paper, we introduced a novel layout approach for planar, multi-layer inductors loosely 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, which yields lower output ripple in our rotating wireless power transfer
application, enabling smaller and lighter secondary-side circuitry and improving efficiency.
basket-wound inductors used in the early days of radio. Our \emph{twisted} inductors generalize several types of
conventional planar inductors including conventional single- or two-layer planar spiral inductors as well as planar
toroidal inductors. For inversion count parameter $k\ge 2$, twisted inductors produce magnetic field distributions that
have better rotational symmetry along the inductor's main axis compared to either single- or two-layer planar spiral
inductors, which yields lower output ripple in Wireless Power Transfer through rotating joints and enables the use of
smaller and lighter secondary-side circuitry, improving efficiency.
Furthermore, besides the advantages twisted inductors show in our particular application, we found that our sample
twisted inductors have improved self-resonant frequency, and slightly increased inductance compared to both conventional
single-layer and two-layer planar inductors. We base this evaluation on laboratory measurements on a set of 39 sample
inductors in total, including 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.
twisted inductors have up to \qty{50}{\percent} improved self-resonant frequency as well as up to \qty{6.5}{\percent}
increased inductance compared to conventional two-layer planar spiral inductors.
We base our evaluation on laboratory measurements on a set of 39 sample inductors in total, including 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, available at the
link at the end of this paper.
\section*{Availability}
This is version \texttt{\input{version.tex}\unskip} of this paper, generated on \today.