958 lines
66 KiB
TeX
958 lines
66 KiB
TeX
\documentclass[journal,12pt,onecolumn,draftclsnofoot]{IEEEtran}
|
|
|
|
\usepackage[T1]{fontenc}
|
|
\usepackage[
|
|
backend=biber,
|
|
style=numeric,
|
|
natbib=true,
|
|
url=false,
|
|
doi=true,
|
|
eprint=false
|
|
]{biblatex}
|
|
\addbibresource{paper.bib}
|
|
\usepackage{amssymb,amsmath}
|
|
\usepackage{eurosym}
|
|
\usepackage{wasysym}
|
|
\usepackage[binary-units]{siunitx}
|
|
\usepackage{commath}
|
|
\usepackage{graphicx,color}
|
|
\usepackage{colortbl}
|
|
\usepackage{subcaption}
|
|
\usepackage{placeins}
|
|
\usepackage{array}
|
|
\usepackage{censor}
|
|
\usepackage{hyperref}
|
|
\usepackage{makecell}
|
|
|
|
\DeclareSIUnit{\baud}{Bd}
|
|
\DeclareSIUnit{\year}{a}
|
|
\DeclareSIUnit{\rpm}{rpm}
|
|
\renewcommand{\floatpagefraction}{.8}
|
|
\newcommand{\degree}{\ensuremath{^\circ}}
|
|
\newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}}
|
|
\newcommand{\partnum}[1]{\texttt{#1}}
|
|
\newcommand{\todo}[1]{\textbf{TODO}\footnote{#1}}
|
|
% Set to 1.0 for final two-column export
|
|
\newlength{\figurescale}
|
|
\setlength{\figurescale}{0.75\textwidth}
|
|
|
|
\begin{document}
|
|
|
|
% TODO
|
|
% Define all acronyms
|
|
% * PCB
|
|
% Use less acronyms
|
|
% Term "twisted"? "interleaved spirals"?
|
|
% Early pic / vis of spirals, somewhere in intro
|
|
% Put explanation of WPT to front of related work
|
|
% One plot instead of big table
|
|
% Move measeurements column to the left?
|
|
% In experiment schematic / setup schema: what is moving, what is stationary?
|
|
% Tone down mentioning of inspiration
|
|
% Go into way more detail on use case
|
|
|
|
\date{November 14 2024}
|
|
\author{\IEEEauthorblockN{Jan Sebastian Götte}\thanks{Jan Sebastian Götte is with the Technical University of Darmstadt,
|
|
64283 Darmstadt, Germany (e-mail: jan.goette@tu-darmstadt.de).}}
|
|
\title{Wireless Power Transfer with a Twist:
|
|
Achieving Rotation-Invariant Coupling using Twisted Multi-Layer PCB Inductors}
|
|
\maketitle
|
|
|
|
\begin{abstract}
|
|
We present \emph{twisted inductors}, a generalization of planar single- and two-layer spiral inductors as well as
|
|
planar toroidal inductors. Compared to conventional planar spiral inductors, twisted inductors generate a magnetic
|
|
field with better rotational symmetry, resulting in decreased output ripple in Wireless Power Transfer (WPT)
|
|
applications with an axially rotating receiver. Additionally, we found that twisted inductors can simultaneously
|
|
yield a significantly improved self-resonant frequency and a higher inductance in the same area as a conventional
|
|
planar spiral inductor, up to \qty{50}{\percent} improved SRF and \qty{6.5}{\percent} increased inductance among our
|
|
test samples. We base our conclusions on several simulations and an extensive set of practical measurements.
|
|
\end{abstract}
|
|
|
|
\section{Introduction}
|
|
|
|
Inductive Wireless Power Transfer (WPT) is a widely used technology supported by a large corpus of research literature
|
|
\cite{awuahNovelCoilDesign2023, batraEffectFerriteAddition2015, curranModelingCharacterizationPCB2015,
|
|
fanSimultaneousWirelessPower2024, leeSimpleWirelessPower2017, liWirelessPowerTransfer2015,
|
|
maierContributionSystemDesign2019, mooreApplicationsWirelessPower2019, mouEnergyEfficientAdaptiveDesign2017,
|
|
mouWirelessPowerTransfer2015, mullenEffectMisalignmentInductive, rezmeritaSelfMutualInductance2017,
|
|
zhangWirelessPowerTransfer2019}.
|
|
While working on an application of Inductive WPT in a Inertial Hardware Security Module (IHSM) as previously
|
|
published by \textcite{gotteCantTouchThis2022}, we found ourselves presented with an unusual set of constraints
|
|
attempting WPT through a rotating joint using a planar inductor implemented in a Printed Circuit Board (PCB)---a set of
|
|
constraints that does not seem to be addressed adequately in the existing literature on inductive WPT yet.
|
|
|
|
Inertial Hardware Security Modules are a hardware security primitive that discourages tampering with a payload (e.g.\ a
|
|
single-board computer) by rotating a tamper-sensing enclosure around the payload. The tamper-sensing enclosure
|
|
continuously monitors itself for tampering using sensors such as tamper-sensing meshes\cite{TamperResistance2020a} and
|
|
accelerometers. When the tamper-sensing enclosure signals a tamper alarm to the payload, the payload immediately
|
|
destroys all sensitive data to prevent the attacker from gaining access to it. In principle, an IHSM is similar to an
|
|
ATM that responds to attempts at opening its vault by dispensing dye over the bank notes within, rendering them unusable.
|
|
|
|
In our IHSM implementation, the tamper-sensing enclosure rotates at \qtyrange{1000}{3000}{\rpm}. The rotating enclosure
|
|
is powered through a pair of WPT inductors located on the IHSM's axis of rotation. The large centrifugal acceleration
|
|
prohibits the use of batteries or liquid electrolyte capacitors on the rotating part, and makes heavy components such as
|
|
large Multilayer Ceramic Capacitors (MLCCs) challenging to balance. To reduce manufacturing cost of both parts, and to
|
|
reduce weight and thereby inertia as well as susceptibility to vibration in the rotating part, we decided to use
|
|
inductors that are directly patterned onto the IHSM's printed circuit boards. The primary constraint that results from
|
|
this choice is that the PCB manufacturing processes' pattern resolution results in a strict upper limit to the turn
|
|
count that can be achieved in an inductor with a given area.
|
|
|
|
Planar inductors are usually considered approximately axisymmetric. In our application, we found that at small turn
|
|
counts, the asymmetry in a planar spiral inductors's field is large enough that the resulting oscillation of the
|
|
coupling coefficient of two such inductors with the inductor's revolution leads to voltage ripple on the secondary side.
|
|
Radial misalignment of the coils further exacerbates this issue.
|
|
|
|
In other inductive WPT systems, this issue is mitigated by one of several factors: First, for this effect to matter in
|
|
the first place, the two coils have to be rotating with respect to one another. In ferrite core inductors, the core is
|
|
the major factor shaping the magnetic field and evens out the small effect of winding asymmetry. In wire-wound
|
|
inductors, the often higher turn count and the tightly packed, circular wires renders this effect negligible. Finally,
|
|
the output ripple caused by this oscillation can be filtered through a voltage regulator or by using a large decoupling
|
|
capacitor on the secondary side if the application can accomodate such components on the rotating part.
|
|
|
|
While there exist a corpus of prior work focusing on efficient power transfer between two coils whose position relative
|
|
to one another cannot be precisely controlled as is the case in wireless phone charging systems as well as in proposed
|
|
WPT electric vehicle chargers,
|
|
% TODO cite
|
|
it is generally assumed that the two coils remain quasi-stationary with respect to one another.
|
|
|
|
There exists a small body of work on inductive power transfer through rotating
|
|
joints\cite{fanSimultaneousWirelessPower2024}, but here the focus lies on higher power budgets than our application
|
|
requires, which in practice requires more space and a ferrite or laminated iron core. Therefore, this paper bridges the
|
|
gap between existing literature on low-power planar WPT inductor design and high-power WPT through rotating joints.
|
|
|
|
\subsection{Twisted inductors}
|
|
|
|
In this paper, 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. To fit our unique use case,
|
|
we applied a principle which the polygonal basket-woven air coils used in early radio sets are based on to an approach
|
|
inspired by contemporary planar inductor layouts. We show that we can layout a twisted inductor for any number of twists
|
|
that is co-prime to the inductor's turn count, and that in fact, our approach opens up a large design space for inductor
|
|
layouts that interpolate between planar spiral inductors on one end, and planar toroidal inductors on the other end. Our
|
|
approach thus generalizes a number of previous approaches to the design of planar inductors.
|
|
|
|
We observe that in high-frequency applications, a moderate number of twists increases the spacing between the beginning
|
|
and end of the inductor's conductor, where the majority of the inductor's AC current flows. This decreases the parasitic
|
|
capacitance of the inductor and raises its self-resonant frequency, raising its maximum possible operating frequency and
|
|
improving its efficiency at lower operating frequencies. We note that the principle behind this reduction in distributed
|
|
capacitance coincides with the intuition that led to the creation of honeycomb or basket-woven inductors in early radio
|
|
sets more than a hundred years ago, before the invention of ferrites.
|
|
|
|
\subsection{Contributions}
|
|
In this paper, we introduce twisted inductors, a novel technique of laying out planar inductors that both improves
|
|
rotational symmetry in rotating wireless power transfer interface as well as quality factor in other applications. We
|
|
provide detailed layout instructions, including a mathematical analysis of the available parameter space and an
|
|
analytical model of both inductance and DC equivalent series resistance of our scheme. Validating our scheme, we provide
|
|
laboratory measurements of the basic parameters of a number of test specimens comparing our scheme to conventional
|
|
techniques. We furhter present the results of FEM simulations to validate our inductance and ESL approximations.
|
|
Finally, to analyze the degree of rotational symmetry in our proposed scheme, we provide the results of a large number
|
|
of automated measurements of coupling between pairs of inductors under various rotations, offsets, distances and load
|
|
conditions.
|
|
|
|
\section{Related Work}
|
|
|
|
% TODO cite fanSimultaneousWirelessPower2024 below (rotating joint)
|
|
% TODO cite \cite{mullenEffectMisalignmentInductive} below (misaligned coils)
|
|
|
|
\subsection{A Short Historical Diversion on Basket-Woven Air Coils}
|
|
|
|
Since the early days of radio engineering, the parasitic capacitance of inductors has been a point of
|
|
concern\cite{nesperHandbuchDrahtlosenTelegraphie1921,flemingPrinciplesElectricWave1910}. Going back to the early days of
|
|
wireless telegraphy after the turn of the twentieth century, coils with high inductance were needed for the construction
|
|
of both transmitters and receivers, but the ferrites that would later permit their compact construction were still being
|
|
developed. The ferromagnetic core material of choice back then was laminated iron, which was only useful at low
|
|
frequencies due to eddy current losses. As a result, the inductors in radio circuits of the era were constructed as
|
|
air-core coils. While air core inductors are immune to core saturation, the poor magnetic permeability of air
|
|
necessitates a large number of wide turns of wire to reach useful inductance values, which for reasons of practicality
|
|
or leakage inductance often could not be wound as a single layer cylindrical coil. This could be resolved by winding an
|
|
inductor with many turns on multiple layers, which improves compactness and leakage inductance, but this in turn gives
|
|
rise to increased distributed capacitance as now turns with a large voltage differential are layered right 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.
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\subcaptionbox{\raggedright A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{
|
|
\includegraphics[width=0.25\figurescale]{figures/saacke-radiotechnik-3-ledionspule.jpg}}
|
|
\subcaptionbox{\raggedright A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{
|
|
\includegraphics[width=0.25\figurescale]{figures/klein-spulen-schwingkreise-korbspule.jpg}}
|
|
\end{center}
|
|
\caption{Illustrations of honeycomb and basket-woven coils from the early days of wireless radio.}
|
|
\textbf{TODO}: Not final graphics. Get proper scans for camera-ready version
|
|
\label{fig_illust_honeycomb_basket}
|
|
\end{figure}
|
|
|
|
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
|
|
\cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
|
|
filbigLehrbuchHochfrequenztechnik1942,
|
|
kleinSpulenUndSchwingungskreise1941,
|
|
meinkeTaschenbuchHochfrequenztechnik1956,
|
|
nottebrockSpulen1950,
|
|
struttVerstarkerUndEmpfanger1951,
|
|
wiggeRundfunktechnischesHandbuch1930,
|
|
zicknerSpulen1927}. The simplest such winding technique is the universal winding as described in depth by
|
|
\textcite{querfurthCoilWindingDescription1954}. In a simple, cylindrical wire-wound inductor, the windings are laid down
|
|
one right next to the other, until the end of the winding area is met, where the winding direction is reversed. One
|
|
layer of such windings forms a helix whose pitch is equal to the wire diameter. A universal winding uses the same
|
|
helical scheme reversing at the coil ends, but uses a helical pitch larger than the wire diameter to form a structure
|
|
similar to a spool of sewing thread.
|
|
|
|
Other winding techniques include honeycomb and basket woven coils, some contemporary examples of which are shown in
|
|
Figure\ \ref{fig_illust_honeycomb_basket}. In a honeycomb coil, like in an universal winding, subsequent winding layers
|
|
are wound at a criss-cross pattern. The characteristic feature of honeycomb coils is that the winding machine is
|
|
adjusted to produce large air gaps between adjacent windings on the same layer. When multiple layers like this are
|
|
stacked, a three-dimensional rhomboid pattern results that is vaguely reminiscent of a honeycomb's structure.
|
|
|
|
In basket-woven coils, a mandrel consisting of an odd number of sticks pointing either radially or axially is used, and
|
|
the wire is woven between adjacent sticks in an alternating direction. While visually similar to honeycomb coils, this
|
|
winding technique is more suited to homebrew construction and less amenable to mass production by machine. In axially
|
|
basket-woven coils, the mandrel can be pulled out after the coil is finished. Like honeycomb coils, the resulting
|
|
structure can be made mechanically stable with some lacquer, with the turns carrying the layers where they cross.
|
|
|
|
Both construction techniques apply similar principles to those leading to the improved high-frequency behavior of
|
|
twisted inductors that we describe in this paper. Interestingly, the winding schemes of both honeycomb and basket-woven
|
|
coils are also governed by the same coprimality condition between the number of turns and the number of inversions
|
|
within each turn that we describe for our twisted inductors below, although we could not find an example in contemporary
|
|
literature where this condition was explicitly stated \cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
|
|
kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querfurthCoilWindingDescription1954}.
|
|
|
|
\subsection{PCB inductor design for wireless power transfer}
|
|
|
|
Air-core inductors with or without ferrite magnetic shielding are the standard solution in inductive WPT links. Since in
|
|
most applications, an air gap of several millimeters between the sending and receiving assemblies is expected, adding a
|
|
ferrite core does not result in a large improvement in coupling. Meanwhile, in many WPT applications, especially for
|
|
charging portable devices or medical implants, some misalignment between the sending and receiving coils is expected.
|
|
Using the available space with an air-core inductor that has a large cross-sectional area reduces the impact of this
|
|
misalignment.
|
|
|
|
Looking at such WPT inductors, they tend to be mostly planar coils with only a few layers, so implementing them in a PCB
|
|
process seems natural. Using a PCB for the inductor has the potential to reduce implementation cost since PCBs are
|
|
cheap, and they can also serve as structural support.
|
|
|
|
Implementing inductors in PCBs has several disadvantages. First, due to the limited layer count of common PCB
|
|
processes, and due to structure size limitations, the number of windings that can be fit into a given volume is much
|
|
lower than in wire-wound inductors. Second, due to a PCB's copper layers being thin compared to its dielectric
|
|
substrate---common values are \qtyrange{15}{30}{\micro\meter} copper thickness and \qtyrange{600}{1600}{\micro\meter}
|
|
substrate thickness---PCB inductors tend to have poor DC resistance, albeit the thin copper layer provides some
|
|
advantage over a solid, round conductors of the same cross-sectional area at higher frequencies due to skin effect.
|
|
However, PCBs can still not approach the performance of litz wire used in high-frequency WPT coils, which commonly use
|
|
wire diameters in the range of tens of micrometer\cite{zhaoDesignOptimizationLitzWire2023}.
|
|
\textcite{lopeFrequencyDependentResistancePlanar2014} and \textcite{nomotoSplittingConductorsCoils2024} propose a
|
|
mitigation that aims to emulate a litz wire's structure in large, high-current PCB inductors, but their mitigation is
|
|
heavily limited by the structure size achievable in common PCB manufacturing
|
|
processes\cite{nguyenReviewComparisonSolid2020}.
|
|
|
|
A further factor that limits the high-frequency performance of PCB inductors is distributed capacitance. Not only do
|
|
large air coils exhibit more parasitic capacitance than much smaller ferrite-core inductors simply due to their size,
|
|
when implemented in a PCB process a large fraction of the electrical fields responsible for this capacitance pass
|
|
through the PCB's substrate, not air. The relative permittivity $\epsilon_r$ of common PCB substrates typically lies in
|
|
the range of $4$ to $5$ \cite{mumbyDielectricPropertiesFR41989}, which increases the distributed capacitance compared to
|
|
a pure air-core inductor by approximately that same factor.
|
|
|
|
\subsection{Twisted Inductors in RFIC Design}
|
|
|
|
Planar inductors are commonly used in radio frequency integrated circuits (RFICs). In RFIC design, the major challenges
|
|
are area optimization and precisely predicting the inductor's characteristics during the design phase. Common
|
|
optimizations include applying a variable trace pitch to reduce distributed
|
|
capacitance\cite{lopez-villegasImprovementQualityFactor2000}, and applying variable trace width to decrease equivalent
|
|
series resistance while preserving total inductance and quality factor\cite{hsuAnalyticalDesignAlgorithm2008}.
|
|
|
|
In RFICs, inductors are commonly designed as \emph{balanced} inductors with a grounded central node. Such designs
|
|
interleave two counter-wound planar spiral inductors on the same layer with the help of some jumper connections on a
|
|
second layer\cite{daneshDifferentiallyDrivenSymmetric2002,martinMultiturnTwistedInductor2016}. The use of such designs
|
|
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 as shown below 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}.
|
|
|
|
% 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.
|
|
|
|
\subsection{Inductive Wireless Power Transfer in Practice}
|
|
|
|
Inductive WPT has been proposed in a large number of
|
|
scenarios\cite{zhangWirelessPowerTransfer2019,mouWirelessPowerTransfer2015}, each of which comes with a set of unique
|
|
constraints. When WPT is used to charge an electric toothbrush, the implementation cost of the system is critical, while
|
|
efficiency and total power output are of little concern. Mechanically, in an electric toothbrush's charging system, the
|
|
position and spacing of the transmitter and receiver coils can easily be controlled down to millimeter precision.
|
|
|
|
In contrast to this, wireless smartphone charging is a much more demanding application. Here, the total cost of the
|
|
system is only secondary, but the receiver's form factor is critical, and total power output as well as efficiency
|
|
become major objectives. At the same time, in wireless smartphone charging, position tolerances are very coarse, and the
|
|
two coils in the charging base and in the phone may be positioned more than a centimeter off-axis, with a gap of several
|
|
millimeters and potentially not even in parallel planes.
|
|
|
|
Power transfer across large distances is even more of a concern in implantable medical
|
|
devices\cite{mooreApplicationsWirelessPower2019}. Where a wireless phone charger must be able to bridge distances of a
|
|
few millimeters, an implantable medical device might be situated underneath several centimeter of tissue and bones. At
|
|
the same time, cost is of (almost) no concern in this medical application, which enables the use of complex
|
|
manufacturing techniques, customized electronic components and exotic materials.
|
|
|
|
While all of the aforementioned applications transfer somewhere between a few hundred milliwatts and several watts of
|
|
power, at the other end of the spectrum there is a large body of research suggesting the use of inductive wireless power
|
|
transfer for the charging of electric vehicles
|
|
(EVs)\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017}. In this application, the wireless power
|
|
transfer system usually replaces the conventional wired charging connector, which improves the systems' user experience
|
|
given the strong force required to seat or unseat these rather large connectors, as well as the heft of the required
|
|
water-cooled cables. In this application, size is of (almost) no concern, but at charging rates up to tens of kilowatt,
|
|
efficiency becomes critical. When charging an EV at a rate of \qty{10}{\kilo\watt}, an efficiency improvement of just
|
|
$0.1\%$ corresponds to a reduction in power dissipation of \qty{10}{\watt}. Besides the monetary cost of the power lost
|
|
this way, each small improvement enables a reduction in size of heat sinks and other cooling components, which directly
|
|
translates to a decrease in cost.
|
|
|
|
\subsection{Air-Core Inductors in WPT}
|
|
|
|
Across application areas, air-core inductors are often used for WPT since in most applications, an air gap of several
|
|
millimeters or more is expected, and adding a ferrite core would not change the system's performance by much in these
|
|
circumstances. A common way to use ferrites in WPT applications is by magnetically shielding the inductor's back side
|
|
with a ferrite plate such that the field does not extend beyond the coil's back side, thereby increasing the intended
|
|
mutual inductance while simultaneously reducing eddy current losses when the WPT coils are placed near metal
|
|
objects\cite{batraEffectFerriteAddition2015,leeSimpleWirelessPower2017,muehlmannMutualCouplingModeling2012}.
|
|
|
|
\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
|
|
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.
|
|
|
|
The trace trajectory of a standard planar spiral inductor can be parameterized in polar coordinates $r, \varphi$ based
|
|
on an Archimedean spiral: \todo{For the lulz, cite Archimedes here}
|
|
|
|
\begin{equation}
|
|
r = a\cdot\varphi
|
|
\label{eqn_arch_spi_basic}
|
|
\end{equation}
|
|
|
|
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
|
|
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\\
|
|
r' &= 1 - t \\
|
|
r &= r_1 + r' \left(r_2 - r_1\right)
|
|
\label{eqn_simple_spiral_ind}
|
|
\end{align}
|
|
|
|
The resulting spiral trace starts at radius $r_2$ on the positive $x$ axis, and spirals inward until it meets $r_1$. In
|
|
its PCB realization, at $r_1$, a via would be placed to connect the end of the spiral trace to a jumper trace on another
|
|
layer of the PCB leading back to the start.
|
|
|
|
To improve layer utilization, a common technique in PCB inductor design is to use both layers of the PCB for the
|
|
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_nk_combined} shows both a simple and a two-layer
|
|
spiral inductor in the first two columns.
|
|
|
|
\begin{align}
|
|
\varphi &= 2\pi n t\\
|
|
r' &= 1 - 2 t \\
|
|
r &= r_1 + \left|r'\right| \left(r_2 - r_1\right)
|
|
\label{eqn_twolayer_spiral}
|
|
\end{align}
|
|
|
|
\subsection{From Spiral to Twisted Inductor}
|
|
|
|
Extending the above parametrization of a spiral inductor's layout, we propose planar \emph{twisted inductors} based on
|
|
two core observations:
|
|
|
|
\begin{description}
|
|
\item[Observation 1.] 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[Observation 2.] In a two-layer spiral inductor (Figure\ \ref{fig_nk_combined}), 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{description}
|
|
|
|
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 $k=0$, 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}
|
|
% \includegraphics[width=\figurescale]{figures/nk_interleave_illust.pdf}
|
|
% \end{center}
|
|
% \caption{single-layer spiral inductor's layout (left), a conventional two-layer planar inductor's layout (middle),
|
|
% and a twisted inductor with two inversions (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}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=\figurescale]{figures/nk_combined.pdf}
|
|
\end{center}
|
|
\caption{Inductor layouts for several sets of turn count $n$ and inversion count $k$. The top row shows the actual
|
|
trace layout in cartesian coordinates, the bottom row visualizes the winding schema.
|
|
}
|
|
\label{fig_nk_combined}
|
|
\end{figure}
|
|
|
|
Figure\ \ref{fig_nk_combined} 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\\
|
|
t &\equiv j \mod k
|
|
\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.
|
|
|
|
%\begin{figure}
|
|
% \begin{center}
|
|
% \includegraphics[width=0.8\figurescale]{figures/nk_chinese_remainder_illust.pdf}
|
|
% \end{center}
|
|
% \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}
|
|
|
|
\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 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
|
|
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.\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~$k$, 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}
|
|
|
|
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:
|
|
|
|
\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
|
|
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}.
|
|
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 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. 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\cite{KiCadEDA}. We integrated the ESR and inductance
|
|
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}\cite{GerbonaraToolsHandle}. 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\cite{geuzaineGmsh3DFinite2009}, the FEM mesher that we chose to interface with Elmer
|
|
FEM\cite{ruokolainenElmerCSCElmerfemElmer2023}.
|
|
|
|
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
|
|
inductor layouts, we performed simulations to determine ohmic resistance and inductance. Due to limitations in our
|
|
gmsh/Elmer toolchain, we were unable to run simulations for parasitic capacitance and self-resonance, or for coupling
|
|
behavior of coil pairs. We found that for these cases which require larger, more complex meshes, gmsh would frequently
|
|
crash during meshing, and where we were able to produce meshes, Elmer would only converge for some of them. While these
|
|
are problems that can be solved through either a more skillful description of the problem in gmsh and Elmer, or by using
|
|
more robust software such as Simulia CST, we decided to instead experimentally measure these quantities instead
|
|
(Section\ \ref{sec_experiments}).
|
|
|
|
\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.
|
|
|
|
\paragraph{Inductance}
|
|
We let Elmer determine inductance by first using its coil solver to determine the volumetric current density in our mesh
|
|
given a test current, then applying its magnetodynamics solver to solve the electromagnetic field. Elmer provides
|
|
routines to derive the total magnetic field energy $U_\text{mag}$ from an EM field solution. Since we have only our
|
|
inductor under test inside the simulation volume, with test current $I_\text{test}$, we can then derive the inductor's
|
|
inductance according to the well-known relation\todo{Find decent source}:
|
|
|
|
\begin{equation}
|
|
L = \frac{2\cdot U_\text{mag}}{I_\text{test}^2}
|
|
\end{equation}
|
|
|
|
\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 $k=0$ (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}.
|
|
|
|
\subsection{Inductance and DC resistance}
|
|
|
|
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_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
|
|
measurements are difficult because the small absolute inductances involved are easily disturbed by stray inductances,
|
|
and ESR measurements are affected by contact and trace resistance even when measurements are taken in four-wire mode.
|
|
|
|
In accordance with our design intuition, we found that for high turn count inductors, the doubled trace width that is
|
|
afforded by splitting a simple spiral inductor across two PCB layers in any two-layer configuration improves ESR by
|
|
approximately a factor of two. Going from a simple single-layer spiral inductor to a simple two-layer spiral inductor
|
|
($k=1$), we observe that the resulting inductance decreases by up to \qty{15}{\percent}. We suspect that the main factor
|
|
leading to this decrease is radial magnetic flux leakage through the PCB material between the inductor's layers.
|
|
Comparing simple two-layer inductors with $k=1$ to the twisted inductors with larger $k$ values that we propose in this
|
|
paper, we observe almost identical performance for $k>1$ with decreases of less than \qty{0.5}{\percent} going from
|
|
$k=1$ 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. We found that going from a single-layer spiral inductor to a two-layer spiral inductor decreases the
|
|
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, 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 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 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}
|
|
\multicolumn{2}{c|}{\textbf{Parameters}}&
|
|
\multicolumn{4}{c|}{\textbf{Design values}}&
|
|
\multicolumn{4}{c|}{\textbf{Simulation results}}&
|
|
\multicolumn{3}{c}{\textbf{Measurements}}\\
|
|
$n$&
|
|
$k$&
|
|
$L \left[\unit{\micro\henry}\right]$&
|
|
Error $\left[\unit{\percent}\right]$&
|
|
$R \left[\unit{\ohm}\right]$&
|
|
Error $\left[\unit{\percent}\right]$&
|
|
$L \left[\unit{\micro\henry}\right]$&
|
|
Error $\left[\unit{\percent}\right]$&
|
|
$R \left[\unit{\ohm}\right]$&
|
|
Error $\left[\unit{\percent}\right]$&
|
|
$L \left[\unit{\micro\henry}\right]$&
|
|
$f_\text{res} \left[\unit{\mega\hertz}\right]$&
|
|
$R \left[\unit{\ohm}\right]$\\\hline
|
|
|
|
\rowcolor[gray]{0.9}
|
|
$1$& $0$& $0.03$& $-86.2$& $0.0076$& $-86.8$& $0.038$& $-42.1$& $0.008$& $-77.5$& $0.054$& $457.585$&$0.0142$\\
|
|
$1$& $3$& $0.03$& $-93.1$& $0.0095$& $-49.9$& $0.039$& $-43.6$& $0.008$& $-78.8$& $0.056$& $\textbf{465.07}$& $\textbf{0.0143}$\\
|
|
$1$& $4$& $0.03$& $-103.4$& $0.0108$& $-38.6$& $0.040$& $-47.5$& $0.008$& $-87.5$& $\textbf{0.059}$& $460.08$& $0.015$\\
|
|
$1$& $5$& $0.03$& $-89.7$& $0.0123$& $-35.3$& $0.041$& $-34.1$& $0.009$& $-84.4$& $0.055$& $460.08$& $0.0166$\\\hline
|
|
\rowcolor[gray]{0.9}
|
|
$2$& $0$& $0.16$& $10.0$& $0.0252$& $-26.7$& $0.126$& $-14.3$& $0.026$& $-22.7$& $0.144$& $266.24$& $0.0319$\\
|
|
\rowcolor[gray]{0.9}
|
|
$2$& $1$& $0.12$& $-28.4$& $0.0253$& $-12.1$& $0.127$& $-17.3$& $0.024$& $-18.3$& $0.149$& $\textbf{245.51}$& $\textbf{0.0284}$\\
|
|
$2$& $3$& $0.12$& $-31.0$& $0.0270$& $-7.9$& $0.128$& $-18.8$& $0.025$& $-16.4$& $\textbf{0.152}$& $240.52$& $0.0291$\\
|
|
$2$& $5$& $0.12$& $-26.7$& $0.0299$& $-0.2$& $0.130$& $-13.1$& $0.027$& $-11.1$& $0.147$& $225.5$& $0.03$\\\hline
|
|
\rowcolor[gray]{0.9}
|
|
$3$& $0$& $0.26$& $-19.6$& $0.0755$& $-5.0$& $0.285$& $-9.1$& $0.077$& $-2.9$& $0.311$& $192.95$& $0.0792$\\
|
|
\rowcolor[gray]{0.9}
|
|
$3$& $1$& $0.26$& $-10.0$& $0.0454$& $-1.6$& $0.262$& $-9.5$& $0.044$& $-4.8$& $\textbf{0.287}$& $\textbf{145.71}$& $0.0461$\\
|
|
$3$& $4$& $0.26$& $-9.6$& $0.0479$& $5.0$& $0.265$& $-7.9$& $0.046$& $1.1$& $\textbf{0.286}$& $\textbf{145.71}$& $\textbf{0.0455}$\\\hline
|
|
\rowcolor[gray]{0.9}
|
|
$5$& $0$& $0.73$& $-9.6$& $0.2357$& $-0.4$& $0.760$& $-5.3$& $0.240$& $1.4$& $0.8$& $125.415$&$0.2366$\\
|
|
\rowcolor[gray]{0.9}
|
|
$5$& $1$& $0.73$& $4.5$& $0.0755$& $-3.1$& $0.670$& $-3.4$& $0.074$& $-5.1$& $\textbf{0.693}$& $61.345$& $0.0778$\\
|
|
$5$& $3$& $0.73$& $4.3$& $0.0763$& $4.7$& $0.671$& $-3.4$& $0.074$& $1.8$& $\textbf{0.694}$& $\textbf{70.285}$& $0.0727$\\
|
|
$5$& $7$& $0.73$& $4.4$& $0.0802$& $16.2$& $0.675$& $-2.8$& $0.077$& $12.7$& $\textbf{0.694}$& $68.05$& $\textbf{0.0672}$\\\hline
|
|
\rowcolor[gray]{0.9}
|
|
$10$& $0$& $2.90$& $-2.4$& $0.7539$& $-2.3$& $2.900$& $-2.4$& $0.761$& $-1.4$& $2.97$& $62.835$& $0.7713$\\
|
|
\rowcolor[gray]{0.9}
|
|
$10$& $1$& $2.90$& $6.3$& $0.2513$& $7.6$& $2.700$& $-0.7$& $0.250$& $7.1$& $\textbf{2.718}$& $24.076$& $0.2322$\\
|
|
$10$& $3$& $2.90$& $6.4$& $0.2520$& $10.5$& $2.700$& $-0.5$& $0.250$& $9.8$& $2.714$& $\textbf{28.571}$& $0.2255$\\
|
|
$10$& $7$& $2.90$& $6.4$& $0.2554$& $16.9$& $2.700$& $-0.5$& $0.252$& $15.8$& $2.713$& $28.072$& $\textbf{0.2122}$\\\hline
|
|
\rowcolor[gray]{0.9}
|
|
$25$& $0$& $18.15$& $1.1$& $3.7693$& $-3.9$& $18.000$& $0.3$& $3.800$& $-3.0$& $17.955$& $24.84$& $3.9156$\\
|
|
\rowcolor[gray]{0.9}
|
|
$25$& $1$& $18.15$& $6.7$& $1.8843$& $9.7$& $16.900$& $-0.2$& $1.900$& $10.4$& $16.938$& $8.84$& $1.7024$\\
|
|
$25$& $3$& $18.15$& $6.8$& $1.8851$& $13.2$& N/A& N/A& N/A& N/A& $16.919$& $8.595$& $1.636$\\
|
|
$25$& $13$& $18.15$& $6.7$& $1.9016$& $18.9$& $16.900$& $-0.2$& $1.900$& $18.8$& $16.931$& $\textbf{10.555}$& $\textbf{1.5429}$\\
|
|
$25$& $37$& $18.15$& $6.0$& $2.0197$& $15.9$& $17.100$& $0.2$& $2.000$& $15.1$& $\textbf{17.066}$& $10.31$& $1.698$\\
|
|
|
|
\end{tabular}
|
|
\caption{Inductor sample design parameters and measured characteristics. All inductors have outer diameter
|
|
\qty{35}{\milli\meter} and inner diameter \qty{15}{\milli\meter}. The missing values in the simulation results
|
|
columns result from the solver failing to converge. Bolded values highlight the best performing two-layer coil
|
|
of each turn count. Shaded rows indicate conventional single-layer ($k=0$) or two-layer ($k=1$) planar
|
|
inductors.}
|
|
\label{tab_coupons}
|
|
\end{table*}
|
|
|
|
\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.
|
|
|
|
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
|
|
\qty{6.5}{\percent} for our largest samples. This behavior indicates that large twisted inductors indeed behave like a
|
|
combination between a conventional planar spiral inductor and a conventional planar toroidal inductor. Comparing the
|
|
magnitude of this increase with the measurements listed in Table\ \ref{tab_wide_coils} for planar toroidal inductors, we
|
|
see that this effect exceeds what one would reach by a simple series configuration of both styles of inductor,
|
|
indicating a contribution from flux linkage.
|
|
|
|
\begin{table}
|
|
\begin{tabular}{cc|cc|ccc|c}
|
|
$d_1$&
|
|
$d_2$&
|
|
$n$&
|
|
$k$&
|
|
$L$&
|
|
$R_\text{ESR}$&
|
|
$f_\text{Res}$&
|
|
$C_\text{p}$\\
|
|
$\left[\unit{\milli\meter}\right]$&
|
|
$\left[\unit{\milli\meter}\right]$&
|
|
&
|
|
&
|
|
$\left[\unit{\micro\henry}\right]$&
|
|
$\left[\unit{\ohm}\right]$&
|
|
$\left[\unit{\mega\hertz}\right]$&
|
|
$\left[\unit{\pico\farad}\right]$\\\hline
|
|
\rowcolor[gray]{0.9}
|
|
$25$&$40$&$1$ &$150$& $5.00$& $11.0$& N/A& N/A\\
|
|
\rowcolor[gray]{0.9}
|
|
$25$&$40$&$53$ &$1$& $120$& $\mathbf{19.6}$& $18.0$& $0.65$\\
|
|
$25$&$40$&$53$ &$50$& $121$& $22.6$& $\mathbf{27.5}$& $\mathbf{0.28}$\\
|
|
$25$&$40$&$53$ &$100$& $123$& $26.9$& $26.5$& $0.29$\\
|
|
$25$&$40$&$53$ &$150$& $\mathbf{125}$& $33.2$& $24.0$& $0.35$\\\hline
|
|
\rowcolor[gray]{0.9}
|
|
$50$&$65$&$1$ &$300$& $10.2$& $21.9$& N/A& N/A\\
|
|
\rowcolor[gray]{0.9}
|
|
$50$&$65$&$53$ &$1$& $270$& $\mathbf{35.7}$& $10.0$& $0.94$\\
|
|
$50$&$65$&$53$ &$100$& $272$& $41.9$& $\mathbf{15.8}$& $\mathbf{0.37}$\\
|
|
$50$&$65$&$53$ &$200$& $277$& $50.1$& $13.3$& $0.52$\\
|
|
$50$&$65$&$53$ &$300$& $\mathbf{280}$& $65.0$& $13.8$& $0.48$\\\hline
|
|
\rowcolor[gray]{0.9}
|
|
$75$&$90$&$1$ &$480$& $17.3$& $35.5$& N/A& N/A\\
|
|
\rowcolor[gray]{0.9}
|
|
$75$&$90$&$53$ &$1$& $441$& $\mathbf{50.7}$& $7.00$& $1.17$\\
|
|
$75$&$90$&$53$ &$160$& $444$& $60.8$& $\mathbf{10.0}$& $\mathbf{0.57}$\\
|
|
$75$&$90$&$53$ &$320$& $461$& $76.2$& $8.75$& $0.72$\\
|
|
$75$&$90$&$53$ &$480$& $\mathbf{470}$& $92.9$& $8.00$& $0.84$\\
|
|
\end{tabular}
|
|
\caption{Parameters and measurement results of a set of larger sample inductors. Bold values indicate best
|
|
performance at a given size. Shaded rows indicate conventional planar toroidal ($n=1$) or two-layer planar
|
|
spiral inductors ($k=1$).}
|
|
\label{tab_wide_coils}
|
|
\end{table}
|
|
|
|
|
|
\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 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}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=.85\figurescale]{figures/test_schematic.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}
|
|
\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 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 impossible to distinguish visually in field plots, the ripple
|
|
induced by rotation is considerable. The sharp dropoff of coupling with radial offset 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}
|
|
\includegraphics[width=\figurescale]{figures/symmetry_3turn_n_twist.pdf}
|
|
\end{center}
|
|
\caption{RMS output voltage of the test circuit from Figure\ \ref{symmetry_test_circuit} for three pairs of matching
|
|
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{fig_symmetry_3turn_n_twist}
|
|
\end{figure}
|
|
|
|
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. 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}
|
|
|
|
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
|
|
twisted inductors improve ripple compared to conventional designs, even at a low inversion count such as $k=3$.
|
|
|
|
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},
|
|
$k=3$ inversions already provided an improvement over standard configurations, with still better performance observed
|
|
for $k=7$ inversions.
|
|
|
|
\todo{concrete coupling factor measurements}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=.85\figurescale]{figures/k_ripple_plot.pdf}
|
|
\end{center}
|
|
\caption{RMS Voltage ripple in a model rotating WPT setup with $R_L=\qty{10}{\ohm}$ as a percentage of total RMS
|
|
output voltage, plotted against inductor inversion count $k$. Measurements were taken with a number of different
|
|
coils with turn count $n$ between a single turn and $25$ turns. Measurements were taken at two different radial coil
|
|
offsets of $r=\qty{1}{\milli\meter}$ and $\qty{4}{\milli\meter}$. Coil distance was $d=\qty{1}{\milli\meter}$ in all
|
|
cases. The shaded area indicates conventional coil layouts, with the remainder of the plot showing twisted
|
|
inductors.}
|
|
\label{fig_k_ripple_plot}
|
|
\end{figure}
|
|
|
|
%\begin{figure}
|
|
% \begin{center}
|
|
% \includegraphics[width=.6\figurescale]{figures/field_plot_3d_n5_k0.pdf}
|
|
% \end{center}
|
|
% \caption{The coupling between a pair of identical coils (here two simple spiral inductors with $n=5$ and $k=0$)
|
|
% visualized in three dimensions. The $x$ and $y$ axis show in-plane displacement, and the $z$ axis shows output
|
|
% amplitude in arbitrary units. Height and rotation are fixed to \qty{1}{\milli\meter} and \qty{0}{\degree},
|
|
% respectively. The most prominent aspects of this plot are that coupling falls off steeply with distance, and that
|
|
% the rotation-dependent variation is small in comparison. The circular valley around the central peak is the region
|
|
% where one inductor is mostly outside the other inductors, and intersects the field lines returning from the other
|
|
% inductor's back, leading to a negative coupling coefficient.}
|
|
% \label{fig_field_plot_3d}
|
|
%\end{figure}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=.75\figurescale]{figures/rms_ripple_double_rotation_n3_r4.pdf}
|
|
\end{center}
|
|
\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\figurescale]{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}
|
|
|
|
\section{Future Work}
|
|
|
|
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 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 WPT 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 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.
|
|
|
|
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]
|
|
|
|
\FloatBarrier
|
|
\appendix
|
|
\section{Supplemental plots}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=\figurescale]{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{fig_symmetry_3turn_n_twist}}
|
|
\label{fig_symmetry_10turn_n_twist}
|
|
\end{figure}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=.75\figurescale]{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\figurescale]{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}
|
|
|
|
\section{Layout examples}
|
|
\label{sec_appendix_layout_examples}
|
|
|
|
\begin{figure*}
|
|
\begin{center}
|
|
\includegraphics[width=.75\textwidth]{figures/nk_complex_illust.pdf}
|
|
\end{center}
|
|
\caption{Layout examples for a number of combinations of turn count $n$ and inversion 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*}
|
|
|
|
\end{document}
|