938 lines
65 KiB
TeX
938 lines
65 KiB
TeX
\documentclass[journal,12pt,onecolumn,draftclsnofoot]{IEEEtran}
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\begin{document}
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\date{November 14 2024}
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\author{\IEEEauthorblockN{Jan Sebastian Götte}\thanks{Jan Sebastian Götte is with the Technical University of Darmstadt,
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64283 Darmstadt, Germany (e-mail: jan.goette@tu-darmstadt.de).}}
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\title{Wireless Power Transfer with a Twist:
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Achieving Rotation-Invariant Coupling using Twisted Multi-Layer PCB Inductors}
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\maketitle
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\begin{abstract}
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We present \emph{twisted inductors}, a generalization of planar single- or two-layer spiral inductors as well as
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planar toroidal inductors. Compared to conventional planar spiral inductors, twisted inductors generate a magnetic
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field with better rotational symmetry, resulting in decreased output ripple in Wireless Power Transfer applications
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with an axially rotating receiver. Additionally, we found that twisted inductors can simultaneously yield a
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significantly improved self-resonant frequency and a higher inductance in the same area as a conventional planar
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spiral inductor, up to \qty{50}{\percent} improved SRF and \qty{6.5}{\percent} increased inductance among our test
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samples. We base our conclusions on several simulations and an extensive set of practical measurements.
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\end{abstract}
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\section{Introduction}
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Inductive Wireless Power Transfer (WPT) is a widely used technology supported by a large corpus of research literature
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\cite{awuahNovelCoilDesign2023, batraEffectFerriteAddition2015, curranModelingCharacterizationPCB2015,
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fanSimultaneousWirelessPower2024, leeSimpleWirelessPower2017, liWirelessPowerTransfer2015,
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maierContributionSystemDesign2019, mooreApplicationsWirelessPower2019, mouEnergyEfficientAdaptiveDesign2017,
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mouWirelessPowerTransfer2015, mullenEffectMisalignmentInductive, rezmeritaSelfMutualInductance2017,
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zhangWirelessPowerTransfer2019}.
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While working on a novel application of Inductive WPT in a Inertial Hardware Security Module (IHSM) as previously
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published in\textcite{gotteCantTouchThis2022}, we found ourselves presented with an unusual set of constraints
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attempting WPT through a rotating joint using a PCB inductor---a set of constraints that does not yet seem to be
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addressed adequately in the existing literature on inductive WPT.
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Our application poses the challenge of transferring power between a stationary and a rotating part of an
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IHSM\cite{gotteCantTouchThis2022} through a pair of WPT inductors located on the IHSM's axis of rotation. To reduce
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manufacturing cost of both parts, and to reduce weight and thereby inertia as well as susceptibility to vibration in the
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rotating part, we decided to use inductors that are directly patterned onto the IHSM's printed circuit boards. The
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primary constraint that results from this choice is that the PCB manufacturing processes' pattern resolution results in
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a strict upper limit to the turn count that can be achieved in an inductor with a given area.
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We found that at such small turn counts, a simple spiral PCB inductors exhibits a \emph{slightly} asymmetric field,
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which means that the coupling coefficient of two such inductors oscillates at one cycle per revolution when the
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inductors are rotated on-axis, even if both inductors are perfectly coaxially aligned.
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In other inductive wireless power transfer systems, this oscillation is mitigated by one of several factors: First, for
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this effect to matter in the first place, the two coils have to be rotating with respect to one another. In ferrite or
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iron-cored inductors, the core is the single major factor shaping the magnetic field, and evens out any small effect
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asymmetric windings might have. In wire-wound inductors, the often higher turn count and the tightly packed, circular
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wires reduce this effect to almost nothing. Finally, the output ripple caused by this oscillation can be filtered
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through a voltage regulator or by using a large decoupling capacitor on the secondary side where those components can be
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accomodated on the rotating part given the centrifugal forces resulting from a concrete design's rate of rotation.
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While there exist a corpus of prior work focusing on efficient power transfer between two coils whose position relative
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to one another cannot be precisely controlled as is the case in wireless phone charging systems as well as in proposed
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WPT electric vehicle charges,
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% TODO cite
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it is generally assumed that the two coils remain (almost) stationary with respect to one another.
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There exists a small body of work on inductive power transfer through rotating
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joints\cite{fanSimultaneousWirelessPower2024},
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but here the focus lies on higher power budgets than our application requires, which often requires ferrite or iron-core
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inductors.
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Our application is unique in that it requires power transfer through a joint that is constantly rotating at high speed,
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while we simultaneously want to avoid heavy components on the (rotating) receiver side. (Liquid) electrolytic capacitors
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cannot be used due to the large centrifugal acceleration that the rotating part experiences, and other heavy components
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such as large ceramic or polymer electrolytic capacitors or ferrite-core power inductors are inadvisable since they will
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exert large stresses onto their solder joints and the surrounding assembly due to the same centrifugal acceleration.
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Any imbalance caused by tolerances in the placement of heavy components or the precise shape of their solder fillets
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can cause detrimental vibration.
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\subsection{Twisted inductors}
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To solve this conundrum, we applied a principle inspired by rectangular or octagonal RFIC inductor design as well as by
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the polygonal basket-woven air coils used in early radio sets. In this paper, we propose a novel way of laying out
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circular PCB inductors that twists the inductor's windings around one another using a ring of vias each on the inside
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and outside of the inductor's windings. Applying some math, we show that we can layout a twisted inductor for any number
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of twists that is co-prime to the inductor's turn count, and that in fact, our approach opens up a large design space
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for inductor layouts that interpolate between planar spiral inductors on one end, and planar toroidal inductors on the
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other end. Our approach thus generalizes a number of previous approaches to the design of planar inductors.
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We observe that in high-frequency applications, a moderate number of twists increases the spacing between the beginning
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and end of the inductor's conductor, where the majority of the inductor's AC current flows. This decreases the parasitic
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capacitance of the inductor and raises its self-resonant frequency, raising its maximum possible operating frequency and
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improving its efficiency at lower operating frequencies. We note that the principle behind this reduction in distributed
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capacitance coincides with the intuition that led to the creation of honeycomb or basket-woven inductors in early radio
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sets more than a hundred years ago, before the invention of ferrites.
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\subsection{Contributions}
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In this paper, we introduce twisted inductors, a novel technique of laying out planar inductors that both improves
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rotational symmetry in rotating wireless power transfer interface as well as quality factor in other applications. We
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provide detailed layout instructions, including a mathematical analysis of the available parameter space and an
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analytical model of both inductance and DC equivalent series resistance of our scheme. Validating our scheme, we provide
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laboratory measurements of the basic parameters of a number of test specimens comparing our scheme to conventional
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techniques. We furhter performed a number of FEM simulations to validate our inductance and ESL approximations. Finally,
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to analyze the degree of rotational symmetry in our proposed scheme, we provide the results of a large number of
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automated measurements of coupling between pairs of inductors under various rotations, offsets, distances and load
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conditions.
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\section{Related Work}
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% TODO cite fanSimultaneousWirelessPower2024 below (rotating joint)
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% TODO cite \cite{mullenEffectMisalignmentInductive} below (misaligned coils)
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\subsection{A Short Historical Diversion on Basket-Woven Air Coils}
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Since the early days of radio engineering, the parasitic capacitance of inductors has been a point of
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concern\cite{nesperHandbuchDrahtlosenTelegraphie1921,flemingPrinciplesElectricWave1910}. Going back to the early days of
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wireless telegraphy after the turn of the twentieth century, coils with high inductance were needed for the construction
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of both transmitters and receivers, but the ferrites that would later permit their compact construction were still being
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developed. The ferromagnetic core material of choice back then was laminated iron, which was only useful at low
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frequencies due to eddy current losses. As a result, the inductors in radio circuits of the era were constructed as
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air-core coils. While air core inductors are immune to core saturation, the poor magnetic permeability of air
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necessitates a large number of wide turns of wire to reach useful inductance values, which for reasons of practicality
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or leakage inductance often could not be wound as a single layer cylindrical coil. This could be resolved by winding an
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inductor with many turns on multiple layers, which improves compactness and leakage inductance, but this in turn gives
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rise to increased distributed capacitance as now turns with a large voltage differential are layered right on top of
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each other.
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Back then, a number of ways were devised to decrease distributed capacitance in multilayer inductors. These methods can
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be divided into two general categories: Optimizing the connecting order of turns to minimize the voltage differential
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between adjacent turns---a technique that is still used to this day\cite{lopeFirstSelfResonantFrequency2021}, and
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optimizing the winding schema to increase the separation between turns. The main technique in the first category
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concerns winding the turns of a cylindrical multilayer inductor not layer by layer, but instead layering them
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diagonally, effectively connecting adjacent turns in a diagonal zigzag pattern. Then as now, wound inductors applying
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this technique were not feasible to manufacture reliably by machine, but the technique can be closely replicated in PCB
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inductors as shown in \textcite{leePrintedSpiralWinding2011}. The main limiting factors in a PCB implementation are the
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requirement for a large number of vias inside the inductor's turns limiting the achievable turn count\footnote{In PCBs,
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as opposed to ICs, vias limit the achievable turn count when they need to be placed in-line inside the turns as opposed
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to on the inside or outside because a PCB's minimum trace/space widths are usually much smaller than the smallest
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feasible via, consisting of a minimum-size drill surrounded by a minimum-size annular ring.} and increasing ESR through
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the thin trace sections that are necessary to accomodate the via structure, as well as the layer pairing limitations
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when blind vias are used in multilayer PCBs.
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\begin{figure}
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\begin{center}
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\subcaptionbox{\raggedright A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{
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\includegraphics[width=0.25\figurescale]{figures/saacke-radiotechnik-3-ledionspule.jpg}}
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\subcaptionbox{\raggedright A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{
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\includegraphics[width=0.25\figurescale]{figures/klein-spulen-schwingkreise-korbspule.jpg}}
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\end{center}
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\caption{Illustrations of honeycomb and basket-woven coils from the early days of wireless radio.}
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\textbf{TODO}: Not final graphics. Get proper scans for camera-ready version
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\label{fig_illust_honeycomb_basket}
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\end{figure}
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This lack of a way to wind high frequency inductors with a machine led to the creation of a number of related winding
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schemes that include honeycomb and basket woven coils
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\cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
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filbigLehrbuchHochfrequenztechnik1942,
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kleinSpulenUndSchwingungskreise1941,
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meinkeTaschenbuchHochfrequenztechnik1956,
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nottebrockSpulen1950,
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struttVerstarkerUndEmpfanger1951,
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wiggeRundfunktechnischesHandbuch1930,
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zicknerSpulen1927}. The simplest such winding technique is the universal winding as described in depth by
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\textcite{querfurthCoilWindingDescription1954}. In a simple, cylindrical wire-wound inductor, the windings are laid down
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one right next to the other, until the end of the winding area is met, where the winding direction is reversed. One
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layer of such windings forms a helix whose pitch is equal to the wire diameter. A universal winding uses the same
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helical scheme reversing at the coil ends, but uses a helical pitch larger than the wire diameter to form a structure
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similar to a spool of sewing thread.
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Other winding techniques include honeycomb and basket woven coils, some contemporary examples of which are shown in
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Figure\ \ref{fig_illust_honeycomb_basket}. In a honeycomb coil, like in an universal winding, subsequent winding layers
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are wound at a criss-cross pattern. The characteristic feature of honeycomb coils is that the winding machine is
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adjusted to produce large air gaps between adjacent windings on the same layer. When multiple layers like this are
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stacked, a three-dimensional rhomboid pattern results that is vaguely reminiscent of a honeycomb's structure.
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In basket-woven coils, a mandrel consisting of an odd number of sticks pointing either radially or axially is used, and
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the wire is woven between adjacent sticks in an alternating direction. While visually similar to honeycomb coils, this
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winding technique is more suited to homebrew construction and less amenable to mass production by machine. In axially
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basket-woven coils, the mandrel can be pulled out after the coil is finished. Like honeycomb coils, the resulting
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structure can be made mechanically stable with some lacquer, with the turns carrying the layers where they cross.
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Both construction techniques apply similar principles to those leading to the improved high-frequency behavior of
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twisted inductors that we describe in this paper. Interestingly, the winding schemes of both honeycomb and basket-woven
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coils are also governed by the same coprimality condition between the number of turns and the number of inversions
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within each turn that we describe for our twisted inductors below, although we could not find an example in contemporary
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literature where this condition was explicitly stated \cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
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kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querfurthCoilWindingDescription1954}.
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\subsection{PCB inductor design for wireless power transfer}
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For wireless power transfer, air-core inductors with or without ferrite magnetic shielding are the standard solution.
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Since in most applications, an air gap of several millimeters between the sending and receiving assemblies is expected,
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adding a ferrite core does not result in a large improvement in coupling. Meanwhile, in many WPT applications,
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especially for charging portable devices or medical implants, some misalignment between the sending and receiving coils
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is expected. Using the available space with an air-core inductor that has a large cross-sectional area reduces the
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impact of this misalignment.
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Looking at such WPT inductors, they tend to be mostly planar coils with only a few layers, so implementing them in a PCB
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process seems natural. Using a PCB for the inductor has the potential to reduce implementation cost since PCBs are
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cheap, and they can also serve as structural support.
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Implementing inductors in PCBs has a number of disadvantages. First, due to the limited layer count of common PCB
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processes, and due to structure size limitations, the number of windings that can be fit into a given volume is much
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lower than in wire-wound inductors. Second, due to a PCB's copper layers being thin compared to its dielectric
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substrate, PCB inductors tend to have poor DC resistance. A PCBs' thin but wide trace cross-section helps with
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skin effect compared to a solid conductor. However, PCBs can still not approach the performance of litz wire used in
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high-frequency WPT coils, which commonly use wire diameters in the tens of micrometer
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range\cite{zhaoDesignOptimizationLitzWire2023}. \textcite{lopeFrequencyDependentResistancePlanar2014} and
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\textcite{nomotoSplittingConductorsCoils2024} propose a mitigation that aims to emulate a litz wire's structure in
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large, high-current PCB inductors, but their mitigation is heavily limited by the structure size achievable in common
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PCB manufacturing processes\cite{nguyenReviewComparisonSolid2020}.
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A further factor that limits the high-frequency performance of PCB inductors is distributed capacitance. Not only do
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large air coils exhibit more parasitic capacitance than much smaller ferrite-core inductors simply due to their size,
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when implemented in a PCB process a large fraction of the electrical fields responsible for this capacitance pass
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through the PCB's substrate, not air. The relative permittivity $\epsilon_r$ of common PCB substrates typically lies in
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the range of $4$ to $5$\cite{mumbyDielectricPropertiesFR41989}, which increases the distributed capacitance compared to
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a pure air-core inductor by approximately that same factor.
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\subsection{Twisted Inductors in RFIC Design}
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Planar inductors are commonly used in RF ICs. In RFIC design, the major challenges are area optimization and precisely
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predicting the inductor's characteristics during the design phase. Common optimizations include applying a variable
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trace pitch to reduce distributed capacitance\cite{lopez-villegasImprovementQualityFactor2000}, and applying variable
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trace width to decrease equivalent series resistance while preserving total inductance and quality
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factor\cite{hsuAnalyticalDesignAlgorithm2008}.
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In RFICs, inductors are commonly designed as \emph{balanced} inductors with a grounded central node. Such designs
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interleave two counter-wound planar spiral inductors on the same layer with the help of some jumper connections on a
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second layer\cite{daneshDifferentiallyDrivenSymmetric2002,martinMultiturnTwistedInductor2016}. The use of such designs
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in RFIC design is mainly focused on their electrical symmetry, so that the two input ports can be fed with a fully
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differential signal, with the inductor loading both driver outputs equally across the inductor's frequency range.
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Setting the inversion count to $k=1$ in our proposed scheme as shown below yields the counterwound scheme that is
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commonly used for two-layer planar
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inductors\cite{lopeFirstSelfresonantFrequency2021,sproHighVoltageInsulationDesign2021,leePrintedSpiralWinding2011}, and
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which has been used to stack planar coils for more than a century\cite{flemingPrinciplesElectricWave1910}.
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% They note that the main point behind the design is electrical symmetry of the two ports to make driving the thing
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% differentially cleaner. We should adopt this observation for our inductors, which likewise are electrically symmetric
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% when compared to a single-layer spiral inductor.
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\subsection{Inductive Wireless Power Transfer in Practice}
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Inductive WPT has been proposed in a large number of
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scenarios\cite{zhangWirelessPowerTransfer2019,mouWirelessPowerTransfer2015}, each of which comes with a set of unique
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constraints. When WPT is used to charge an electric toothbrush, the implementation cost of the system is critical, while
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efficiency and total power output are of little concern. Mechanically, in an electric toothbrush's charging system, the
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position and spacing of the transmitter and receiver coils can easily be controlled down to millimeter precision.
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In contrast to this, wireless smartphone charging is a much more demanding application. Here, the total cost of the
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system is only secondary, but the receiver's form factor is critical, and total power output as well as efficiency
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become major objectives. At the same time, in wireless smartphone charging, position tolerances are very coarse, and the
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two coils in the charging base and in the phone may be positioned more than a centimeter off-axis, with a gap of several
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millimeters and potentially not even in parallel planes.
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Power transfer across large distances is even more of a concern in implantable medical
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devices\cite{mooreApplicationsWirelessPower2019}. Where a wireless phone charger must be able to bridge distances of a
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few millimeters, an implantable medical device might be situated underneath several centimeter of tissue and bones. At
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the same time, cost is of (almost) no concern in this medical application, which enables the use of complex
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manufacturing techniques, customized electronic components and exotic materials.
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While all of the aforementioned applications transfer somewhere between a few hundred milliwatts and several watts of
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power, at the other end of the spectrum there is a large body of research suggesting the use of inductive wireless power
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transfer for the charging of electric vehicles
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(EVs)\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017}. In this application, the wireless power
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transfer system usually replaces the conventional wired charging connector, which improves the systems' user experience
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given the strong force required to seat or unseat these rather large connectors, as well as the heft of the required
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water-cooled cables. In this application, size is of (almost) no concern, but at charging rates up to tens of kilowatt,
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efficiency becomes critical. When charging an EV at a rate of 10 kW, an efficiency improvement of just $0.1\%$
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corresponds to a reduction in power dissipation of 10 W. Besides the monetary cost of the power lost this way, each
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small improvement enables a reduction in size of heat sinks and other cooling components, which directly translates to a
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decrease in cost.
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\subsection{Air-Core Inductors for Inductive Power Transfer}
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Across application areas, air-core inductors are often used for wireless power transfer since in most applications, an
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air gap of several millimeters or more is expected, and adding a ferrite core would not change the system's performance
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by much in these circumstances. A common way to use ferrites in WPT applications is by magnetically shielding the
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inductor's back side with a ferrite plate such that the field does not extend beyond the coil's back side, thereby
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increasing the intended mutual inductance while simultaneously reducing eddy current losses when the WPT coils are
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placed near metal
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objects\cite{batraEffectFerriteAddition2015,leeSimpleWirelessPower2017,muehlmannMutualCouplingModeling2012}.
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\section{Twisted Inductor Design}
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In this section, we will provide a detailed derivation of the layout of twisted inductors. We can approach this layout
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by construction. Let us first consider a simple, planar, circular spiral coil with a fixed pitch. We will ignore trace
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width for now, and consider the trace a thin wire. We will assume the inductor's ports are both located on the positive
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$x$-Axis. We can rotate it so its first port aligns with the $x$-Axis. To minimize the loop area of the inductor's
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connections, inductors are usually designed with both ports close to one another, so we can also assume its second port
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aligns with the $x$-Axis.
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The trace trajectory of a standard planar spiral inductor can be parameterized in polar coordinates $r, \varphi$ based
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on an Archimedean spiral: \todo{For the lulz, cite Archimedes here}
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\begin{equation}
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r = a\cdot\varphi
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\label{eqn_arch_spi_basic}
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\end{equation}
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An Archimedean spiral defined this way always starts at the origin, and it continues to infinity. Let us re-parameterize
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this spiral to a curve parameter $t$ with range $\left[0,1\right]$, such that $t=0$ corresponds to the start of the
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inductor and $t=1$ corresponds to its end. As is customary in PCB inductors, we place the inductor's start on its outer
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circumference. To make handling of this easier, we introduce a variable $r' \in \left[0,1\right]$ representing the
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radius normalized to the spiral's width. Let $n$ be the turn count of our inductor. The resulting parametrization is:
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\begin{align}
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\varphi &= 2\pi n t\\
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r' &= 1 - t \\
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r &= r_1 + r' \left(r_2 - r_1\right)
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\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_interleave_illust} shows both a simple and a two-layer
|
|
spiral inductor.
|
|
|
|
\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{itemize}
|
|
\item 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 In a two-layer spiral inductor as shown in Figure\ \ref{fig_nk_interleave_illust}, 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{itemize}
|
|
|
|
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}
|
|
\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}
|
|
|
|
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\\
|
|
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. 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
|
|
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}\todo{Cite gerbonara}. 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\todo{Cite gmsh}, the
|
|
FEM mesher that we chose to interface with Elmer FEM\todo{Cite Elmer}.
|
|
|
|
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{Inductor sample design parameters and measured characteristics for a number of physically larger,
|
|
ring-shaped inductors. $L$ and $R_\text{ESR}$ have been measured with a Keysight U1733C handheld LCR meter.
|
|
$f_\text{Res}$ has been measured with a LiteVNA VNA. $C_p$ has been calculated for the simple parallel LC
|
|
resonator model from $f_\text{Res}$ and $L$. $f_\text{Res}$ was not be measured for the $n=1$ case since these
|
|
are just planar toroidal inductors, which show different resonance characteristics compared to planar spiral or
|
|
multi-turn twisted inductors. Bolded values highlight the best performance among the coils of one 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. 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}
|
|
\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 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 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}
|