918 lines
65 KiB
TeX
918 lines
65 KiB
TeX
\chaptertitle{Rotation-Invariant Envelope Power Supply}
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% Twisted Inductor paper
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A central engineering challenge in inertial HSMs is transferring power and data between the payload and the rotating
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mesh cage. Industrially, power and data transfer through rotating joints is usually done using slip ring assemblies. A
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slip ring consists of one or more contacts that wipe on a rotating circular surface. Industrially, metal spring contacts
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plated with hard gold or other common surface coatings are used for transferring small currents and data signals, and
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carbon brushes are used for higher currents. Slip rings are widely used in motors and other rotating machinery.
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For use in IHSMs, slip rings have several limitations. First, they are complex precision-machined components and thus
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are rather expensive. Beyond cost, they also have performance limitations. Generally, slip rings are most well-suited to
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slow rotation, as high rotation increases the wear of the contacts. The design target of \qty{1000}{rpm} we use in IHSMs
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are at the upper end of what commercial slip rings usually support. A third disadvantage is that they are sensitive, and
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any misalignment or contamination by dust can increase wear and cause intermittant contact.
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An IHSM's data link can easily be realized using optical communication. Although power transfer using light is also
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possible---and we have in fact demonstrated it in our first prototype IHSM---it comes at the disadvantage of a heavy
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rotating assembly since large solar cells are needed, and it has poor end-to-end efficiency. For the large-scale meshes
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needed in a high-performance IHSM tailored to SMPC applications, we engineered a better solution: A rotation-invariant
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inductive Wireless Power Transfer link.
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While Wireless Power Transfer (WPT) is widely used and can be implemented in many different ways~\cite{
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awuahNovelCoilDesign2023,
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batraEffectFerriteAddition2015,
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curranModelingCharacterizationPCB2015,
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fanSimultaneousWirelessPower2024,
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leeSimpleWirelessPower2017,
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liWirelessPowerTransfer2015,
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maierContributionSystemDesign2019,
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mooreApplicationsWirelessPower2019,
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mouEnergyEfficientAdaptiveDesign2017,
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mouWirelessPowerTransfer2015,
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zhangWirelessPowerTransfer2019}.
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Most WPT variants link the primary and secondary side primarily through the magnetic component of the
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electromagnetic field, and coils are used as the transmitting and receiving antenna. Such \emph{inductive} WPT uses low
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frequency, which reduces circuit complexity, and it is well-suited for transferring high power across short distances.
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The electronic realization of a WPT link is usually similar to that of a DC/DC converter, except that in place of the
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inductor or flyback transformer, the pair of transceiver coils is used. Compared to a flyback transformer, the WPT
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link's transceiver coil pair has a lower coupling coefficient that varies with distance.
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A challenge in WPT links is the strong dependency between link inductor coupling coefficient and distance. In a naïve
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implementation that uses the link coils as a simple transformer, link efficiency would drop sharply with distance. To
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decrease the impact of this distance dependency, almost all WPT implementations combine the transceiver coils with
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capacitors to form a pair of tuned tank circuits that are driven like they would be in a resonant converter. Like in
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resonant converters, a variety of topologies such as series, parallel, or series-parallel LC are used for these tuning
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circuits.
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\section{Construction Approach}
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\begin{figure}
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\begin{center}
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\subcaptionbox{\raggedright A classic planar spiral inductor}{
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\includegraphics[width=0.28\textwidth]{svg_vis_paper_plain.png}}
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\subcaptionbox{\raggedright A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{
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\includegraphics[width=0.18\textwidth]{saacke-radiotechnik-3-ledionspule.jpg}}
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\subcaptionbox{\raggedright A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{
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\includegraphics[width=0.18\textwidth]{klein-spulen-schwingkreise-korbspule.jpg}}
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\subcaptionbox{\raggedright Our proposed inductor layout}{
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\includegraphics[width=0.28\textwidth]{svg_vis_paper.png}}
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\end{center}
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\caption{Illustration of our proposed inductor layout compared to contemporary conventional planar inductors and
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honeycomb as well as basket-woven coils from the early days of wireless radio.}
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\label{fig_illust_honeycomb_basket}
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\end{figure}
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\todo{Not final graphics. Get proper scans for camera-ready version}
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In the WPT link powering the rotating mesh of an IHSM presentsan unusual set of constraints, which does not seem to be
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addressed adequately in the existing literature on inductive WPT yet. To reduce the need for custom-wound inductors, we
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settled on using a planar inductor implemented in a Printed Circuit Board (PCB). Such planar PCB inductors are limited
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by the structure size limits of the PCB process, resulting in rotational asymmetry due to the trace width. Planar
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inductors are usually considered approximately axisymmetric. In our application, we found that the field asymmetry in
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feasible PCB inductors is large enough that axial rotation of two such inductors results in an oscillation of their
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coupling coefficient that leads to voltage ripple on the secondary side, especially when the coils are
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misaligned.
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The large centrifugal acceleration on an IHSM mesh prohibits the use of batteries or liquid electrolyte capacitors on
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the rotating part, and makes heavy components such as large Multilayer Ceramic Capacitors (MLCCs) and ferrite-core
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inductors challenging to balance. As a result, the secondary-side voltage ripple poses a significant issue since the
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conventional ways of efficiently filtering such ripple through large bypass capacitors or through a secondary-side
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switchmode power supply are difficult to implement due to their mass.
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In other inductive WPT systems, this issue is mitigated by one of several factors: First, for this effect to matter in
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the first place, the two coils have to be rotating with respect to one another. In ferrite core inductors, the core is
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the major factor shaping the magnetic field and evens out the small effect of winding asymmetry. In wire-wound
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inductors, the often higher turn count and the tightly packed, circular wires render this effect negligible. Finally,
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the output ripple caused by this oscillation can be filtered through a voltage regulator or by using a large decoupling
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capacitor on the secondary side if the application can accomodate such components on the rotating part.
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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 chargers~\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017}, it is generally
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assumed that the two coils remain quasi-stationary with respect to one another.
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There exists a body of work on inductive power transfer through rotating joints but here the focus often lies on higher
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power budgets than our application requires, which in practice requires more space and a ferrite or laminated iron
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core~\cite{
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fanSimultaneousWirelessPower2024,
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songRotationLightweightWirelessPower2019,
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wangCoaxialNestedCouplersBased2020,
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}.
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Often, these rotating joint WPT systems use coaxial structures, but segmented approaches exist, too~\cite{
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wangNovelRotatingWireless2024,
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yanFreeRotationWirelessPower2023,
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xiaRotaryWirelessPower2024,
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liWirelessPowerTransfer2021,
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}.
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In lower-power applications, segmented approaches are more common. A key challenge in segmented approaches is the
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reduction of secondary-side ripple induced when the segments' alignment changes throught one revolution~\cite{
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zhangWirelessSensorPower2024,
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}, which usually requires additional secondary-side circuitry. In this work, we introduce a planar coil topology for WPT
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through a rotating joint using a single planar PCB coil on both the transmitting and the receiving side that improves
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rotation ripple at low turn counts.
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\subsection{Twisted inductors}
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To solve these issues, we propose a layout for circular PCB inductors that uses a number of series-connected interleaved
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spirals to achieve a topological equivalent to a torus knot from mathematical knot theory. Our layout twists the
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inductor's windings around one another by connecting the interleaved spiral segments with a ring of vias each on the
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inside and outside of the inductor's windings. Our approach provides better performance beyond our particular use case,
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and improves over conventional contemporary planar inductors applying similar principles to those which inspired the
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polygonal basket-woven air coils used in early radio sets. We show that we can layout a twisted inductor for any number
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of layer inversions that is co-prime to the inductor's turn count. Our approach opens up a design space for inductor
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layouts that interpolate between planar spiral inductors on one end, and planar toroidal inductors on the other end. Our
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approach thus generalizes a super-set to 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 layer inversions increases the spacing between the
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beginning and end of the inductor's conductor, where the majority of the inductor's AC current flows. This decreases the
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parasitic capacitance of the inductor and increases its Self-Resonant Frequency (SRF), raising its maximum possible
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operating frequency and improving its efficiency at lower operating frequencies.
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\subsection{Contributions}
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Our contributions on this matter include:
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\begin{itemize}
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\item We introduce twisted inductors, a planar inductor layout that improves rotational symmetry in WPT through
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rotating joins, and promises improved high-frequency behavior in other applications.
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\item We provide detailed instructions for the construction of such layouts, including a mathematical analysis of
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the available parameter space.
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\item We provide an analytical model of inductance and DC equivalent series resistance of our scheme.
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\item Validating our scheme, we provide laboratory measurements of the basic parameters of 39 test specimens
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comparing our scheme to conventional layouts.
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\item We further present the results of Finite Element Method (FEM) simulations to validate our inductance and ESR
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approximations.
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\item Finally, to analyze the degree of rotational symmetry in our proposed scheme, we provide the results of a
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large number of automated measurements of coupling between pairs of inductors under various rotations, offsets,
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distances and load conditions.
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\end{itemize}
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\section{Related Work}
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\subsection{Inductive WPT 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 little concern, but at charging rates up to tens of kilowatt,
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efficiency becomes critical.
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%When charging an EV at a rate of \qty{10}{\kilo\watt}, an efficiency improvement of just
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%$0.1\%$ corresponds to a reduction in power dissipation of \qty{10}{\watt}. Besides the monetary cost of the power lost
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%this way, each small improvement enables a reduction in size of heat sinks and other cooling components, which directly
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%translates to a decrease in cost.
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\subsection{Core materials in WPT}
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Across application areas, air-core inductors are often used for WPT since in most applications, an air gap of several
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millimeters or more is expected~\cite{curranModelingCharacterizationPCB2015}. Especially in low-power application such
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as mobile device charging, the size and weight of ferrites is an obstacle to their use, and at lower power levels losses
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are less of a concern.
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A common way to use ferrites in WPT applications is by magnetically shielding the inductor's back side with a ferrite
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plate such that the field does not extend beyond the coil's back side, thereby increasing the intended mutual inductance
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while simultaneously reducing eddy current losses when the WPT coils are placed near metal
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objects~\cite{batraEffectFerriteAddition2015,leeSimpleWirelessPower2017,muehlmannMutualCouplingModeling2012}. Similar to
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how the trace layouts of planar WPT coils are optimized to improve power transfer efficiency, the layout of ferrite
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components has been proposed for optimization~\cite{batraEffectFerriteAddition2015}.
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\subsection{PCB inductor design for wireless power transfer}
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Today, air-core inductors are the standard solution in inductive WPT links. Since in most WPT applications an air gap of
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several millimeters between the sending and receiving assemblies is expected, adding a ferrite core does not result in a
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large improvement in coupling. Instead, the impact of this misalignment is reduced by maximizing the area of the
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air-core inductors used, or by tiling multiple
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inductors~\cite{curranModelingCharacterizationPCB2015,wangNovelRotatingWireless2024,zhangDynamicWirelessPower2025}.
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WPT inductors tend to be mostly planar coils with only a few layers, so implementing them in a PCB process seems
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natural. Using a PCB for the inductor has the potential to reduce implementation cost since PCBs are cheap, and they can
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also serve as structural support. However, implementing inductors in PCBs has several disadvantages. First, due to the
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limited layer count of common PCB processes and due to structure size limitations, the number of windings that can be
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fit into a given volume is much lower than in wire-wound inductors. Second, due to a PCB's copper layers being thin
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compared to its dielectric substrate\footnote{common values are \qtyrange{15}{30}{\micro\meter} copper thickness and
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\qtyrange{600}{1600}{\micro\meter} substrate thickness} PCB inductors tend to have poor DC resistance, albeit the thin
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copper layer decreases skin effect losses compared to a solid, round conductors of the same cross-sectional area.
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However, PCBs can still not approach the performance of litz wire used in high-frequency WPT coils, which commonly use
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wire diameters in the range of tens of micrometer~\cite{zhaoDesignOptimizationLitzWire2023}.
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\textcite{lopeFrequencyDependentResistancePlanar2014} and \textcite{nomotoSplittingConductorsCoils2024} propose a
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mitigation that aims to emulate a litz wire's structure in large, high-current PCB inductors, but their mitigation is
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heavily limited by the structure size achievable in common PCB manufacturing
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processes~\cite{nguyenReviewComparisonSolid2020}.
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A further factor that limits the high-frequency performance of PCB inductors is distributed capacitance. Not only does a
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large air coil exhibit more parasitic capacitance than an equivalent, smaller ferrite-core inductor simply due to its
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size, 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{Planar Inductors in RFIC Design}
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Beyond WPT, planar inductors are commonly used in radio frequency integrated circuits (RFICs). In RFIC design, the major
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challenges are area optimization and precisely predicting the inductor's characteristics during the design phase. Common
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optimizations include applying a variable trace pitch~\cite{lopez-villegasImprovementQualityFactor2000} and variable
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trace width~\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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% Note: They note that the main point behind the design is electrical symmetry of the two ports to make driving the
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% thing differentially cleaner. We should adopt this observation for our inductors, which likewise are electrically
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% symmetric when compared to a single-layer spiral inductor.
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\subsection{A Brief 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
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of wireless telegraphy after the turn of the twentieth century, coils with high inductance were needed for the
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construction of both transmitters and receivers, but the ferrites that would later permit their compact construction
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were still being developed. The ferromagnetic core material of choice back then was laminated iron, which was only
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useful at low frequencies due to eddy current losses. As a result, the inductors in radio circuits of the era were often
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constructed as air-core coils. While air-core inductors are immune to core saturation, the poor magnetic permeability of
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air necessitates a large number of wide turns of wire to reach useful inductance values, which for reasons of
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practicality or leakage inductance often could not be wound as a single layer cylindrical coil. This could be resolved
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by winding an inductor with many turns on multiple layers, which improves compactness and leakage inductance, but this
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in turn gives rise to increased distributed capacitance as now turns with a large voltage differential are layered right
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on top of each other.
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Before the invention of ferrites, a number of ways were devised to decrease distributed capacitance in multilayer
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inductors. These methods can be divided into two general categories: Optimizing the connecting order of turns and
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optimizing the winding schema of turns. Both aim at increasing spacing between parts of the coil that have a large
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voltage differential.
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The connecting order of turns was optimized at the assembly level by stacking coils in a particular
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way~\cite{flemingPrinciplesElectricWave1910} and at the component level by winding coils in a particular way to minimize
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the voltage differential between adjacent turns---a technique that is still used to this
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day~\cite{lopeFirstSelfResonant2021}. The main winding optimization in the first category concerns winding the
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turns of a cylindrical multilayer inductor not layer by layer, but instead layering them diagonally, effectively
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connecting adjacent turns in a diagonal zigzag pattern. Then as now, wound inductors applying this technique were not
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feasible to manufacture reliably by machine, but the technique can be closely replicated in PCB inductors as shown in
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\textcite{leePrintedSpiralWinding2011a}. The main limiting factors in a PCB implementation are the requirement for a
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large number of vias inside the inductor's turns limiting the achievable turn count\footnote{In PCBs, as opposed to
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integrated circuits (ICs), vias limit the achievable turn count when they need to be placed in-line inside the turns as
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opposed to on the inside or outside because a PCB's minimum trace/space widths are usually much smaller than the
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smallest feasible via, consisting of a minimum-size drill surrounded by a minimum-size annular ring.} and increasing
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equivalent series resistance (ESR) through the thin trace sections that are necessary to accomodate the via structure,
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as well as the layer pairing limitations when blind vias are used in multilayer PCBs.
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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 historic examples of which are shown in Figure\
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\ref{fig_illust_honeycomb_basket}. In a honeycomb coil, like in an universal winding, subsequent winding layers are
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wound at a criss-cross pattern. The characteristic feature of honeycomb coils is that the winding machine is adjusted to
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produce large air gaps between adjacent windings, resulting in a three-dimensional rhomboid pattern that is vaguely
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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 chapter.\footnote{Interestingly, the winding schemes of both honeycomb and
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basket-woven coils are also governed by the same coprimality condition between the number of turns and the number of
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inversions within each turn that we describe for our twisted inductors below, although we could not find an example in
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historic literature where this condition was explicitly stated~\cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
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kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querfurthCoilWindingDescription1954}.}
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\section{Twisted Inductor Design}
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In this section, we present a detailed derivation of the layout of twisted inductors. We approach this layout by
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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 on top of one another on different layers, which also helps to minimize the loop area of the inductor's
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connections.
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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:
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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.
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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.
|
|
|
|
Let $n$ be the turn count of our inductor. The resulting parametrization is:
|
|
|
|
\begin{align}
|
|
\varphi &= 2\pi n t\\\nonumber
|
|
r &= r_1 + \left|1 - 2 t\right| \left(r_2 - r_1\right)
|
|
\label{eqn_twolayer_spiral}
|
|
\end{align}
|
|
|
|
The resulting spiral trace starts at radius $r_2$ on the positive $x$ axis, and spirals inward until it meets $r_1$,
|
|
where the sign indicates a layer change, and the trace reverses to continue back to $r_2$ on another layer. 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 the other layer
|
|
of the PCB leading back to the start.
|
|
|
|
\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.]\hfill\\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.]\hfill\\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}
|
|
|
|
Setting the inversion count to $k=1$ in our proposed scheme yields the conventional two-layer counterwound
|
|
scheme~\cite{lopeFirstSelfResonant2021,sproHighVoltageInsulationDesign2021,leePrintedSpiralWinding2011a}.
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=\textwidth]{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}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=\textwidth]{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}
|
|
|
|
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 ($k=1$), which results in a
|
|
conventional two-layer inductor, and with $k=2$ inversions, creating two interleaved spirals on both the top and the
|
|
bottom layer of the PCB. Figure\ \ref{fig_nk_complex_illust} shows additional layout examples for other values of $n$
|
|
and $k$. For $k=\frac{1}{2}$, we get a standard single-layer planar spiral inductor for any turn count $n$, and for
|
|
$k=1$ we get a standard two-layer planar spiral inductor for any turn count $n$. In this chapter, we will call all
|
|
layouts with $k\ge 2$ \emph{Twisted Inductors}. The coordinate description of Equation\ \ref{eqn_twolayer_spiral} thus
|
|
becomes:
|
|
|
|
\begin{align}
|
|
\varphi &= 2\pi n t\\\nonumber
|
|
r &= r_1 + \left|1 - \left( 2 k t \mod 2 \right) \right| \left(r_2 - r_1\right)
|
|
\label{eqn_twisted_spiral}
|
|
\end{align}
|
|
|
|
%\begin{figure}
|
|
% \begin{center}
|
|
% \includegraphics[width=\figurescale]{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}
|
|
|
|
Topologically, the shape of our inductors can be described as a $(k, n)$-torus knot. From knot theory, we know that such
|
|
a torus knot exists if and only if both $n$ and $k$ are co-prime. Figure\ \ref{fig_nk_combined} illustrates a derivation
|
|
of the coprimality requirement. If we plot the spiral in polar coordinates on a cartesian plot we observe that for a
|
|
$n$-turn coil with $k$ inversions, the trace crosses the $\varphi$ axis once for each inversion, wrapping around $r$.
|
|
Likewise, it crosses the $r$ axis once for each turn of the inductor, wrapping around $\varphi$. Based on this, we can
|
|
re-label the angular axis in steps from $0$ to $k$, and re-label the radial axis in steps from $0$ to $n$. Labelling the
|
|
new angular axis $i$ and the new radial axis $j$, in the resulting integer lattice, the trace has slope $1$. We can
|
|
state the trace's trajectory as a function of a curve parameter $t \in [0, nk]$ as $f(t) = (i, j) = (t \mod n, t \mod
|
|
k)$. To produce a valid inductor, the trace must not intersect anywhere. Thus, the system of congruences
|
|
|
|
\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 if and only if $k$ and $n$ are coprime.
|
|
|
|
In the following paragraphs, we will derive analytical expressions for Ohmic resistance and inductance of inductors
|
|
derived under this schema.
|
|
|
|
%\begin{figure}
|
|
% \begin{center}
|
|
% \includegraphics[width=0.8\figurescale]{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}
|
|
|
|
\paragraph{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 nor its dimensions, the combined arc length of all traces of the twisted
|
|
inductor does not change. Twisted inductors require two additional vias per inversion, which will increase DC resistance
|
|
slightly, but the contribution of these vias will remain small in practical applications since the overall number of
|
|
vias is still no more than a couple per turn, and since each via only bridges the short distance between the inductor's
|
|
layers.
|
|
|
|
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}
|
|
|
|
\paragraph{Inductance}
|
|
|
|
Even for geometrically simple inductors, analytically calculating their inductance is a surprisingly hard problem whose
|
|
complexity quickly escalates when geometrically complex inductors are analyzed, when realistic wire shapes as opposed to
|
|
thin wire or current sheet approximations are used, and when taking into account differing magnetic permeabilities of
|
|
air or dielectrics and core materials. Instead of precise analytical models, a number of approximations are commonly
|
|
used. A commonly referenced approximation for the inductance of planar spiral inductors is given by
|
|
\textcite{mohanSimpleAccurateExpressions1999}, whose current-sheet approximation for circular planar spiral inductors we
|
|
will use here to estimate our inductor's inductance. The current-sheet approximation from
|
|
\textcite{mohanSimpleAccurateExpressions1999} reads:
|
|
|
|
\begin{equation}
|
|
\label{eqn_mohan_approx}
|
|
L = \frac{\mu n^2 d_\text{avg} c_1}{2}\left(\ln\left(c_2/\rho\right)+c_3\rho+c_4\rho^2\right)
|
|
\end{equation}
|
|
|
|
In this equation, $c_{1-4}$ denote four empirically determined coeficcients that are specific to the coil's shape. The
|
|
values for circular coils are $c_{1-4}=(1.00, 2.46, 0.00, 0.20)$. $\mu$ is the magnetic permeability of air (for an
|
|
air-core inductor), $n$ is the number of turns, $d_\text{avg}$ is the \emph{average} turn radius, i.e. $d_\text{avg} =
|
|
2\frac{r_1 + r_2}{2} = r_1 + r_2$. $\rho = \frac{r_2-r_1}{r_2+r_1}$ is the planar spiral inductor's \emph{fill ratio}.
|
|
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
|
|
chapter, 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}.
|
|
|
|
\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 (cf.\
|
|
Section\ \ref{sec_experiments}). While our measurements only cover a small number of inductor samples, their results are
|
|
more reliable than results from FEM and can serve as a baseline for future work on such simulations.
|
|
|
|
We conducted our FEM simulations as follows:
|
|
|
|
\paragraph{Ohmic Resistance}
|
|
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~\cite{meeekerFiniteElementMethod2015}:
|
|
|
|
\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 24 test coupons with a number of variations
|
|
of twisted inductors with winding count $n$ between $1$ and $25$, and twist count ranging from $k=\frac{1}{2}$ (simple
|
|
single-sided spiral inductor) to $k=37$. All test inductors had an inner diameter of \qty{15}{\milli\meter} and an outer
|
|
diameter of \qty{35}{\milli\meter} corresponding to the space available in our IHSM implementation.
|
|
|
|
\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
|
|
chapter, 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, we decided to evaluate the high-frequency performance of twisted inductors. It is well-known that self-resonant
|
|
frequency decreases when going from a single-layer spiral inductor to a two-layer spiral inductor while keeping
|
|
inductance and dimensions constant~\cite{zhangImprovedCompensationMethod2025}. Our measurements show this effect, with
|
|
it being more pronounced with higher turn count. Intuitively, this makes sense if we consider the mechanism of inductor
|
|
self-resonance. The primary contributor to self resonance, particularly in higher turn count inductors, is capacitive
|
|
coupling between the inductor's windings. In a single-layer spiral inductor, this effect gets partially mitigated since
|
|
the strongest coupling exists between adjacent windings, which here have only a small voltage differential as only a
|
|
fraction of the inductor's total voltage appears across each winding. Compared to this, when the inductor is constructed
|
|
as a simple two-layer inductor with $k=1$, now the start and end windings of the inductor, which have the highest
|
|
voltage differential, are located right on top of each other with the substrate in between. Making things worse, common
|
|
PCB substrates have a relative permittivity much larger than air (usually around $4$).
|
|
|
|
We observe that this decrease in high-frequency performance is eventually counteracted by increasing inversion count
|
|
$k$. While our test samples focused on smaller turn counts, we observe a notable increase from a self-resonant frequency
|
|
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 15 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{sidewaystable}
|
|
\centering
|
|
\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}
|
|
$1$& $3$& $0.03$& $-93.1$& $0.0095$& $-49.9$& $0.039$& $-43.6$& $0.008$& $-78.8$& $0.056$& $\textbf{465.07}$& $\textbf{0.0143}$\\
|
|
$1$& $4$& $0.03$& $-103.4$& $0.0108$& $-38.6$& $0.040$& $-47.5$& $0.008$& $-87.5$& $\textbf{0.059}$& $460.08$& $0.015$\\
|
|
$1$& $5$& $0.03$& $-89.7$& $0.0123$& $-35.3$& $0.041$& $-34.1$& $0.009$& $-84.4$& $0.055$& $460.08$& $0.0166$\\
|
|
\hline\rowcolor[gray]{0.9}
|
|
$2$& $1$& $0.12$& $-28.4$& $0.0253$& $-12.1$& $0.127$& $-17.3$& $0.024$& $-18.3$& $0.149$& $\textbf{245.51}$& $\textbf{0.0284}$\\
|
|
$2$& $3$& $0.12$& $-31.0$& $0.0270$& $-7.9$& $0.128$& $-18.8$& $0.025$& $-16.4$& $\textbf{0.152}$& $240.52$& $0.0291$\\
|
|
$2$& $5$& $0.12$& $-26.7$& $0.0299$& $-0.2$& $0.130$& $-13.1$& $0.027$& $-11.1$& $0.147$& $225.5$& $0.03$\\
|
|
\hline\rowcolor[gray]{0.9}
|
|
$3$& $1$& $0.26$& $-10.0$& $0.0454$& $-1.6$& $0.262$& $-9.5$& $0.044$& $-4.8$& $\textbf{0.287}$& $\textbf{145.71}$& $0.0461$\\
|
|
$3$& $4$& $0.26$& $-9.6$& $0.0479$& $5.0$& $0.265$& $-7.9$& $0.046$& $1.1$& $\textbf{0.286}$& $\textbf{145.71}$& $\textbf{0.0455}$\\
|
|
\hline\rowcolor[gray]{0.9}
|
|
$5$& $1$& $0.73$& $4.5$& $0.0755$& $-3.1$& $0.670$& $-3.4$& $0.074$& $-5.1$& $\textbf{0.693}$& $61.345$& $0.0778$\\
|
|
$5$& $3$& $0.73$& $4.3$& $0.0763$& $4.7$& $0.671$& $-3.4$& $0.074$& $1.8$& $\textbf{0.694}$& $\textbf{70.285}$& $0.0727$\\
|
|
$5$& $7$& $0.73$& $4.4$& $0.0802$& $16.2$& $0.675$& $-2.8$& $0.077$& $12.7$& $\textbf{0.694}$& $68.05$& $\textbf{0.0672}$\\
|
|
\hline\rowcolor[gray]{0.9}
|
|
$10$& $1$& $2.90$& $6.3$& $0.2513$& $7.6$& $2.700$& $-0.7$& $0.250$& $7.1$& $\textbf{2.718}$& $24.076$& $0.2322$\\
|
|
$10$& $3$& $2.90$& $6.4$& $0.2520$& $10.5$& $2.700$& $-0.5$& $0.250$& $9.8$& $2.714$& $\textbf{28.571}$& $0.2255$\\
|
|
$10$& $7$& $2.90$& $6.4$& $0.2554$& $16.9$& $2.700$& $-0.5$& $0.252$& $15.8$& $2.713$& $28.072$& $\textbf{0.2122}$\\
|
|
\hline\rowcolor[gray]{0.9}
|
|
$25$& $1$& $18.15$& $6.7$& $1.8843$& $9.7$& $16.900$& $-0.2$& $1.900$& $10.4$& $16.938$& $8.84$& $1.7024$\\
|
|
$25$& $3$& $18.15$& $6.8$& $1.8851$& $13.2$& 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 coil of each
|
|
turn count. Shaded rows indicate conventional two-layer planar inductors ($k=1$).}
|
|
\label{tab_coupons}
|
|
\end{sidewaystable}
|
|
|
|
\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 that were larger (up to \qty{90}{\milli\meter} outer diameter) and that had a higher turn count (up to 53)
|
|
compared to our initial set of samples. The parameters of these new samples and our measurement results are shown in
|
|
Table\ \ref{tab_wide_coils}. In these results, we can identify three clear trends. First, the ESR of twisted inductors
|
|
is generally poorer when compared to two-layer spiral inductors. This increase in ESR is due to the large number of vias
|
|
used in these sample inductors. It should be noted that while twisted inductors have worse ESR compared to conventional
|
|
two-layer inductors, in our first set of test coupons we saw that their ESR is still better than that of a single-layer
|
|
inductor because the traces can be made wider. Our second observation is that in every set of samples from this second
|
|
run of physically larger inductors, twisted inductors outperform conventional planar inductors in self-resonant
|
|
frequency by a considerable margin with an increase in SRF of up to \qty{58}{\percent} from our
|
|
$d_2=\qty{65}{\milli\meter}$ sample going from $k=1$ to $k=100$.
|
|
|
|
Our third observation is that unlike in the smaller inductors from Table\ \ref{tab_coupons}, in these larger instances,
|
|
twisted inductors show increased inductance by approximately \qty{3.7}{\percent} for our smallest samples, and
|
|
\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}
|
|
\centering
|
|
\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}
|
|
|
|
To evaluate twisted inductors in our WPT application, we measured the variation of the coupling between a pair of
|
|
inductors using an automated measurement 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.
|
|
|
|
\todo{pics of 3d printer test setup}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=.65\textwidth]{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 approximate our application, we loaded the secondary inductor with a \qty{10}{\ohm} resistor 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 Root Mean Square (RMS) mode.
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=0.8\textwidth]{symmetry_3turn_n_twist.pdf}
|
|
\end{center}
|
|
\caption{RMS output voltage of the test circuit from Figure\ \ref{fig_test_schematic} for three pairs of matching
|
|
inductors with one inductor rotating w.r.t.\ the other. The inductors have $n=3$ turns each and $k=\frac{1}{2}$,
|
|
$k=1$, 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}
|
|
|
|
Figure\ \ref{fig_symmetry_3turn_n_twist} shows the ratio between input and output voltage of our test link for a set of
|
|
three-turn inductors with multiple inversion numbers $k$ when one inductor is rotated. In practical WPT setups, the
|
|
transmitter and receiver coils are rarely aligned perfectly, so we show measurements across a range of radial offsets.
|
|
In line with our inductance measurements, coupling is lower at $k>0$ compared to a single-layer spiral inductor. Across
|
|
one revolution, we find that the single-layer spiral inductor exhibits the most voltage ripple, with simple two-layer
|
|
inductors with $k=1$ already improving ripple. For $k$ above $1$, ripple amplitude stay sconstant, but energy is shifted
|
|
into higher frequencies that are easier to passively filter on the WPT link's secondary side in our application.
|
|
|
|
Expanding our measurements in the previous section, we performed a series of measurements rotating both inductors. In
|
|
these measurements, the coils' distance is fixed \qty{1}{\milli\meter} and the radial offset is set to a worst-case
|
|
value of \qty{4}{\milli\meter}. Figure\ \ref{fig_rms_ripple_n3} shows the normalized output voltage of a WPT link made
|
|
from three-turn inductors with rotation of one inductor shown on the horizontal axis, and the rotation of the other
|
|
shown on the vertical axis.
|
|
|
|
We performed similar measurements on 24 of our test coupons at \qty{1}{\milli\meter} and \qty{4}{\milli\meter} radial
|
|
offsets. Figure\ \ref{fig_k_ripple_plot} shows the combined results of these measurements, with worst-case voltage
|
|
variation plotted across inversion count $k$ for multiple turn counts $n$ and radial offsets $r$. In this graph, we see
|
|
that twisted inductors improve ripple compared to conventional designs, even at a low inversion count such as $k=3$.
|
|
|
|
Concluding our measurements, we achieved our primary objective of reducing coupling variation under rotation, with
|
|
twisted inductors ($k>1$) improving over conventional two-layer spiral inductors, which perform better than simple
|
|
single-layer spiral inductors. This improvement is greatest for inductors with low turn count and consequentially coarse
|
|
pitch, as their turns deviate the furthest from a set of ideal, concentric circles.
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=.65\textwidth]{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]{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
|
|
% $k=\frac{1}{2}$)
|
|
% 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=.65\textwidth]{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 chapter 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}.}
|
|
\label{fig_rms_ripple_n3}
|
|
\end{figure}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=.65\textwidth]{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=.65\textwidth]{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}
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=.65\textwidth]{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}
|
|
|
|
Our derivation of twisted inductors opens up a space for future research. 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. Furthermore, the precise
|
|
contribution of vias to the twisted inductor's parasitics is interesting, especially for layouts with large values of
|
|
inversion count $k$. We suspect that via influence will be frequency dependant as vias and traces have distinct DC
|
|
resistances, and skin effect will affect both to a differring extent.
|
|
|
|
\section{Conclusion}
|
|
|
|
In this chapter, we introduced a novel layout approach for planar, multi-layer inductors. 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
|
|
conventional 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 in the supplementary
|
|
material to this thesis.
|
|
|
|
Applied to an IHSM design, a wireless power transfer system using twised inductors to power the rotating mesh improves
|
|
efficiency by reducing losses due to stray capacitance and reduces secondary-side ripple. The reduced secondary-side
|
|
ripple allows the use of smaller filtering components, reducing board mass and mitigating heavy components as a possible
|
|
fault location. Additionally, the reduced ripple allows the use of secondary-side voltage regulators with less voltage
|
|
headroom, further reducing power transfer losses. By directly embedding twisted inductors into the IHSM's secondary side
|
|
mesh monitoring PCB, construction is simplified. The resulting assembly is lighter and smaller, which reduces motor load
|
|
and enables the implementation of compact IHSM meshes.
|
|
|
|
%\begin{figure}
|
|
% \begin{center}
|
|
% \includegraphics[width=\figurescale]{symmetry_10turn_n_twist.pdf}
|
|
% \end{center}
|
|
% \caption{Coupled RMS output voltage of three pairs of matching inductors with $n=10$ turns each and
|
|
% $k=\frac{1}{2}$, $k=1$, and $k=3$, respectively, shown as in Figure\ \ref{fig_symmetry_3turn_n_twist}}
|
|
% \label{fig_symmetry_10turn_n_twist}
|
|
%\end{figure}
|
|
|
|
|