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IEEE TRANSACTIONS ON MAGNETICS, VOL. 4, APRIL 20Fractal InductorsNathan Lazarus1, Christopher D. Meyer2, and Sarah S. Bedair21Oak Ridge Associated Universities Fellowship Program, U.S.
Army Research Laboratory, Adelphi, MD 20783 USA2Sensors and Electron Devices Directorate, U.S. Army Research Laboratory, Adelphi, MD 20783 USAThis paper investigates the performance of planar inductors based on space filling curves, a family of fractals with the propertyof completely filling a bounded area. Fractal-based inductor design is a method for obtaining a very long trace lengthsand thusinductance densitiesin 2-D space as a replacement for the serpentines currently used in one layer inductors. Because of theintricate course created by a fractal curve, these types of inductors are particularly well suited for stretchable electronics, wherea tortuous path relieves mechanical stress and creates a more compliant structure. Inductors based on seven common space fillingcurves, all bounded within a one square millimeter area, were both simulated and measured experimentally and found to varybetween 3.0 and 7.1 nH. Lower order fractals were found to give comparable performance to serpentine inductors with similarinductance density. More complicated fractals, after more than two iterations, were found to have lower inductance density thansimilar resistance serpentines.
Mechanical simulations demonstrate a reduction in stress by a factor of 10 or more compared withthe loop and serpentine designs.Index Terms Fractals, inductor geometries, inductors, space filling curves, stretchable electronics.I. INTRODUCTIONCERTAIN types of fractals, geometries with the propertyof self-similarity 1, have the interesting property ofhaving an extraordinarily long path or perimeter length, thatin the limit approaches infinity, while maintaining a fixedbounded area. Electrical devices based on fractals have beenpreviously demonstrated, with fractal capacitors shown to givehigh capacitance density in a single layer process 2 and frac-tal antennas to have useful multiband frequency behavior 3.Previous studies 4, 5, both based on a fractal known asthe Hilbert curve, have suggested creating inductors based onspace filling curves, fractals with the property of completelyfilling a 2-D surface.
As with a meander or serpentine inductor611, a fractal inductor can be used to create a highinductance density in a single-layer inductor in circumstanceswhere access to multiple metal layers is limited due to costconsiderations or unusual fabrication processes.The fractal inductor is interesting as a potential design fora stretchable electronics, where highly deformable inductorshave applications in medical monitoring 12 and strain sens-ing 13. One of the most important challenges in stretchableelectronics is the creation of conductive traces able to undergovery large deformations without permanent deformation orbreakage. Metal electrodes are typically embedded or pat-terned on an elastomer, such as rubber or polydimethylsiloxanethat is able to stretch to several times the initial dimensions ofthe material. Unlike an elastomer, a straight metal electrodecan only stretch minimally without permanent damage andbreakage.
An inductor such as a planar coil or serpentine hasa series of parallel metal electrodes that create a very rigidmechanical structure unable to survive significant deforma-tions. Adding frequent kinks and changes in direction, inherentManuscript received September 26, 2013; revised October 30, 2013;accepted November 1, 2013. Date of publication November 11, 2013; dateof current version April 4, 2014. Corresponding author: N. Lazarus (e-mail:[email protected]).Color versions of one or more of the figures in this paper are availableonline at Object Identifier 10.1109/TMAG.20Fig. Construction of Hilbert curve (a) zeroth, (b)(d) first, and (e) and (f)second iterations.in a fractal design, creates a more compliant structure able toundergo large deformations without permanent damage.This paper investigates inductors based on seven differentwell-known space filling curves. Since each type of fractalfollows a different path, there is the potential for significantlydifferent performance between the fractals.
The mechanicalbehavior is modeled first and demonstrates an order of magni-tude reduction in mechanical stress compared with serpentinedesigns. The electrical behavior is then modeled, followedby fabrication and impedance characterization of each design.Although significant variation was found between the differentfractal designs, a number of fractals were found to givecompetitive performance with the serpentines while providingsignificantly greater mechanical compliance.II. SPACE FILLING CURVESA space filling curve can be defined as a trace that, in thelimit, fills every point in a bounded area 14. These curves aretypically defined by an iterative construction process based onan initial curve.
Each iteration creates a longer and more ornatecurve, with the individual curves identified by the number ofiterations. 1 shows the first stages of one of the most0018-9464 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See for more information.8400708 IEEE TRANSACTIONS ON MAGNETICS, VOL. 4, APRIL 2014Fig. (d) Sierpinski.
(e) Luxberg I.(f) Luxberg II. (g) R space filling curves.well-known space filling curves, the Hilbert curve 14. Thecurve begins with a simple upside-down U-shape, the zerothiteration Fig.
The space is then divided into four quad-rants, each containing a copy of the zeroth iteration Fig. 1(b).The top two copies are oriented the same as the initial curve,while the bottom right and bottom left are rotated by 90 and270, respectively. The quadrants are then connected togetherFig. 1(c) to form the first iteration Fig.
All of thesegment lengths are held constant at a third the original sidelength to remain within the same bounding area.This process is then repeated, with the first iteration againcopied in each quadrant with similar rotations Fig. 1(e).The copies are connected in the same order (bottom left totop left, top left to top right, and top right to bottom right)to form the second iteration Fig. 1(f), with the segmentlength again scaled down, this time to one seventh the originalside length.
The process can be repeated indefinitely, creatingTABLE ICURVE PROPERTIESincreasingly intricate and complex patterns as the iterationnumber increases.A large number of possible space filling curves havebeen discovered with different characteristics, some ofwhich include intersections or numerous sharp turns that areless desirable for inductors. This paper focuses on sevenwell-known curves with exclusively right angle turns and nointersections (Fig. Many of the curves have multiple namesin the literature, but will be called the Hilbert Fig. 2(a),Peano S 14 Fig. 2(b), Moore 15 Fig. 2(c),Sierpinski 15 Fig. 2(d), Luxberg I 16 Fig.
2(e),Luxberg II 16 Fig. 2(f), and R 16 Fig. 2(g) curves here.Table I shows the dimensions of each of the space fill-ing curves considered. Only two of the curves, Moore andSierpinski, form a complete loop, beginning and ending atapproximately the same location; the remaining curves end atother corners of the square, similar to a serpentine design. Fourof the curves, the Peano S, Luxbergs I and II, and R, are builtusing repetition on a three by three grid, resulting in a morerapid increase in both length and complexity. Since the third-order versions of these curves surpassed the lithographic limitsof several micrometers for the inductor fabrication processused, only the first two iterations were considered.
Due to thediamond shape of the Sierpinski curve, this curve was rotatedby 45 to fill a comparable area with the other curves.III. MECHANICAL BEHAVIORThe frequent bends in a fractal inductor result in a signif-icantly more compliant structure compared with a serpentineor loop. This behavior can be illustrated by comparing asimple one-turn serpentine inductor with an example fractalinductor. A serpentine or meander is relatively flexible inthe direction perpendicular to the major trace legs, but verystiff in the other direction due to the parallel metal lines.Fig. 3(a) and (b) shows a mechanical FEM simulation inCOMSOL of a 20 m-wide serpentine copper trace embeddedwithin poly(dimethylsiloxane) (PDMS); the outer dimensionsare 1 mm on a side. A uniaxial strain condition was appliedusing a fixed displacement along the opposing edge of thestructure.
PDMS was chosen as an elastomer commonlyused as a substrate for stretchable electronics. The YoungsModuli of PDMS and copper used were 1.84 MPa 17 andLAZARUS et al.: FRACTAL INDUCTORS 8400708Fig. Stress in copper inductor in PDMS undergoing 20% uniaxial strainin (a) vertical and (b) horizontal direction for serpentine and (c) vertical and(d) horizontal direction for Luxberg I first-order fractal inductors. The PDMSmatrix is dark due to the low stresses compared with the embedded coppertrace.1.20 GPa 18, respectively. When the inductor is stretchedlengthwise by 20% Fig.
3(b), the center conductor is per-pendicular to the applied force and is able to bend to relievethe stress in the material, resulting in a maximum stress of0.97 GPa. When the inductor is stretched by 20% in thevertical direction, however, the force is parallel to the tracesand forces the metal to stretch. This results in a peak stressof close to 30 GPa, 30 times higher. A square-loop inductor,with solid parallel traces in both directions, will have similarlyhigh stresses upon stretching.One method for allowing a trace to undergo large defor-mations is to deliberately use a tortuous path 21. By incor-porating periodic switchbacks or kinks, the overall structurebecomes more compliant and able to undergo larger strainswithout reaching the yield strength of the metal. One exampleof this effect is in the Luxberg I first-order fractal induc-tor. This Luxberg I inductor can be considered a one-turnserpentine incorporating a series of kinks along the verticallines.
These kinks serve to relieve stress in the inductor whena vertical stress is applied Fig. 3(c), dropping the peakstress from 30 to 2.0 GPa, a factor of 15. The horizontalstiffness remains low, with only a 30% increase in peak stressFig. 3(d) compared with the original serpentine.
The Hilbert,Sierpinski and Moore first-order inductors similarly behave assimple loops with kinks along the outer electrode for stressrelease.Mechanical simulations of the effects of 20% uniaxialstrains were performed for each design and the peak stress ineach case was extracted (Fig. A single-loop inductor andserpentine inductors with one to 10 turns were also simulated.In all cases, a 20 m width trace was embedded in PDMS,with outer dimensions of 1 mm on a side. The fractals arealigned, as in Fig.
2, with the exception of the Sierpinskifractals, which are rotated by 45 to fill the same area; x isthe horizontal and y the vertical dimension in that alignment.A 20 m gap was used for the closed-loop structures (thesimple square loop and Sierpinski designs).Due to rigid traces in both the applied stress directions,the square-loop structure experienced the highest stress levels.All of the serpentine designs behaved highly anisotropically,with a low peak stress in the x-direction (along the serpentinelength) and a high stress in the perpendicular direction. Withthe exception of the two designs with a long straight tracealong a boundary, the R zeroth and Moore first iterations, thefractals experienced relatively low stresses for applied strainsin both directions, 8400708 IEEE TRANSACTIONS ON MAGNETICS, VOL. 4, APRIL 2014Fig. (a) Simulated peak stress for 20% applied uniaxial strain in thex-direction (x-axis value) and y-direction (y-axis value) and (b) magnifiedview of lower stress region. Fractals are designated by the name first letterand order number with exception of Sierpinski (Si) and the Luxbergs I and II(designated as LI and LII, respectively).(all in centimeters) is given by 22Lself = 2l(ln(2lw + t)+ 0.5 +(w + t3l)).
(1)The mutual inductance between the two parallel wiresegments with lengths l1 and l2, aligned, as shown inFig. 5, is 24Lmut = sinh1 d sinh1 d sinh1 d2 + d2+2 + d2+ 2 + d22 + d2 (2)where = l1 + l2 +, = l1 +, and = l1 +.Fig. Illustration of two parallel wire segments.If the currents flowing in the wire segments are in the samedirection, positive mutual coupling results; negative mutualcoupling results when the currents flow in opposite directions.The total sum of the mutual inductance is added to the totalself-inductance to arrive at the total inductance of each fractaldesign (Fig. The inductances for a single loop and serpen-tine inductors with one through 10 turns were also calculatedusing the same model. In all cases, the outer dimension of theinductor was constrained to fit in a one millimeter square, withtrace width and thickness of 20 m and 10 m, respectively.Since the neighboring traces for the designs analyzed areprimarily carrying currents in opposite directions, negativemutual inductance results with a magnitude dependent on thewire spacing. For the simpler lower order designs, the mutualinductances are small, typically LAZARUS et al.: FRACTAL INDUCTORS 8400708Fig.
Modeled inductance. For each space filling curve, higher resistancecorresponds to higher iteration number; the R curve begins with zerothiteration, the remainder with the first iteration. For the serpentines, higherinductance corresponds to more turns; one turn through ten turns are plotted.by converting the S-parameters to complex impedanceparameters 19Z = R + j X (3)Fig. (a) First-order Luxberg I.
(b) Third-order Hilbert. (c) Second-orderMoore.
(d) Second-order Peano S space filling curve inductors.where R and X are the resistance and reactance, respectively.For the single-port inductors (the single loop, Moore andSierpinski designs), the single port impedance input impedancewas used. For the remaining designs, the scattering parameterswere converted to ABCD parameters 20 and the impedancewas calculated from B/D, giving the impedance of the networkat one port with the other port shorted. The inductance L andquality factor Q were then calculated usingL = X(4)Q = XR(5)where is the angular frequency. The extracted inductanceand quality factor for the Moore-type space filling curveis shown in Fig.
8, along with the characteristics for theloop inductor. Since the inductance and resistance extractedfrom the impedance are frequency dependent, the terms beginto vary from the low frequency values as the frequencyapproaches self-resonance and the capacitance impedancebegins to play a role, resulting in the rise in apparent induc-tance shown in Fig. The inductance and resistance wastherefore extracted at least an order of magnitude in frequencybelow the resonant peak where the parasitic capacitance effectsare minimal. The measured low frequency inductances are3.61, 4.68, and 5.82 nH for the first, second, and thirditerations, respectively, compared with 3.23 nH for the loopinductor.Five copies per inductor type were measured, with standarddeviation in inductor values 8400708 IEEE TRANSACTIONS ON MAGNETICS, VOL. 4, APRIL 2014Fig. Measured (a) inductance and (b) quality factor of Moore-type spacefilling curve inductors.comparable serpentines.
The resistance was extracted at lowfrequencies well-below resonance of the devices. At higherfrequencies, the proximity effect, the reduction in ac resis-tance due to magnetic fields from neighboring conductors, isexpected to play a role 25. Although difficult to analyzeanalytically 22, the proximity effect is known to be heav-ily dependent on the wire spacing with lower wire spacingresulting in larger magnetic fields in neighboring wires anda higher ac resistance 26. The more complex inductors,both fractal and serpentine, with finer line spacing and largermutual inductance are expected to have more pronouncedproximity effects.
9(b) and (c) show the average peakquality factor and self-resonant frequency, respectively, foreach inductor. The inductors ranged in value from 3.23 nH,for the simple square-loop inductor, to 7.87 nH for a 10 turnserpentine inductor. The self-resonant frequency was foundto vary from roughly 12 GHz, for the loop and one turnserpentine inductors, to 6.8 GHz for the 10 turn serpentineinductor. The self-resonant frequency for a number of thefractal designs, notably the Sierpinski and Moore fractals, wasFig.
(a) Low frequency inductance (b) peak quality factor and (c) selfresonant frequency for space filling inductors. For each space filling curve,higher inductance corresponds to higher iteration number; the R curve beginswith zeroth iteration, the remainder with the first iteration. For the serpentines,higher inductance corresponds to more turns; one turn through ten turns areplotted.lower than the serpentine for a given inductance; this suggeststhat the intricate traces add parasitic capacitance that degradesthe high frequency performance.LAZARUS et al.: FRACTAL INDUCTORS 8400708TABLE IIMODELED AND MEASURED INDUCTANCE COMPARISON TABLEThe overall performance was comparable with the predic-tions of the theoretical model (Table II). There were severalnotable differences, however. For the serpentine and loopinductors, the predicted inductances were within 12% ofthe expected values; the model performed well due to therelatively simple geometry, with a small number of segmentsarranged in a relatively simple pattern. The fractal geometries,with a much more complicated arrangement of line segments,had larger variation.
In of the most cases, the measured induc-tance was significantly higher than predicted by the model,with increases of as much as 30%. The one exception was theSierpinski-based inductors, which had lower inductance thanpredicted by between 10% and 30%. These variations from themodel may result from the calculations of mutual inductancebased on the assumption of filamentary traces, rather than thereal wire dimensions. Errors of tens of percent are not unusualfor simple analytical models of inductors, with the originalGreenhouse paper on this modeling technique having errorsas large as 22% for square planar coil inductors 23.As in the theoretical model, the serpentine inductors had arelatively high inductance for a given resistance compared withmost of the fractal designs, particularly for higher inductancesabove 6 nH. In this range, the inductance/resistance ratiorolls off more rapidly for the higher iteration space fillingcurves, with none competitive with the serpentine designs.
Inthe range from 4 to 6 nH several of the space filling inductordesigns behave similarly to the serpentine designs, includingthe Luxbergs I and II, Peano, and R first iteration and Hilbertfirst and second iteration designs. Certain designs of the spacefilling inductors do exceed serpentine performance below4 nH, with the R zeroth and Moore first-order designs showing5% and 8.5% higher peak quality factors, respectively, thanthe single turn serpentine design. Due to the large variationswithin each fractal geometry one family of curves did notconsistently outperform the other designs. The Hilbert curveswere found to give the most consistent results, with the firstand second iterations competitive with the serpentine and thethird iteration dropping off fairly slowly. All three measuredcurves in the Sierpinski family, predicted by the model togive a relatively good performance, were found to have lowinductance for a given length compared with the other designsmeasured.V. CONCLUSIONSeven common space filling curves were investigated forcreating single-layer inductors. Due to the complex natureof space filling curves, there is a significant variation bothfrom one class of curve to the other, and within a single typeof curve as the iteration number changes.
From an electricalstandpoint, the fractal inductor designs are most competitiveat lower iteration numbers; as the inductor became moreornate, the inductance does increase but at a lower rate thancomparable serpentine designs. There were clear benefits in themechanical domain, however, with almost all of the fractalsmore compliant than the more conventional designs, with peakstresses dropping by factors of between five and 15 dependingon the specific design used.ACKNOWLEDGMENTThis work was supported by the Army Research Laboratoryunder Cooperative Agreement W911NF-12-2-0019.REFERENCES1 E. Berenschot, H. Jansen, and N. Tas, Fabrication of 3D frac-tal structures using nanoscale anisotropic etching of single crystalline sil-icon, J. Microeng., vol. Nasserbakht, andT.
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