Deckmyn, Thomas

Rogier, Hendrik

Rossi, Marco

Vande Ginste, Dries

English

Thomas Deckmyn

Marco Rossi

Dries Vande Ginste

Hendrik Rogier

Crossref

Stochastic modeling framework for wireless power transfer in the radiative near-field

Ghent University Academic Bibliography

Stochastic Modeling Framework forWireless Power Transfer in the Radiative Near-FieldThomas Deckmyn, Marco Rossi, Dries Vande Ginste, Hendrik RogierGhent University-imec: Dept. of Information Technology, Technologiepark Zwijnaarde 15,9052 Ghent, Belgium, thomas.deckmyn@ugent.beAbstract—Given the growing interest in wireless power trans-fer, there is a need to quickly evaluate the efficiency of suchlinks for different power source and receiver separations andorientations. Moreover, a designer is not only interested inthe nominal operating conditions, but also in the variationsencountered due to variability and uncertainty in the systemparameters and deployment conditions. We therefore propose astochastic modeling framework to evaluate the efficiency of awireless power transfer system in both the radiative near-fieldand far-field. Nominal performance and variations in the figuresof merit of the system are rapidly accounted for by leveraginga vector spherical wave expansion, Wigner-D rotations and astochastic testing procedure.Index Terms— wireless power transfer, wireless channel mod-eling, stochastic testing.I. INTRODUCTIONThe interest in wireless power transfer (WPT) has increasedexplosively the last years, fueled by recent evolutions suchas the Internet of Things and Electrical Vehicles. It is clearthat, for the safe transfer of large power levels, inductivepowering via the near-field is to be preferred. However, alarge plethora of applications exist where huge amounts ofwireless sensors nodes need to be powered wirelessly at ratherlow power levels. These nodes are typically extremely difficultto access, distributed over large areas and deployed in hugequantities. The only option to recharge them is via WPT linksthat sweep over the coverage area in a rather quick manner,without paying particular attention to each individual node.Still, although it is impossible to provide individual care toeach node separately, the contribution of every sensor is crucialto maintaining detailed coverage of the area to be monitored.Therefore, there is a particular need for a modeling tool thatevaluates not only the nominal power transfer efficiency (PTE)of a WPT system, but that also studies all potential randomvariations and uncertainties due to random system tolerancesand receiver/transmitter locations. Moreover, WPT proves tobe more efficient at smaller distances. Therefore, besides beingbased on the conventional far-field approximations, such amodeling framework must also be able to accurately predictthe PTE in radiative near-field conditions.In this contribution, we present a novel modeling paradigmto evaluate the nominal PTE and its statistical variations forlinks in the Fresnel and Fraunhofer regions. The transmit andreceive antenna systems are described by commercial elec-tromagnetic (EM) field simulators. These component modelsare complemented by a hybrid plane-wave/vector-spherical-wave expansion based on a Green-function addition theoremto model the radiowave propagation characteristics down todistances of a few wavelengths and a generalized polynomialchaos expansion (gPC) to include variability and uncertainty.Once the radiation patterns of the different antennas areknown, the framework allows to model repositioning andreorientation of the transmit and receive systems without re-quiring any full-wave simulation. This, in turn, yields efficientuncertainty and variability modeling for varying deploymentconditions and random fabrication errors.In Section II, we describe the deterministic model thatfirst creates component models for the power source with thetransmit antenna and the receive rectenna. We introduce thewireless channel model that accurately describes far-field aswell as radiative near-field power transfer between the transmitand receive antennas. It is based on a plane-wave expansion ofthe Green function through an addition theorem. This yields adescription for the interaction between the radiation patternsof transmit and receive antenna by a translation operator,which is easily and accurately calculated for any set ofantenna positions in the radiative near-field. Next, the radiationpatterns are expanded into vector-spherical modes, enablingeasy rotation of both antennas through Wigner-D matrices.In Section III, we introduce the stochastic framework, whichoperates around the deterministic model in a non-intrusivemanner. To obtain the probability density function of the PTE,instead of performing Monte Carlo simulations, which onlyconverge with the inverse of the square root of the numberof processed realizations [1], in Section IV we leverage gPCexpansions [2], [3], to assess the impact of antenna variabilityand positioning uncertainties on the PTE. Given the large num-ber of random variables present in the complete wireless link,we introduce Stochastic Testing [4], [5] to reduce the numberof deterministic simulations needed to gather the statisticaldata. In Section V, the approach is validated on a simpleWPT system consisting of a transmitting horn antenna anda receiving Industrial, Scientific and Medical (ISM) flexibleantenna operating at 2.45 GHz.II. MODELING THE NOMINAL PTEConsider a WPT system whose building blocks are schemat-ically shown in Fig. 1. Note that, although in the scheme asingle transmit and receive antenna are assumed, the schemeVgZgZTX ZRXIscPower source/transmit antennaThevenin equivalentZLDCL RL VoutMatchingcircuitReceive antennaNorton equivalentRectifiersectionWireless channelradiative near-field modelFig. 1. Building blocks of the WPT system model for evaluation of the power transfer efficiency.can be extended to antenna arrays at both sides of the link [6].On the one hand, the power source is represented by itsThevenin equivalent, connected to a transmit antenna withinput impedance ZTX . On the other hand, the receive antennais described by its Norton equivalent, composed of the antennainput impedance ZRX and its short-circuit current source ISC ,related to the incoming field byISC = − 1C∫ΩT (rRX , rTX , kˆ)FRX(kˆ) · FTX(kˆ) dkˆ, (1)where the integration extends over the Ewald sphere Ω, with Ca normalization constant, F(kˆ) the radiation pattern of transmitor receive antenna, and the translation operator given byT (rRX , rTX , kˆ) =N∑n=0(2n+ 1)j−nh(2)n (krRX,TX)Pn(kˆ · rˆRX,TX), (2)with rRX,TX = |rRX − rTX | and rˆRX,TX = rRX−rTXrRX,TX .h(2)n (·) denotes the spherical Hankel function of the secondkind and order n, and Pn(·) represents the Legendre polyno-mial of degree n. The relative positions of the transmit andreceive antenna can be easily modified without requiring full-wave simulations, since the antenna radiation patterns remainunchanged. Only (2) must be updated to account for thenew positions. Moreover, the distance between transmit andreceive antenna may be reduced, bringing both antennas ineach other’s radiative near field, at the cost of more terms(larger order N ) in (2). To reorient both antennas withoutperforming any EM field simulation, we expand their radiationpatterns into vector spherical harmonics and apply Wigner-Drotations to these expansion coefficients. After reassemblingboth patterns, we can evaluate (1) in a straightforward manner,given these reoriented radiation patterns. For all mathematicaldetails about this model, we refer to [7].III. INCLUDING VARIABILITY AND UNCERTAINTYDuring WPT, variability in the antenna dimensions anduncertainty in deployment conditions, modifying the antennapositions and orientations, will affect the PTE in a randommanner. To quantify the statistical distribution resulting fromall random variables in the link, we leverage a a generalizedpolynomial chaos expansion of the formPTE ≈L∑l=0yxl φxl (x). (3)with x a vector containing all input random variables, relatedto the PTE through a multivariate polynomial of degree L.Prior to this expansion, the random variables xk in the vectorhave been made statistically independent through a Choleskidecomposition. An exponentially converging expansion is ob-tained when and orthogonal set of expansion polynomialsφxl (x) is chosen, with orthogonality defined by∫Ωkφxki,k(xk)φxkj,k(xk)dPxk =∣∣∣φxki,k(xk)∣∣∣2 δij . (4)In order to perform the stochastic analysis without requiringthe modify the deterministic approach outlined in Section II,we apply the statistical framework in a non-intrusive manner.To calculate yXk in expansion (3), we exploit (4) and obtainyxl = E[PTE(x)φxl (x)]=∫Ω1,...,ΩLPTE(x)φx1l1,1(x1) . . . φxKlK ,K(xK)dPx1 . . . dPxK .(5)This integral may be evaluated by a multidimensional Gaussquadrature rule, such as a tensor product rule or a Stroudcubature [8]. Yet, for a large number of random variables, thisapproach requires an excessively large number of WPT link re-alizations. The Stochastic Testing algorithm [4], [9] drasticallyreduces the number of tensor product Gaussian quadraturepoints needed to construct the probability density function ofthe PTE. First, the initial set of quadrature points is sortedaccording to their corresponding quadrature weights. Then,after selecting the point with the most dominant weight, ofthe remaining points, only those are chosen that significantlycontribute to the quadrature rule via a thresholding procedureoutlined in [4].IV. STOCHASTIC WPT LINK MODELWe now outline the stochastic framework to quantify thevariability and uncertainty in the WPT setup shown in Fig. 1.The nominal PTE is evaluated according to the method out-lined in Section II, for each radiative near-field link realizationat given transmit and receive antenna positions and orienta-tions. The rectifier network of the received antenna is analyzedby a harmonic balance simulation in Keysight’s AdvancedDesign System to obtain PTE= ηlink ·ηmatch ·ηrect as a productof the WPT link efficiency ηlink, matching network efficiencyηmatch and rectifier efficiency ηrect.A. Accounting for antenna variabilityTo simplify the analysis, in this contribution we will assumethat only the receive antenna is subject to variability. Variationsin the transmit antenna may be included in the framework bya straightforward extension of the theory. Random changes inantenna performance may be due to uncertain antenna dimen-sions and variations in relative permittivity r. Their effectson the radiation impedance ZRX = Zre + jZim and radiationpattern are described by polynomial chaos expansions, givenbyZre,im =LZre,im∑l1=0yxl1φxl1(xVAR) (6)Are,impq =LAre,impq∑l2=0yxl2φxl2(xVAR) (7)Bre,impq =LBre,impq∑l3=0yxl3φxl3(xVAR), (8)with xVAR the vector of (statistically independent) random vari-ables related to the antenna variability, and Apq = Arepq+jAimpqand Bpq = Brepq + jBimpq the spherical harmonics expansioncoefficients of the antenna’s radiation pattern. The latter areexploited to perform the antenna translations and rotations andevaluate the receive antenna circuits parameters in Fig. 1.B. Evaluating antenna position and orientation uncertaintyThe final gPC expansion of the overall WPT system PTEincludes both antenna variability and uncertainty in positionand orientation, accounted for byPTE =L∑l=0yxl φxl (xWPT) (9)with xWPT now a vector including all random variables in thelink. First, we determine (9) in MPTE stochastic testing pointsxWPTm . For each link realization corresponding to a quadraturepoint, we exploit the expansions (6)–(8) constructed in Sec-tion IV-A to describe both the antenna radiation impedancesand their radiation patterns. Second, application of the WPTmodel of Section II yields the position and orientation uncer-tainties xWPTm in the WPT link. Finally, the PTE is evaluatedfor each WPT link realization. This multi-stage approach limitsthe number of EM field simulations needed to construct theantenna macromodels. In addition, these macromodels arereused when antenna elements in the link are deployed atdifferent positions.V. EXAMPLEConsider a WPT link between a power source connected toan MI-212-1.72.45 GHz horn antenna, which emits 10 mWof power at 2.45 GHz to a flexible dual-polarized rectenna,as shown in Fig. 2. In nominal conditions, the WPT systemoperates at a distance d = 0.6 m ≈ 5λ (being in the radiativenear field), and both antennas are located in the xy-planewith their phase centers aligned. The receive antenna [10] hasa nominal radiation impedance ZRX = 49.91 − 1.93i Ω ateach feed, with more than 15dB of isolation better the feeds.On this antenna, a rectifier, voltage doubler and matchingnetwork are deployed (Fig. 3) to generate DC power. Thereceive antenna suffers from variability in its dimensions andsubstrate properties. The most significant design parametersL, W and r (Fig. 2 for more details) are found [11],[12] to vary as independent Gaussian random variables withstandard deviations, σL = 0.1268 mm, σL = 0.1628 mm andσr = 0.03190, respectively. Moreover, there is uncertaintyon antenna positions d, x, y and orientation angles θ and φ.All again vary as Gaussian random variables with standarddeviations σd = 16.66 mm, σy = σx = 6.66 mm andσφ = σθ = 10◦.yxdθφ(-xf,yf)LWLS WS(xf,yf)Fig. 2. WPT link between standard gain horn power source and flexible(substrate thickness 3.94 mm, r = 1.5259) dual-polarized rectenna at2.45 GHz. L=45.3854 mm, W = 44.4516 mm, LS = 14.88 mm,WS = 1 mm, (±)(xf , yf ) = (±5.7, 5.7) mm.To account for the effect of all normal random variables,we construct the polynomial chaos expansions (6), (7), (8)and (9) using Hermite polynomials. The stochastic testingprocedure constructs polynomial expansions (6)–(8) of degree6 by performing 84 ADS-Momentum full-wave simulationsof receive antenna realizations. Next, we concentrate on thepolynomial expansion (9) in eight random variables. To obtainan expansion of degree L = 4, stochastic testing requires495 quadrature points xLINKm . The PTE is evaluated very ef-ficiently by leveraging the approach outlined in Section II.Next, a Monte Carlo analysis based on 10000 realizationsis performed to validate the stochastic WPT link model. Forboth approaches, the cumulative distribution function (CDF)is shown in Fig. 4. A Kolmogorov-Smirnov test confirms thatVgZgZTXGenerator+standard gain hornThevenin equivalentWireless channelradiative near-field modelZRXIscLm C1 Lp DCpLpDCpC2 RL VoutReceive antennaNorton equivalentMatchingcircuitVoltage doublerand rectifier sectionFig. 3. Schematic of the example WPT link. Lm = 5 nH, diodes are HSMS-2850 Schottky diodes with Cp = 0.08 pF and Lp = 2 nH, C1 = C2 = 100 pF,RL = 100Ω.both curves indeed belong to the same distribution, with a sig-nificance level α = 5%. In the complete stochastic modelingframework, the electromagnetic field simulations turn out tobe the most costly step, since one ADS Momentum full-waveantenna simulation costs 15 s of CPU-time, requiring a total of21 min to generate the expansions (6)–(8). By leveraging theseresults, only an additional 54 s are needed to construct the finalPTE expansion (9), resulting in a total CPU time of 22 minfor the full model. In contrast, a straightforward applicationof (9), based on direct full-wave simulations without use ofthe efficient model outlined in Section II, would require 2 hof CPU time whereas the Monte Carlo approach needs 41 hto evaluate 10000 realizations.PTE (%)CDF0 0.2 0.4 1.00.6 0.8 1.20.40.20.60.81.0Monte Carlonew frameworkFig. 4. CDFs of the power transfer efficiency (PTE) of the WPT link: MonteCarlo simulation versus the novel stochastic framework.VI. CONCLUSIONWe have presented a new modeling framework to evaluatethe power transfer efficiency of a WPT link in the radiativenear field for both nominal and randomly varying operatingconditions. The technique cleverly exploits an addition theo-rem for the Green function, vector spherical harmonics andWigner-D rotations to describe the WPT channel for arbitrarypositions and orientations of the transmit and receive antennas.This is combined with a stochastic testing procedure to derivethe statistical distribution of the PTE from the distributions ofthe random variables characterizing the WPT link.ACKNOWLEDGMENTThis research was partially funded by BELSPO and by theResearch Foundation Flanders (FWO-Vlaanderen).REFERENCES[1] G. Fishman, Monte Carlo: concepts, algorithms, and applications.Springer Science & Business Media, 2013.[2] D. Xiu and G. E. Karniadakis, “The wiener–askey polynomial chaos forstochastic differential equations,” SIAM Journal on Scientific Computing,vol. 24, no. 2, pp. 619–644, 2002.[3] D. Xiu, “Fast numerical methods for stochastic computations: a review,”Communications in computational physics, vol. 5, no. 2-4, pp. 242–272,2009.[4] Z. Zhang, T. El-Moselhy, I. Elfadel, and L. Daniel, “Stochastic testingmethod for transistor-level uncertainty quantification based on general-ized polynomial chaos,” Computer-Aided Design of Integrated Circuitsand Systems, IEEE Transactions on, vol. 32, no. 10, pp. 1533–1545,2013.[5] P. Manfredi, D. Vande Ginste, D. De Zutter, and F. Canavero, “Gen-eralized decoupled polynomial chaos for nonlinear circuits with manyrandom parameters,” Microwave and Wireless Components Letters,IEEE, vol. 25, no. 8, pp. 505–507, 2015.[6] G.-J. Stockman, S. Lemey, H. Rogier, and D. Vande Ginste, “Full-Wave Modeling of Interacting Multiport Devices With Arbitrary RelativePositions and Orientations for Efficient EMI Assessment,” IEEE Trans.on Electromagnetic Compatibility, vol. 58, no. 4, 2, SI, pp. 1322–1330,AUG 2016.[7] G. J. Stockman, H. Rogier, and D. Vande Ginste, “Efficient modeling ofinteractions between radiating devices with arbitrary relative positionsand orientations,” IEEE Transactions on Electromagnetic Compatibility,vol. 56, no. 6, pp. 1313–1321, Dec 2014.[8] D. Xiu and J. S. Hesthaven, “High-order collocation methods fordifferential equations with random inputs,” SIAM Journal on ScientificComputing, vol. 27, no. 3, pp. 1118–1139, 2005.[9] M. Rossi, G.-J. Stockman, H. Rogier, and D. Vande Ginste, “Stochasticanalysis of the efficiency of a wireless power transfer system subject toantenna variability and position uncertainties,” Sensors, vol. 16, no. 7,2016.[10] L. Vallozzi, H. Rogier, and C. Hertleer, “Dual polarized textile patchantenna for integration into protective garments,” IEEE Antennas andWireless Propagation Letters,, vol. 7, pp. 440–443, May 2008.[11] M. Rossi, A. Dierck, H. Rogier, and D. Vande Ginste, “A stochasticframework for the variability analysis of textile antennas,” Antennas andPropagation, IEEE Transactions on, vol. 62, no. 12, pp. 6510–6514,2014.[12] M. Rossi, S. Agneessens, H. Rogier, and D. Vande Ginste, “Stochasticanalysis of the impact of substrate compression on the performanceof textile antennas,” IEEE Transactions on Antennas and Propagation,vol. 64, no. 6, pp. 2507–2512, June 2016.

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Stochastic modeling framework for wireless power transfer in the radiative near-field

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Ghent University Academic Bibliography