Robust Extended-Range Wireless Power Transfer Using a Higher-Order PT-Symmetric Platform
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PHYSICAL REVIEW RESEARCH 2, 013152 (2020) Robust extended-range wireless power transfer using a higher-order PT-symmetric platform Maryam Sakhdari, Mehdi Hajizadegan, and Pai-Yen Chen * Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA (Received 4 October 2019; accepted 6 December 2019; published 12 February 2020) A fundamental challenge for the nonradiative wireless power transfer (WPT) resides in maintaining the stable power transmission with a consistently high efficiency under dynamic conditions. Here, we propose and experimentally demonstrate that a frequency-locked WPT system satisfying the higher-order parity-time (PT) symmetry can achieve a near-unity power transfer efficiency that is resilient to effects of distance variation and misalignment between coils, and impedance fluctuations in electric grids. In specific higher-order PT electronic systems, a purely real-valued and coupling-invariant (nonbifurcated) eigenfrequency would enable the robust and efficient wireless charging without frequency hopping or adaptive impedance matching, even for midrange operation. We envision that this WPT technique may provide reliable, fast and efficient power delivery for a variety of consumer electronics, electric vehicles, and medical devices. DOI: 10.1103/PhysRevResearch.2.013152 I. INTRODUCTION when the coupling factor (as a function of distance and alignment between coils) changes, the operating frequency Radio-frequency (RF) wireless power transfer (WPT) has must be adjusted accordingly to maintain a high transfer experienced a rapid and widespread growth in recent years, efficiency [20,21]. Besides, the range of inductive power as driven by the rising demand for wireless charging in transfer remains a principal challenge. In spite of advances in consumer electronics, medical devices, sensors, and automo- various coil designs and the capability to create spatial Bessel tive applications [1–10]. The concept of transferring power beams [22], it remains difficult to overcome the performance without any physical contact was put forward soon after the deterioration due to the poor tolerance in coil misalignment, proposition of Faraday’s law of induction [11]. In late 1890s, especially in the weak-coupling regime [23,24]. Nikola Tesla conducted pioneering experiments with lighting Very recently, a new WPT mechanism based on the concept electric bulbs wirelessly through electrodynamic induction of parity-time (PT) symmetry incorporating a nonlinear gain- (also known as resonant inductive coupling) [12]. Although saturation element has been projected to address this long- his efforts appear to have met with little success, nowadays, existing challenge [7]. Specifically, this system is engineered considerable progress in WPT has been made in the realm to have a subtle balance of energy flowing in (i.e., gain) of nonradiative transfer that employs the near-field magnetic and out (i.e., loss), and in its exact PT-symmetric phase, a coupling to efficiently transmit electrical energy from a power nearly perfect transfer efficiency that is independent of the source to electrical loads [3]. While the WPT technology has coupling strength can be achieved for the short-range oper- motivated considerable research and development in the past ation. Even more importantly, this PT-symmetric nonlinear two decades [13–15], there are still several fundamental issues WPT system does not need the dynamic frequency recon- which need further investigation to maximize the potential figuration. In contrast, when the standard second-order PT of this technology. Traditional WPT systems are not robust system is adopted for wireless powering applications [7], the against alteration of distance and misalignment between coils system requires adaptive frequency tuning (although within [5,16], and variations in the terminating impedance of an a narrow range) and switch-mode amplifiers that cover the electric power grid or battery over time [1]. To date, several full range of operating frequencies. Adjusting the operating techniques have been proposed to optimize the efficiency frequency may increase the design complexity and cost, since of power transfer, which include dynamic adjustment of the system requires tunable filters with high quality factor and operating frequency (resonant-frequency tracking) [17,18], synchronization of convertor/invertor in the transmitter and combination of multiple receivers and repeaters for adaptive receiver. impedance matching [19], and adding nonlinear tuning ele- However, in the standard (second-order) PT-symmetric ments in circuits [7,10]. For these often used WPT schemes, electronic systems [25–34], the exact PT-symmetric phase requires a moderately strong coupling strength [25,28], and this makes the system fail to maintain a satisfactory efficiency *Corresponding author: [email protected] at weak/loose coupling (e.g., midrange power transfer). In addition, perturbing the load resistance of receiver could lead Published by the American Physical Society under the terms of the to a symmetry-breaking phase transition [29,34] and a low Creative Commons Attribution 4.0 International license. Further efficiency is obtained in such broken PT-symmetric phases. distribution of this work must maintain attribution to the author(s) In the following, we will present a new nonradiative WPT and the published article’s title, journal citation, and DOI. system with high-order PT symmetry, as shown in Fig. 1, 2643-1564/2020/2(1)/013152(11) 013152-1 Published by the American Physical Society SAKHDARI, HAJIZADEGAN, AND CHEN PHYSICAL REVIEW RESEARCH 2, 013152 (2020) FIG. 1. (a) Schematics for (i) standard, (ii) third-order, and (iii) generalized higher-order PT-symmetric electronic systems consisting of electronic molecules with gain (red), neutral (gray), and loss (blue) properties. (b) Third-order PT-symmetric wireless power transfer system, where a robot car (loss) is wirelessly charged by a platform consisting of a repeater (neutral) and an active transmitter (gain). The inductive coupling strength between the intermediate and receiver coils (κ) is random, and that between the transmitter and intermediate coils (κ)is tuned by rotating the transmitter coil and a feedback algorithm, such that κ = κ and PT symmetry is preserved. which can further overcome the drawback of the standard cal [36,37], acoustic [38] and electronic [25–34] platforms, PT-symmetric WPT system. Higher-order PT-symmetric elec- and has aroused intense interest for real-life applications, tronic systems in Fig. 1(a) exhibit a unique eigenspectral such as lasers [39–41], sensors [28–32,42–46], cloaks [47,48], feature: a purely real eigenfrequency that is insensitive to and nonreciprocal devices [49]. In electronics, the lumped- variations in the coupling strength and the gain-loss parameter element circuitry could be the simplest possible configuration (related to the receiver’s load impedance). Such a property for observation of exotic physics of PT-symmetric Hamilto- can ensure a transfer efficiency approaching 100% over a nian [30]. In analogy to optical cases, spatially-distributed, wide range of distances and misalignment levels, thereby balanced gain and loss can be readily realized with an active offering an unprecedented robustness and freedom in wireless negative-resistance converter [50,51] and a resistor, which charging. are connected through either capacitive or inductive links, as illustrated in Fig. 1(a). Figure 1(b) presents the proposed third-order (TO) PT-symmetric WPT systems with an equiv- alent circuit model shown in Fig. 1(a) (type ii). In this case, II. HIGHER-ORDER PT-SYMMETRIC the transmitter, repeater, and receiver are represented by the WIRELESS POWERING −RLC, LC, and RLC oscillators, respectively. Here, inductive In 1998, Bender and Boettcher introduced a class of couplings between the neighboring oscillators are identical, non-Hermitian Hamiltonians [35] satisfying PT symmetry. and the coupling between the transmitter and receiver is Counterintuitively, if a quantum system is invariant under the assumed to be ignorable. These coupled the −RLC, LC, combined action of space inversion (P ) and time reversal (T), and RLC oscillators are analogous to “electronic” molecules it exhibits either entirely real eigenspectra or complex conju- with gain, neutral, and loss characteristics, whose relations gate eigenvalue pairs. Although PT-symmetry was originally can be described by a PT -symmetric Hamiltonian. Applying considered as one of interesting theoretical findings in quan- Kirchhoff’s laws to this circuit, the system can be described tum physics, it has been experimentally demonstrated in opti- by Liouvillian formalism: ⎛ ⎞ 000 100 ⎜ ⎟ ⎜ 000 010⎟ ⎜ 000 001⎟ d ⎜ 2 2 2 2 ⎟ L = , L = ⎜− 1−κ κ − κ 1 1−κ 0 − 1 κ ⎟, (1) dτ ⎜ 1−2κ2 1−2κ2 1−2κ2 γ 1−2κ2 γ 1−2κ2 ⎟ ⎜ κ − 1 κ − 1 κ 1 κ ⎟ ⎝ 1−2κ2 1−2κ2 1−2κ2 γ 1−2κ2 0 γ 1−2κ2 ⎠ − κ2 κ − 1−κ2 1 κ2 − 1 1−κ2 1−2κ2 1−2κ2 1−2κ2 γ 1−2κ2 0 γ 1−2κ2 T ω . where ≡ (q1, q2, q3, q˙1, q˙2, q˙3) , qk andq ˙k correspond to measured in units of 0 The effective Hamiltonian can be the charge stored on the capacitor and the displacement cur- written as Heff = iL(Heff = i∂τ ), which is non-Hermitian † = rent in the kth oscillator (the subscripts 1, 2, and 3 denote (Heff Heff ) and symmetric with respect to the PT transfor- , = the active, neutral, and passive tanks, respectively), τ ≡ ω0t, mation, namely [PT Heff ] 0, with the natural frequency of the neutral LC√tank, the gain-loss − parameter (or non-Hermiticity) γ = R 1 L/C, the coupling J 0 I 0 strength characterized by the rescaled mutual inductance = and = , (2) √ P 0 J T 0 −I K κ = M/L, ω0 = 1/ LC, and all (angular) frequencies are 013152-2 ROBUST EXTENDED-RANGE WIRELESS POWER … PHYSICAL REVIEW RESEARCH 2, 013152 (2020) FIG. 2. Evolution of the real (top) and imaginary (bottom) parts of eigenfrequencies as a function the gain-loss parameter γ and the coupling strength κ, for the (a) third-order and (b) standard PT-symmetric electronic systems in Fig. 1(a) (see types i and ii).