Graduate Student Member, IEEE Icaro V Soares

Mingxiang Gao

Graduate Student Member, IEEE, Zvonimir Šipuš , Senior Member, IEEE Erdem Cil

Anja K Skrivervik

Member, IEEE John S Ho

Member, IEEE Denys Nikolayev
email: denys.nikolayev@deniq.com

Zvonimir Šipuš

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1 Wireless Powering Efficiency of Deep-Body Implantable Devices

Keywords: Biomedical electronics, implantable devices, pacemakers, radiation efficiency, wireless power transfer (WPT)

The wireless power transfer (WPT) efficiency to implanted bioelectronic devices is constrained by several frequency-dependent physical mechanisms. Recent works have developed several mathematical formulations to understand these mechanisms and predict the optimal operating conditions. However, the existing approaches rely on simplified body models, which are unable to capture important aspects of WPT. Therefore, this article proposes the efficiency analysis approach in anatomical models that can provide insightful information on achieving the optimum operation conditions. First, this approach is validated with a theoretical spherical wave expansion (SWE) analysis, and the results for a simplified spherical model and a human pectoral model are compared. The results show that although a magnetic receiver outperforms an electric one for near-field operation and both the sources could be equally use in the far-field range, it is in the mid-field that the maximum efficiency is achieved with an optimum frequency between 1 and 5 GHz depending on the implantation depth. The receiver orientation is another factor that affects the efficiency, with a maximum difference between the best and worst case scenarios around five times for the electric source and over 13 times for the magnetic one. This approach is used to analyze the case of a deep-implanted pacemaker wirelessly powered by an on-body transmitter and subjected to stochastic misalignments. We evaluate the efficiency and exposure, and we demonstrate how a buffered transmitter can be tailored to achieve maximum powering efficiency. Finally, design guidelines that lead to optimal implantable WPT systems are established from the results obtained with the proposed approach.

I. INTRODUCTION

W IRELESS power transfer (WPT) has been the subject of many works and publications mainly due to its technological and commercial appeal. It was founded in the works of Nikola Tesla, who first proposed the use of electromagnetic waves to wirelessly power electric devices. Even though the theoretical aspects have been studied throughout the years and are well-consolidated, the physical realization and the proposition of different techniques for its application have seen renewed interest with the rise of portable electronic devices [START_REF] Shinohara | History and innovation of wireless power transfer via microwaves[END_REF].

One way to classify the WPT techniques that use electromagnetic waves as the energy carrier is by the radiation region that comprises the distance between the transmitter and the receiver. For instance, the near-field techniques are based on the near-field coupling between antennas that can be either inductive or capacitive [START_REF] Dai | A survey of wireless power transfer and a critical comparison of inductive and capacitive coupling for small gap applications[END_REF].

Consequently, due to the short distance between antennas, these techniques can achieve high efficiency, and they are able to deliver high power levels [START_REF] Karimi | Wireless power and data transmission for implanted devices via inductive links: A systematic review[END_REF]. This efficiency can be even further increased by electromagnetic resonance [START_REF] Heidarian | A design technique for optimizing resonant coils and the energy transfer of inductive links[END_REF]. On the other side, the radiative or far-field techniques can cover a wide distance range with the drawback of lower efficiency. Therefore, they usually use highly directive antennas to reduce the power dispersed in free space and thus increase the received power [START_REF] Shen | Joint waveform and beamforming optimization for MIMO wireless power transfer[END_REF]. A new mid-field WPT technique has been proposed to achieve higher efficiency than the far-field techniques and larger distance ranges than their near-field counterparts. In the mid-field WPT, the distance between the transmitter and the receiver is in the order of a wavelength, and, in this case, the electromagnetic fields are composed of reactive and propagating components [START_REF] Ma | Midfield wireless power transfer for bioelectronics[END_REF], [START_REF] Nguyen | Midfield wireless power transfer for deep-tissue biomedical implants[END_REF].

In the context of biomedical engineering, the application of WPT for implantable bioelectronic and biosensor devices is envisaged to monitor physiological processes [START_REF] Shah | Radiative near-field wireless power transfer to scalp-implantable biotelemetric device[END_REF], drug delivery [START_REF] Joo | Soft implantable drug delivery device integrated wirelessly with wearable devices to treat fatal seizures[END_REF], and organ stimulation through electric signals [START_REF] Kozielski | Nonresonant powering of injectable nanoelectrodes enables wireless deep brain stimulation in freely moving mice[END_REF], to name a few. Therefore, different WPT techniques have already been applied for wireless powering bioelectronic implants [START_REF] Barbruni | Miniaturised wireless power transfer systems for neurostimulation: A review[END_REF]. For instance, near-field WPT is used for subcutaneous pacemakers [START_REF] Liu | An effective sandwiched wireless power transfer system for charging implantable cardiac pacemaker[END_REF], [START_REF] Campi | Wireless power transfer charging system for AIMDs and pacemakers[END_REF], [START_REF] Wang | A 403 MHz wireless power transfer system with tuned split-ring loops for implantable medical devices[END_REF], neural and spinal cord stimulators [START_REF] Ma | Double split rings as extremely small and tuneable antennas for brain implantable wireless medical microsystems[END_REF], [START_REF] Yeon | Feasibility study on active back telemetry and power transmission through an inductive link for millimeter-sized biomedical implants[END_REF], [START_REF] Mirbozorgi | Robust wireless power transmission to mm-sized free-floating distributed implants[END_REF], cochlear [START_REF] Hong | Cochlear implant wireless power transfer system design for high efficiency and link gain stability using a proposed stagger tuning method[END_REF], and ocular implants [START_REF] Qian | TSV based orthogonal coils with high misalignment tolerance for inductive power transfer in biomedical implants[END_REF], among other. All these applications share the same characteristic of having a small implantation depth. Conversely, far-field techniques are used for ultralow-power implants usually applied for biotelemetry [START_REF] Shah | Radiative near-field wireless power transfer to scalp-implantable biotelemetric device[END_REF]. However, for powering deep-implanted devices, the mid-field WPT is mostly used. For example, it has been applied for cardiac implants and pacemakers [START_REF] Ho | Wireless power transfer to deep-tissue microimplants[END_REF], [START_REF] Das | A multiband antenna associating wireless monitoring and nonleaky wireless power transfer system for biomedical implants[END_REF], [START_REF] Le-Huu | Bipolar spiral midfield wireless power transfer for cardiac implants application[END_REF], gastrointestinal endoscopy capsules [START_REF] Yousaf | Compacted conformal implantable antenna with multitasking capabilities for ingestible capsule endoscope[END_REF], [START_REF] Abid | Wireless power transfer to millimeter-sized gastrointestinal electronics validated in a swine model[END_REF], and neural implants [START_REF] Tanabe | High-performance wireless powering for peripheral nerve neuromodulation systems[END_REF].

Several physical, technical, and legal aspects impose limitations on the in-body WPT application. First, the body tissues are composed of dynamic, highly heterogeneous, dispersive, and lossy media that deteriorate the efficiency of WPT techniques which in other applications present a reasonable efficiency. The two prominent factors that reduce WPT efficiency are the electromagnetic wave attenuation in these lossy tissues, which increases with the operating frequency, and the reflection at the interface between different media [START_REF] Nikolayev | Electromagnetic radiation efficiency of body-implanted devices[END_REF]. Moreover, the miniaturized implantable receivers are usually both physically and electrically small, further reducing the maximum achievable efficiencies [START_REF] Nikolayev | Optimal radiation of body-implanted capsules[END_REF], not to mention other sources of losses such as impedance mismatching, misalignments, and internal resonances in the body organs. Another major concern is the human body exposure to electromagnetic fields, which must comply with regulatory limits for human use [START_REF] Campi | EMF safety and thermal aspects in a pacemaker equipped with a wireless power transfer system working at low frequency[END_REF]. Therefore, the WPT system parameters-such as operating frequency, nature of the source, and implantation depth-must be judiciously chosen to achieve the highest efficiency possible, taking into account the exposure-level regulations.

Several models have been proposed to mathematically assess the maximum radiation efficiency and optimal frequency for implantable bioelectronic devices. Simplified analytical models can help making conclusions about the optimal parameters of the system independently of the source and receiver design [START_REF] Berkelmann | Antenna optimization for WBAN based on spherical wave functions de-embedding[END_REF]-these conclusions are difficult to make with the 3-D full-wave simulations because it cannot search through the entire design space. The planar stratified heterogeneous model [START_REF] Poon | Optimal frequency for wireless power transmission into dispersive tissue[END_REF], [START_REF] Ho | Midfield wireless powering for implantable systems[END_REF] is a simple approach that allows an analytical formulation that includes the wave reflection between different layers. In this model, the equivalence principle replaces a volumetric surface current distribution (SCD) for an in-plane electric current density source that produces the same electromagnetic fields. Although this model is mathematically more straightforward and provides physical insight of the problem under analysis, it considers the human body as planar and infinite media. Therefore, it disregards the effects of the human body's shape and dimensions on radiation performance. Besides, the assumption of an infinite medium may lead to inaccuracies in the maximum efficiency prediction.

On the other hand, the spherical wave expansion (SWE) [START_REF] Skrivervik | Fundamental limits for implanted antennas: Maximum power density reaching free space[END_REF], [START_REF] Bosiljevac | Propagation in finite lossy media: An application to WBAN[END_REF], [START_REF] Gao | Rules of thumb to assess losses of implanted antennas[END_REF] is a semi-analytical approach that represents the human body as a spherical model composed of a homogeneous or concentric multilayered medium. In this case, the source is represented by an electric or magnetic dipole encapsulated by a small sphere positioned anywhere inside the spherical phantom. Then, the electric and magnetic fields are decomposed in vector spherical harmonics, and, from orthogonality relationships, the problem is mathematically solved by a linear system. From the computational point of view, this approach is less costly than a full-wave analysis and can lead to insightful results once the implantation depth mainly influences the level of achievable power density just outside the body. Furthermore, due to the high electric permittivity of the biological tissues, the near-field region is shorter than in free space; therefore, the radiation pattern of mm-sized implantable antennas is less sensitive to the shape of the phantom itself [START_REF] Gao | Rules of thumb to assess losses of implanted antennas[END_REF]. However, this approach is difficult to implement for more complex sources and realistic body models. Apart from that, the standing wave pattern in spherical models differs from the anatomical-shaped ones. Therefore, this difference is also verified in the WPT efficiency values obtained with both the phantoms. For this intent, a full-wave computation needs to be carried out to solve the inhomogeneous wave equation numerically given an arbitrary finite-sized source [START_REF] Nikolayev | Optimal radiation of body-implanted capsules[END_REF].

This work aims to investigate the electromagnetic energy exchange between an optimal on-body transmitter and an implantable receiver, as well as to analyze the influence of the system's parameters on WPT efficiency. We develop a general 2-D-axisymmetric (2-DA) full-wave approach, providing the best case scenario for the energy transfer efficiency as a function of frequency, implantation depth, and nature of the receiver, i.e., electric or magnetic. The basis for this problem formulation and some results for a simplified spherical phantom model have been presented in [START_REF] Soares | Physical bounds on implant powering efficiency using body-conformal WPT systems[END_REF]. However, the proposed approach is further extended and validated with a theoretical model based on SWE in this article. Then, both the methods are used to evaluate the frequency-dependent radiation efficiency for several implantation depths considering electric and magnetic receivers, and these results surpass the efficiency levels previously presented in the literature with other approaches.

Previous studies on modeling electromagnetic wave propagation in the human body mainly focused on wireless communications with implantable antennas [START_REF] Bosiljevac | Propagation in finite lossy media: An application to WBAN[END_REF], [START_REF] Teshome | A review of implant communication technology in WBAN: Progress and challenges[END_REF], medical treatments based on electromagnetic waves such as hyperthermia [START_REF] Guérin | Computation of ultimate SAR amplification factors for radiofrequency hyperthermia in non-uniform body models: Impact of frequency and tumour location[END_REF], [START_REF] Ho | Design and analysis of novel focused hyperthermia devices[END_REF], [START_REF] He | A hexagonal focused array for microwave hyperthermia: Optimal design and experiment[END_REF], [START_REF] Roca | Effect of frequency and field amplitude in magnetic hyperthermia[END_REF], or dosimetry [START_REF] Hirata | Assessment of human exposure to electromagnetic fields: Review and future directions[END_REF]. However, the investigation of wave propagation for WPT applications [START_REF] Ho | Wireless power transfer to deep-tissue microimplants[END_REF], [START_REF] Poon | Optimal frequency for wireless power transmission into dispersive tissue[END_REF] involves a specific focus on the electromagnetic interaction between an on-body transmitter and an in-body receiver. Therefore, the optimal operating conditions identified for in-body biotelemetry and hyperthermia applications might differ. Specifically, for the in-body biotelemetry, the goal is to maximize the omnidirectional radiation performance through the high-contrast tissue-air interface [START_REF] Nikolayev | Electromagnetic radiation efficiency of body-implanted devices[END_REF]. For hyperthermia, the main objective is to control the local and nonlinear [i.e., ε(E)] absorption of energy close to the source [START_REF] Ho | Design and analysis of novel focused hyperthermia devices[END_REF]. As a result, most deep-body implantable devices operate in the MedRadio band (401-457 MHz) [START_REF] Nikolayev | Immune-to-detuning wireless in-body platform for versatile biotelemetry applications[END_REF]; the optimal band for hyperthermia is between 130 and 500 MHz [START_REF] Bellizzi | On the optimal matching medium and the working frequency in deep pelvic hyperthermia[END_REF], whereas for WPT, the maximum efficiency is achieved around 1-3 GHz [START_REF] Poon | Optimal frequency for wireless power transmission into dispersive tissue[END_REF], [START_REF] Ho | Midfield wireless powering for implantable systems[END_REF], [START_REF] Soares | Physical bounds on implant powering efficiency using body-conformal WPT systems[END_REF], [START_REF] Nikolayev | Reconfigurable dual-band capsule-conformal antenna array for inbody bioelectronics[END_REF]. This work studies the electromagnetic exchange in an in-/on-body WPT system, derives, and demonstrates the optimal operating conditions in terms of efficiency and A muscle-equivalent spherical phantom with radius R p and dispersive complex permittivity εr ( f ), surrounded by a lossless matched medium with ε r ( f ), encloses the implant. Both (a) and (b) share the same axis and are surrounded by the same matched medium. For (a) 2-DA model, this implant is insulated by a capsule-shaped volume with ε r ( f ) and the receiving electric or magnetic antenna is modeled as their corresponding current distribution J s on the cylindrical aperture, whereas for (b) SWE model, the source is a lossless bubble with ε r ( f ) and radius a, and the receiver is an elementary electric or magnetic dipole. (c) Anatomical 2-D model of a human pectoral region comprising nine dispersive tissues. An on-body source represents the transmitter, and the receiver is a deep-tissue pacemaker implanted inside the heart at a depth of 37 mm from the closest skin interface and filled by a lossless dielectric with ε r ( f ), matched to the wave impedance in the myocardium. exposure as a function of implantation depth and nature of the power source antenna.

Moreover, this study applies for the first time the 2-DA full-wave formulation to assess the WPT efficiency in an anatomical scenario considering a deep-implanted mm-sized pacemaker in a transverse cut model for the human pectoral. The proposed approach has the advantage of taking into account the complexities of the human body regarding its shape, dimensions, and electromagnetic heterogeneity, being less computationally costly than a tridimensional full-wave analysis. In this way, it is possible to analyze the real-life application of WPT for powering a cardiac stimulator as presented in [START_REF] Ho | Wireless power transfer to deep-tissue microimplants[END_REF]. In addition, different configurations for electric and magnetic on-body sources are investigated, and the results for the anatomical model are compared with the simplified spherical phantom. Finally, the origin for the losses is discussed from these results, and ways to mitigate them and achieve maximum efficiency are investigated from the parametric analysis of the on-body transmitter that wirelessly powers the implanted pacemaker.

II. EFFICIENCY ANALYSIS WITH SIMPLIFIED

SPHERICAL MODELS The problem under analysis consists of a WPT system composed of an implantable bioelectronic device that acts as a receiver and a transmitter structure conformal to the surface of the body. To determine the achievable WPT efficiency, the human body is modeled as a dispersive homogeneous spherical phantom of radius R p with complex permittivity εr ( f ) equivalent to the human muscle tissue [START_REF] Gabriel | The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues[END_REF], as shown in Fig. 1(a). The phantom radius R p is directly related to the implantation depth, and its evaluated range embraces most deep-body implantable WPT applications. Inside this phantom, the implantable device is represented as a lossless ε r ( f ) capsule-shaped volume that is matched to the wave impedance in the surrounding tissue and has length L and radius R c . This spherical model allows the results obtained with the proposed approach be compared with the semi-analytical solutions based on SWE. In addition, this simplified model disregards the shape complexities and heterogeneities leading to more general results that are better suitable for an initial analysis of implantable WPT systems.

To eliminate losses due to reflection at the interface between the body and the external medium and thereby ensure the maximum WPT efficiency, the permittivity of the region around the phantom is considered as the real part of εr ( f ), i.e., the external medium is lossless ε r ( f ) and matched to the wave impedance in the muscle. In this way, the on-body transmitter is well-matched to the tissue, which can be physically realized using a high dielectric constant layer. This model disregards any impedance mismatching in the antennas and any system-level losses in electronic circuitry.

To mathematically formulate the problem, an equivalent problem can be proposed based on reciprocity [START_REF] Jackson | Classical Electrodynamics[END_REF]. For this intent, the radiation source is modeled as the electric current density distribution J s inside the phantom, as indicated in Fig. 1(a). The reciprocity principle can be applied since the medium is assumed linear. Consequently, the radiated fields induce a power flow in the lossy phantom and, from energy conservation, the transmitted power P t is given by

P t = P r + P d + i 2ω(W m -W e ) (1)
where P r and P d are, respectively, the received and dissipated powers, and W m and W e are the time-averaged magnetic and electric energies, respectively. From the time-harmonic electric E and magnetic H fields, as well as the source electric J s and magnetic M s current densities, each one of the total power components can be calculated as follows:

P t = s 1 2 E × H * • ds (2a) P r = p 1 2 E × H * • ds (2b) P d = 1 2 p σ |E| 2 dv ( 2c
)
where σ is the phantom's electric conductivity which is related to the imaginary part of the phantom's complex permittivity

[ε r ( f )] through σ ( f ) = 2πε 0 f [ε r ( f )]
. The integration domains s and p indicate that the integral is performed on the surface of the source and the phantom, respectively. Besides, the integrals (2a)-(2c) can be considerably simplified due to the problem's axial symmetry. Finally, the WPT efficiency η is defined as

η ≡ (P r ) (P t ) = 1 - (P d ) (P t ) (3)
with being the real part of the complex variable inside the brackets.

Hereafter, the implantable bioelectronic device is referred to as the receiver antenna, whereas the transmitter antenna is the radiating on-body structure.

A. Full-Wave 2-DA Analysis

To numerically evaluate the WPT efficiency using the proposed approach, this model was implemented using the fullwave 2-DA formulation available in COMSOL Multiphysics (assuming no variation in the fields in the ϕ direction). The electric and magnetic sources with size L = 1 cm and R c = L/3 are considered in this computation, which corresponds to the standard encapsulation size used for implantable and ingestible bioelectronic devices [START_REF] Nikolayev | Immune-to-detuning wireless in-body platform for versatile biotelemetry applications[END_REF]. Each source is represented by the electric J E s (r, ϕ, z) and magnetic J H s (r, ϕ, z) surface current densities given by [27]

J E s = 0, 0, cos π z L (4a) J H s = [0, 1, 0]. (4b)
Then, the WPT efficiency in (3) can be calculated from the evaluated integrals in (2a)-(2c).

B. SWE Analysis

The results from a semi-analytical approach based on spherical wave modal expansion are used as a reference to validate the proposed model. In this model depicted in Fig. 1(b), the electromagnetic fields E and H in a spherical structure free of charges can be expressed using vector spherical harmonics M and N [START_REF] Jackson | Classical Electrodynamics[END_REF]

E -i ηH = n,m a mn M mn N mn + b mn N mn M mn (5)
with

M mn = ∇ × rψ mn (6a) N mn = 1 k ∇ × ∇ × rψ mn (6b)
where k denotes the wavenumber of the considered media, r is the unit position vector in the r -direction, and ψ mn is the solution to the Helmholtz differential equation

ψ mn = 1 kr Z n (kr )P m n (cos θ)e imϕ . ( 7
)
The Schelkunoff-type spherical Bessel or Hankel functions are denoted by Z n , whereas P m n are the associated Legendre functions of the first kind [START_REF] Harrington | Time-Harmonic Electromagnetic Fields[END_REF]. From the orthogonality relationships for spherical harmonics, (5) becomes a linear system whose solution provides the modal representation for the electromagnetic fields. More details about this implementation can be found in [START_REF] Bosiljevac | Propagation in finite lossy media: An application to WBAN[END_REF].

The integrals in (2a)-(2c) can be evaluated from the spherical model coefficients and vectors and, thus, the efficiency can be obtained through [START_REF] Karimi | Wireless power and data transmission for implanted devices via inductive links: A systematic review[END_REF]. This theoretical approach can also be applied to off-centered sources [START_REF] Skrivervik | Fundamental limits for implanted antennas: Maximum power density reaching free space[END_REF], [START_REF] Bosiljevac | Propagation in finite lossy media: An application to WBAN[END_REF].

C. Experimental Validation

To experimentally evaluate the WPT efficiency of devices deeply implanted in lossy media, the measurement setup described in Fig. 2 is proposed. It consists of a transmitter positioned on the outside wall of a cubic glass tank filled with water. The choice of water as the medium for the measurements is based on the fact that its dielectric properties are well-characterized across all the analyzed frequency bands; therefore, it is best suitable for validating the proposed approach. In contrast, the receiver is placed inside the tank at a distance d from the transmitter, with aligned polarization. For these measurements, two kinds of transmitters were used: 1) a dipole and 2) a loop antenna with the dimensions shown in Fig. 2(b) and (c), respectively. On the other side, the receiver design is described in [START_REF] Nikolayev | Reconfigurable dual-band capsule-conformal antenna array for inbody bioelectronics[END_REF]. The results obtained with this experiment setup are expected to be representative of wirelessly powered devices implanted in biological media primarily composed of soft human tissues. At the same time, the glass permittivity can numerically approximate those of hard tissues.

The scattering parameters were measured using a two-port vector network analyzer (VNA) calibrated in the frequency range from 100 MHz to 5 GHz, in which the transmitter was connected to port 1 and the receiver to port 2. Based on these measurements, the WPT efficiency η can be calculated as

η = |S 21 | 2 1 -|S 11 | 2 (8)
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. in which the forward transmission coefficient S 21 is de-embedded from the input reflection coefficient S 11 . These measurements were carried out for distances d ranging from 50 to 200 mm.

ACCEPTED MANUSCRIPT / CLEAN COPY

D. Results and Discussion

The WPT efficiencies for the electric η E and magnetic η H sources were evaluated from the electromagnetic fields calculated through the 2-DA model and the SWE model. In Fig. 3, the efficiency for both the sources is shown as a function of the frequency for different phantom radius R p , which is asymptotically equivalent to the variation in the implantation depth. Besides, the chosen range comprises most of the applications for wireless-powered implants, from subcutaneous to deeply implanted devices. In addition, the measured efficiencies defined as η = |S 21 | 2 from different I. WPT approaches in the literature are indicated in Fig. 3 and they are detailed in terms of source type, operating frequency, and implantation depth in Table I. First, the results obtained from the two approaches previously described are compared. As can be seen in Fig. 3, the efficiency evaluated with the 2-DA model agrees with the one calculated with SWE, especially at the vicinity of the optimum frequency where the error between them is 0.64% for the electric receiver and 2.07% for the magnetic one. However, for lower frequencies, an offset between them is noticeable mainly due to the difference in the source formulation. Apart from that, the maximum efficiencies theoretically calculated with the proposed model corroborate with those experimentally achieved in previous works.

Regarding the efficiency behavior over the frequency, for near-field operations at lower frequencies, the magnetic source outperforms the electric one for all the evaluated implantation depths. For instance, Table II shows that the optimum frequency for the magnetic receiver is below the far-field region, roughly starting at 1 GHz. In addition, at the nearfield region, η E sharply increases until reaching its maximum, whereas η H initially increases and then reaches a plateau at the maximum efficiency. Conversely, both the receivers present similar decaying behavior in the far-field region, and η E asymptotically approaches η H at frequencies above the optimum one for the electric receiver.

As the reflection losses can be disregarded, this overall behavior for both the receivers can be explained by the fact that in low frequencies, the near-field losses are higher, whereas in the far-field region, the attenuation losses become more significant [START_REF] Skrivervik | Fundamental limits for implanted antennas: Maximum power density reaching free space[END_REF]. Thus, the optimum frequency corresponds to the tradeoff point between both the loss mechanisms. Moreover, once the near-field losses affect more electric than magnetic sources, the efficiency at low frequencies is significantly lower for this receiver.

The analysis carried out in this article considers fixed dimensions for the receiver. However, a further study on the peak efficiency and optimum frequency dependence on the receiver dimensions is presented in [START_REF] Nikolayev | Optimal radiation of body-implanted capsules[END_REF]. It has shown that the receiver length L is the parameter that impacts the efficiency the most; for instance, the larger the electric source, the higher the efficiency. On the other hand, for a magnetic source, the peak efficiency is reached within the range 1.5Rc < L < 3Rc. In addition, the fundamental limit on the electrically small antennas efficiency states that a reduction in the receiver size will also lead to a reduction in its radiation efficiency. Therefore, the receiver size considered in the previous analysis not only is in the range that leads to maximum efficiency but also corresponds to the size of practical implementation of implantable and ingestible antennas [START_REF] Nikolayev | Immune-to-detuning wireless in-body platform for versatile biotelemetry applications[END_REF].

The experimental results for the maximum efficiency and the optimum frequency considering an electric (dipole antenna) transmitter and a magnetic (loop antenna) transmitter are presented in Table III for distances d between the transmitter and the receiver from 50 to 200 mm. The measurements agree with the theoretical conclusions obtained with the proposed model and the SWE approach, showing that as the distance between the transmitter and the receiver (implantation depth) increases, the optimal frequency decreases for both the sources. However, while the electric transmitter leads to an efficiency peak around 1 GHz which corresponds to the mid-field region, the magnetic transmitter achieves maximum efficiency for near-field operation. Moreover, the measurements were carried out in a solution with 1 g of sodium chloride (NaCl) per liter. Although there are no significant changes in the optimum frequency, the increase in the medium conductivity leads to a reduction in the efficiency. The difference in the efficiency values is due to the fact that the theoretical ones represent the maximum achievable efficiency taking into account only the intrinsic loss of the channel. In contrast, other losses such as the scattering phenomena at the interface between the phantom and external media affect the experimental results (note that the antenna reflection losses are deembedded from the results; see Section II-C).

III. EFFICIENCY ASSESSMENT USING THE ANATOMICAL PECTORAL MODEL

To analyze the WPT efficiency in a more realistic scenario, we implemented a 2-D heterogeneous model of the human pectoral region presented in Fig. 1(c) with nine dispersive media with complex permittivity equivalent to those in the different human tissues [START_REF] Gabriel | The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues[END_REF]. Based on the wirelessly powered pacemaker proposed in [START_REF] Ho | Wireless power transfer to deep-tissue microimplants[END_REF], the receiver is represented by a circular region with a radius of 1 mm inside the heart with implantation depth d equal to 37 mm from the closest skin interface.

Similarly, as described for the spherical model in Section II, the electric permittivity of the region external to the body is the same as a lossless skin ε skin ( f ), whereas the lossless medium inside the pacemaker ε muscle ( f ) is also perfectly matched to the wave impedance of the myocardium. In this way, the reflection losses can be disregarded. For the analysis of this problem as a 2-D model, it is assumed that the implantable receiver and the on-body transmitter are perfectly aligned, and therefore, the z-axis invariance is valid, i.e., E(x, y, z) = E(x, y)e ik z z , where k z is the out-of-plane propagation constant. Moreover, due to the high tissue dispersivity, the radiated power is attenuated at short distances from the transmitter. Therefore, the computational cost can be reduced by analyzing only the cross section of the body region in which the receiver is implanted.

The same approach described in Section II was applied to the 2-D pectoral model to calculate the maximum WPT efficiency in a realistic scenario and compare these results with those obtained with the simplified spherical model. The reciprocity theorem is again invoked for calculation purposes, and the radiation source is considered inside the heart. This premise is still valid once the different media are linear and there is no polarization mismatch. Consequently, the obtained efficiency calculated with (3) is valid for the original problem of powering the implantable device.

However, differently from the 2-DA case, different receiver configurations and orientations can be used in the realistic planar model, as shown in Fig. 4. The first possibility is to assign an SCD over the line of length L (in the 2-D simulation) for the electric J E s (x, y, z) and magnetic J H s (x, y, z) receivers according to

J E s = ⎧ ⎪ ⎨ ⎪ ⎩ cos π y L A/m ŷ, if vertically oriented cos π x L A/m x, if horizontally oriented (9a) J H s = 1 V/m ŷ, if vertically oriented 1 V/m x, if horizontally oriented. (9b)
Henceforward, vertically oriented means that the receiver is oriented in the y-axis direction (ŷ), whereas horizontally in the x-axis direction (x), according to the coordinate system depicted in Fig. 4.

Subsequently, the next alternative is to define the receiver as an electric point dipole (EPD) or magnetic point dipole (MPD). In each one, the electric n E p or magnetic n H p dipole moment vector is assigned as follows:

n E p = 1 A•m ŷ, if vertically oriented 1 A•m x, if horizontally oriented (10a) n H p = 1 A•m 2 ŷ, if vertically oriented 1 A•m 2 x, if horizontally oriented. (10b)
Finally, it is also possible to define an out-of-plane magnetic current (MC) for modeling the electric receiver or an out-ofplane line current (LC) for the magnetic one. Respectively for each case, the electric or magnetic fields are orthogonal to the current distribution direction and the position vector. Therefore, the out-of-plane magnetic I M = 1 V and electric I E = 1 A currents must be assigned following the direction and orientation shown in Fig. 4(c) and (f), respectively.

By taking into account the three source types and the vertical and horizontal orientations, six configurations for each electric and magnetic receiver are possible. Finally, the WPT efficiency was evaluated for all the described scenarios, and the results are presented in Fig. 5. It is possible to see that the efficiencies evaluated with different sources agree between them, which demonstrates the flexibility of the proposed model for dealing with different source configurations.

The source orientation is the factor that mainly affects the radiation efficiency. In the best case scenario, the sources are horizontally oriented, whereas the vertical orientation corresponds to the worst position for the receiver. Since the receiver is vertically oriented, the radiation beam propagates toward the direction parallel to the chest boundary, which is the closest to the receiver and, therefore, the region that contributes the most for the integral in (2b). This behavior was also pointed out in [START_REF] Nikolayev | Electromagnetic radiation efficiency of body-implanted devices[END_REF] but, from the results shown in Fig. 5, it is possible to evaluate the difference between best and worst case scenarios quantitatively. Namely, for the best positioning, the electric source presents an optimum frequency 590 MHz higher with a maximum η E approximately five times the one verified with a vertically oriented source. A similar offset in efficiency magnitude can be verified between a horizontal magnetic source and a vertical magnetic source. For lower frequencies, the best positioning leads to an efficiency around 2.5 times higher; however, the maximum difference between them surpasses 13 times in the far-field region.

It is also noticeable that the radiation efficiency behavior for an electric or magnetic receiver obtained with the realistic pectoral model agrees with those obtained with the simplified spherical model. Namely, for the horizontally oriented electric sources, η E increases in the near-field range with a peak of 0.77% at 1.74 GHz, close to the optimum frequency shown in Table II for the same implantation depth of 37 mm, and then decreasing for higher frequencies. In contrast, η H reaches Fig. 5. Efficiency was evaluated with the realistic pectoral model for an (a) electric η E and (b) magnetic η H receiver implanted in the heart. Different receiver orientations (vertical and horizontal) and configurations (SCD, EPD and MPD, out-of-plane MC and LC) are compared. its maximum at lower frequencies, remaining almost constant around 10% in the near-field, before decreasing for frequencies above 1 GHz. The sharp reduction in radiation efficiency by a magnetic source in the far-field region is explained by the fact that this kind of source achieves its optimal performance as an electrically small antenna [START_REF] Nikolayev | Dielectric-loaded conformal microstrip antennas for versatile in-body applications[END_REF], [START_REF] Nikolayev | Minimizing electromagnetic exposure for wireless mm-size neural implants[END_REF], i.e., when L < λ/10 which is satisfied only in the near-field region.

As expected, there is a significant reduction in the maximum efficiencies obtained with the realistic model compared with the simplified spherical model. First, the latter considers a homogeneous medium equivalent to the human muscle tissue, whereas in the pectoral model, several organs with different electromagnetic properties are taken into account. Two predominant loss sources are responsible for this efficiency deterioration: the increased attenuation, mainly in the fat and muscle layers where the power is dissipated before reaching the receiver in the heart; and the reflection losses due to the wave impedance contrast between the media that compose the tissues. In this last case, with the multiple reflections, most of the reflected power is dissipated in the tissue reducing the amount of power that reaches the receiver encapsulation. Therefore, this loss is accounted on (1) also through P d . Moreover, in a realistic scenario, the shape of the organs may lead to resonance modes responsible for local peaks and valleys in efficiency for some frequencies, such as 3.63 GHz for the electric source and around 1.74 GHz for the magnetic source.

IV. EFFICIENCY ASSESSMENT USING A PHYSICALLY REALIZABLE MODEL

The case study of a wirelessly powered pacemaker is analyzed to establish the optimum parameters that lead to the maximum efficiency in a physically realizable application. In this case, the human pectoral model in Fig. 1(c) is used. However, the transmitter is now localized in the chest, as shown in Fig. 6(a). This transmitter comprises a substrate with the same dielectric properties as the medium external to the body and a lossless superstrate (or buffer) with thickness d and relative electric permittivity ε b r . The electric transmitter in Fig. 6(b) is defined by assigning an SCD J E s (x, y, z) given by

J E s = cos π x L , 0, 0 (11)
in which L is the transmitter length. Concurrently, the magnetic transmitter is modeled as a unitary out-of-plane electric current I E = 1 A on the edge of the source with opposite phases, as illustrated in Fig. 6(c). As seen from the field distribution for the electric and magnetic transmitters, they are, for instance, equivalent to a dipole and loop antenna, respectively.

A. WPT Efficiency as a Function of Transmitter Parameters

In this configuration, the two main transmitter parameters that can be changed to achieve the maximum WPT efficiency are the buffer thickness d and its permittivity ε b r . Therefore, a parametric analysis was carried out for both the parameters within the ranges: 1 ≤ d ≤ 50 mm and 1 ≤ ε b r ≤ 80. Furthermore, the lower and upper bounds for ε b r were chosen so that this range comprises the permittivities of all the nine tissues in the frequency range from 100 MHz to 5 GHz. Consequently, the effective permittivity for this pectoral region is also in the chosen interval.

The operating frequency must be chosen and fixed to carry out this parametric study. Based on the results shown in Fig. 5, the maximum theoretical WPT efficiency for a deep-implanted pacemaker charged by an electric source transmitter is achieved at the frequency of 1.74 GHz. Considering the frequency bands allocated for industrial, scientific, and medical (ISM) usage, the closest band to the optimal frequency is centered at 2.45 GHz. Henceforward, this is the frequency chosen for all the following analyses, and the transmitter length L was set as λ/2 at this frequency in free space.

To evaluate the WPT system for this configuration, (3) can still be applied with the results from integrals in (2a)-(2c); however, the integration domains need to be reformulated. In this case, s indicates that the integral is performed on the boundary of the pacemaker, whereas p represents the transmitter's boundary.

The contour levels for the efficiency as a function of the buffer thickness and permittivity are shown in Fig. 7(a) for electric and Fig. 7(b) for magnetic sources. As can be seen, the buffer parameters that lead to the maximum efficiency are d = 7 mm and ε b r = 80. The fact that the maximum efficiency is obtained with the highest permittivity can be explained once the wave impedance mismatching at the buffer-skin transition is practically eliminated. In addition, taking into account this permittivity, the maximum occurs exactly at λ/2. As proved in [START_REF] Alexópoulos | Fundamental superstrate (cover) effects on printed circuit antennas[END_REF] by considering the superstrate as a transmission line, it corresponds to the resonance condition, once the impedance seen in the transmitter-skin transition is equal to the impedance seen in the source, which leads to a high voltage at the source position, thus increasing the radiation efficiency. Moreover, it can be seen that as the buffer's thickness increases, it decreases the optimum ε b r in such a manner that a λ/2 periodicity is obeyed, as it also happens with a lossless transmission line.

On the other side, the maximum efficiency obtained with a magnetic source is approximately an order of magnitude lower when compared with the electric one. It means that at the frequency of 2.45 GHz, the magnetic source is suboptimal, as it was also predicted in the graph of Fig. 5(b), once it does not satisfy the electrically small antenna condition at this frequency. However, the maximum efficiency is achieved with d = 6 mm and ε b r = 46.5. It is important to mention that in this case, this maximum occurs in a significantly narrow distance range, which in practice would require precise positioning of the source.

B. Maximum Efficiency Assessment Due to Stochastic Antenna Misalignment

To classify and quantify the losses, the optimal scenario in which the transmitter is composed of an electric source and a buffer with d = 7 mm and ε b r = 80 is considered. Three main loss mechanisms can be identified: 1) reflection losses due to the wave impedance contrast between the on-body transmitter and the body tissue; 2) antenna near-field losses; and 3) the attenuation losses in the tissues. The first two losses can be mitigated by adequately designing the WPT system, whereas the latter cause is unavoidable. For instance, approximately 7% of the transmitted power is lost due to reflection in this optimum configuration, whereas 93% is due to attenuation and near-field losses combined. However, the optimal buffer parameters justify the low reflection losses and reduce the near-field region and its associated losses.

Apart from that, the maximum efficiency levels are significantly affected by fluctuations in the implantation depth caused by the intrinsic movements of the human body. For instance, the heart movements lead to fluctuations in the implant position, resulting in misalignment between the transmitter and the receiver. Therefore, to evaluate these efficiency variations, the full-wave simulation was performed at 200 random implant locations uniformly distributed within the heart. After that, the WPT efficiency for each pacemaker position is calculated, considering the optimal electric transmitter. The obtained results for maximum, minimum, and average efficiency are shown in Fig. 8. In addition, the standard deviations on the efficiency values σ η are shown for the ISM frequencies of 433 MHz, 915 MHz, and 2.45 GHz.

The results in Fig. 8 show that the difference between the minimum and maximum efficiency values increases with the frequency; however, the average efficiency and standard deviation remain within a close range for the frequency bands commonly used in biomedical applications. This variation in the optimum frequency is explained by the fact that the optimum frequency and the maximum efficiency decrease as the implantation depth increases, as previously described in Fig. 3.

Finally, the fact that the efficiency obtained in this case is significantly lower when compared with the optimum efficiency presented in Fig. 5 shall be addressed. It can be explained by the fact that the transmitter is considered to be a conformal structure covering the entire body surface in the ideal scenario described in Section III. In contrast, the current analysis takes into account a mono-antenna transmitter, leading to a much smaller integration domain p thus reducing the received power. In practical terms, it means that the single antenna mid-field WPT solutions currently in the literature are suboptimal by many orders of magnitude. Therefore, to physically achieve the maximum efficiency theoretically obtained, further developments in mid-field WPT are required based on conformal antenna arrays such as in [START_REF]Conformal phased surfaces for wireless powering of bioelectronic microdevices[END_REF] and techniques for electromagnetic focusing in lossy media that have already been investigated for other biomedical applications [START_REF] Iliopoulos | Enhancement of penetration of millimeter waves by field focusing applied to breast cancer detection[END_REF].

C. Electromagnetic Exposure Assessment

Another major concern of in-body WPT applications is the electromagnetic exposure levels. The surface profiles for the local specific absorption rate (SAR) normalized by the transmitted power are shown in Fig. 9 for the optimal electric and magnetic sources. First, the SAR profiles follow the field distribution shown in Fig. 6(b) and (c), being higher in the center for the electric source, whereas it reaches its maximum around the source edges for the magnetic. In addition, in both the cases, the peak SAR is located in the skin layer, and no significant SAR values are observed in the vital organs. However, by comparing both the SAR levels, it is noted that the magnetic source leads to a larger SAR, almost twice the value for the electric source, even with much lower efficiency levels. Since the SAR is related to the attenuation in the tissues that cannot be physically mitigated, it is possible to conclude that the efficiency is maximized by properly choosing the transmitter parameters, and the electromagnetic exposure is minimized.

V. GUIDELINES FOR DESIGNING OPTIMAL IMPLANTABLE WPT SYSTEMS

The analysis and results described in Sections II-IV lead to useful insights that can be applied as guidelines to design implantable WPT systems that are able to achieve efficiency close to the expected optimal values. For instance, given the implantation depth of the receiver, the following points can be considered.

A. WPT Technique and Operating Frequency

For subcutaneous implants, near-field techniques such as inductive and capacitive coupling can achieve higher efficiency levels. However, as the implantation depth increases, the maximum efficiency is reached at frequencies between 900 MHz and 3 GHz. Therefore, the mid-field and far-field techniques can show better performance for powering of deep-implanted miniature bioelectronics. Moreover, given the implantation depth and the operating frequency, the maximum efficiency values are given in Fig. 3.

B. Antenna Type and Orientation

At frequencies below 300 MHz, magnetic loop antennas are able to reach significantly higher efficiencies than electrictype antennas. On the other side, the high permittivity of the human tissues electrically increases the effective aperture size of dipole or patch antennas, making them more suitable for implantable WPT systems operating at frequencies higher than the sub-GHz band. The maximum efficiency values for both the electric and magnetic antennas as a function of the frequency are presented in Fig. 3 whereas the effect of the source orientation is shown in Fig. 5.

C. On-Body Transmitter Design

For single on-body transmitter applications, higher WPT efficiencies can be achieved by designing a buffer structure that separates the transmitter antenna and the skin, as shown in Fig. 7. In this case, the relative permittivity and the thickness of this superstrate must be chosen to reduce the wave impedance mismatch at the transmitter-skin interface [START_REF] Rice | High-contrast low-loss antenna: A novel antenna for efficient into-body radiation[END_REF]. For instance, for an electric source, the maximum WPT efficiency is achieved for the buffer thickness equal to λ/2. Apart from that, the standard deviation on the efficiency due to stochastic misalignments between the on-body transmitter and the in-body receiver caused by body movements is shown in Fig. 8 for the main ISM frequencies used in biomedical applications. Moreover, the local SAR patterns for an electric and magnetic on-body transmitter can be seen in Fig. 9. Even though the single-antenna configuration is constructively simpler, it exhibits suboptimal performance for implantable WPT systems. Therefore, to achieve efficiency closer to the levels given in Fig. 3 and lower exposure levels, conformal transmitter arrays should be used in addition to applying techniques for electromagnetic focusing in lossy media.

D. In-Body Receiver Design

Although the constraints on the shape and dimensions of in-body antennas reduce the degrees of freedom in their design, some points can be observed to enhance the receiver performance. First, an appropriate type of the antenna (electric or magnetic) has to be chosen depending on the receiver size and its operating frequency [START_REF] Nikolayev | Optimal radiation of body-implanted capsules[END_REF]. A high-permittivity encapsulation material can be used to increase the efficiency of an electric-type receiver [START_REF] Nikolayev | Robust ultraminiature capsule antenna for ingestible and implantable applications[END_REF]. The dielectric loading electrically increases the aperture size, reduces the near-field losses, and an efficiency close to the fundamental bounds can be achieved [START_REF] Nikolayev | Optimal radiation of body-implanted capsules[END_REF]. Other alternatives to enhance the efficiency are filling the receiver encapsulation with a low-loss dielectric material and using the thickest possible substrate [START_REF] Nikolayev | Immune-to-detuning wireless in-body platform for versatile biotelemetry applications[END_REF]. Finally, reconfigurable receiver antennas can be implemented to achieve near-optimal radiation performance across a broad range of conditions [START_REF] Nikolayev | Reconfigurable dual-band capsule-conformal antenna array for inbody bioelectronics[END_REF], [START_REF] Bao | Conformal capsule antenna with reconfigurable radiation pattern for robust communications[END_REF].

VI. CONCLUSION To wirelessly power implantable bioelectronic devices, several techniques can be used depending on the application characteristics. However, the combination of different physical mechanisms imposes a limit on maximum achievable wireless powering efficiency. Among them, the principal mechanism is the electromagnetic attenuation in the dispersive tissues, the reflection losses, and the fundamental efficiency limitation of electrically small antennas. In addition, there are system-level losses and resonant modes in the body cavities that also contribute to efficiency deterioration. Therefore, this work focused on evaluating the wireless powering efficiency of deep-body implanted devices and studying how the system's parameters could be tuned to reach the highest efficiency possible in a practical application.

Toward this goal, a mathematical formulation for this problem was proposed based on reciprocity and T -symmetry principles as well as considering an on-body conformal source, perfectly matched to the human tissue. Such hypotheses lead to optimum energy focusing and, thus, to the highest possible efficiency. To validate this approach, a semi-analytical SWE analysis was carried out, and the results for a simplified spherical representation for the human body were compared.

This first analysis indicates that a magnetic receiver outperforms the electric one in the near-field frequency range, whereas in the far-field, both the receivers show similar performance. Therefore, the optimum frequency is in the mid-field region and depends on the source type and implantation depth, leading to the highest efficiency.

Once the proposed approach was validated, it was applied to an anatomical model focused on a deep-implanted wirelesspowered pacemaker application [START_REF] Ho | Wireless power transfer to deep-tissue microimplants[END_REF]. This analysis verified that the overall behavior for both the sources agrees with the one predicted by the simplified spherical model, including the optimum operation in the mid-field range. Furthermore, this optimum frequency also agrees with the results presented in the literature obtained through different approaches [START_REF] Nikolayev | Electromagnetic radiation efficiency of body-implanted devices[END_REF], [START_REF] Nikolayev | Optimal radiation of body-implanted capsules[END_REF], [START_REF] Poon | Optimal frequency for wireless power transmission into dispersive tissue[END_REF], [START_REF] Ho | Midfield wireless powering for implantable systems[END_REF], [START_REF] Skrivervik | Fundamental limits for implanted antennas: Maximum power density reaching free space[END_REF], [START_REF] Bosiljevac | Propagation in finite lossy media: An application to WBAN[END_REF], [START_REF] Gao | Rules of thumb to assess losses of implanted antennas[END_REF]. Even though there is a significant efficiency reduction in the realistic model results, the wave impedance contrast between the different tissues imposes reflection losses that were disregarded in the simplified model. Moreover, with this model, it was verified that the receiver orientation is another practical aspect that can reduce efficiency. Considering the source positioning, the maximum difference in efficiency between the best and worst case scenarios is around five times for an electric receiver, whereas for the magnetic one, this difference can be above 13 times. The effect of the source orientations is justified once in the best case, i.e., for a horizontally oriented receiver, the power flows toward the chest line, which is the closest point to the receiver, contributing more to the overall received power. It is important to mention that the flexibility in analyzing different source types is another contribution of the proposed approach compared with other methods of analysis.

Finally, a WPT system considering a deep-implanted pacemaker as the receiver and a multilayered (substrate-sourcebuffer) transmitter positioned in the chest was considered to analyze the impact of the transmitter's parameters on the powering efficiency and to provide information on how these parameters could be tuned to achieve the maximum efficiency. This parametric study has shown that in the 2.4-GHz ISM band, the electric transmitter leads to efficiency over a degree of magnitude higher than a magnetic one. In addition, the maximum efficiency can be achieved with a lossless highpermittivity buffer, matched to the skin permittivity, and thickness equal to λ/2 in this medium. Furthermore, other local maxima can be obtained for thicker and lower permittivity superstrates, respecting a λ/2 periodicity.

To summarize, the approach proposed in this article is able to predict the maximum achievable powering efficiency considering simplified or anatomical body models as well as to provide information on the parameters that can be optimized to achieve this maximum efficiency. In addition, even though the case study of a deep-implanted pacemaker was analyzed in this article, the same approach could also be used to evaluate the powering efficiency for other deep-body implantable devices.

Fig. 1 .

 1 Fig. 1. Geometrical representation of the problem under analysis. (a) and (b) Simplified spherical model (not to scale). A muscle-equivalent spherical phantom with radius R p and dispersive complex permittivity εr ( f ), surrounded by a lossless matched medium with ε r ( f ), encloses the implant. Both (a) and (b) share the same axis and are surrounded by the same matched medium. For (a) 2-DA model, this implant is insulated by a capsule-shaped volume with ε r ( f ) and the receiving electric or magnetic antenna is modeled as their corresponding current distribution J s on the cylindrical aperture, whereas for (b) SWE model, the source is a lossless bubble with ε r ( f ) and radius a, and the receiver is an elementary electric or magnetic dipole. (c) Anatomical 2-D model of a human pectoral region comprising nine dispersive tissues. An on-body source represents the transmitter, and the receiver is a deep-tissue pacemaker implanted inside the heart at a depth of 37 mm from the closest skin interface and filled by a lossless dielectric with ε r ( f ), matched to the wave impedance in the myocardium.

Fig. 2 .

 2 Fig. 2. Experimental evaluation of WPT efficiency in lossy media. (a) Transmitter is positioned outside a cubic glass tank filled with water, whereas the receiver designed in [44] is inside the tank at a distance d from the transmitter. Then, the scattering parameters are measured with a two-port VNA considering (b)-(d) electric (dipole antenna) and (c)-(e) magnetic (loop antenna) transmitters. All the dimensions are in millimeters.

Fig. 3 .

 3 Fig. 3. WPT efficiency for an (a) electric η E receiver and (b) magnetic η H receiver as a function of frequency and phantom radius R p (asymptotically equivalent to implantation depth) evaluated through the 2-DA and the SWE models. The red dots indicate the efficiency of physically realized WPT systems proposed in the literature and detailed in TableI.

Fig. 4 .

 4 Fig. 4. Different receiver configurations are used for the realistic model analysis. An electric receiver can be defined through (a) surface electric current distribution J E s , (b) EPD n E p , and (c) out-of-plane MC I M . Alternatively, a magnetic receiver can be modeled as (d) surface MC distribution J H s , (e) MPD n H p , and (f) out-of-plane LC I E . The contour lines depict the electric E and magnetic H field distributions in arbitrary units.

Fig. 6 .

 6 Fig. 6. Analysis setup for the parametric analysis of the efficiency in a wireless-powered pacemaker application. (a) Close-up view of the on-body transmitter with length L composed of a radiation source positioned apart from the skin by a lossless superstrate (buffer) with thickness d and permittivity ε b r and with a substrate matched to the medium external to the body. (b) Electric source modeled as a surface current density J E s with the electric field distribution indicated by the arrows and the magnetic field by the isolines. (c) Magnetic source modeled as an out-of-plane current I E with the magnetic field distribution indicated by the arrows and the electric field by the isolines.

Fig. 7 .

 7 Fig. 7. Contour plots for the efficiency as a function of the buffer thickness d and permittivity ε b r for (a) electric source η E and (b) magnetic source η H . The star indicates the global maximum position, and the dotted lines indicate the position corresponding to λ/2 its integer multiples as a function of ε b r .

Fig. 8 .

 8 Fig. 8. WPT efficiency considering an optimal electric transmitter. The continuous blue line is the efficiency averaged (avg η E ) over the 200 pacemaker positions. Besides, the blue area represents the maximum (max η E ) and minimum (min η E ) efficiency values. Finally, the red bars show the standard deviation on the efficiency σ η at the ISM frequencies of 433 MHz, 915 MHz, and 2.45 GHz.

Fig. 9 .

 9 Fig. 9. Surface plot of the local SAR (point SAR) in the region between the on-body transmitter and the deep-implanted receiver normalized by the transmitted power considering an optimal (a) electric and (b) magnetic source.

TABLE I COMPARISON

 I BETWEEN DIFFERENT IMPLANTABLE WPT SYSTEMS PROPOSED IN THE LITERATURE

TABLE II MAXIMUM

 II EFFICIENCY AND OPTIMUM FREQUENCY FOR THE 2-DA MODEL AS A FUNCTION OF THE PHANTOM'S RADIUS (ASYMPTOTIC TO THE IMPLANTATION DEPTH)

TABLE III MEASURED

 III MAXIMUM EFFICIENCY AND OPTIMUM FREQUENCY AS A FUNCTION OF THE DISTANCE d BETWEEN TRANSMITTER AND RECEIVER

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.ACCEPTED MANUSCRIPT / CLEAN COPY

This work was supported in part by the French Agence Nationale de la Recherche (ANR) through the Project "MedWave" under Grant ANR-21-CE19-0045 and in part by the French Region of Brittany through the Stratégie d'attractivité durable (SAD) Project "EM-NEURO."


Maxime Juston
email: maxime.juston@sncf.fr

Nicolas Damay

Christophe Forgez

Extracting the diffusion resistance and dynamic of a battery using pulse tests

Keywords: Battery, electrical characterization, open-circuit voltage, GITT, diffusion

The modeling of a battery by an equivalent circuit model requires the determination of its parameters. This determination can be done by exploiting the transient response of a battery to a current pulse (often called GITT: galvanostatic intermittent titration technique). A classical approach is to first separate the open circuit voltage (OCV) and the overvoltage, and then to extract the model parameters from the latter. However, the estimation of the OCV can be difficult, which can lead to errors on the overvoltage, especially for slow dynamics such as diffusion. We propose here a method to estimate the OCV during GITT and a method to estimate the overvoltage which allow the extraction of parameters associated with slow dynamics. The results brought by the proposed method are compared with more classical approaches.

Introduction

Modelling a battery using an equivalent circuit model requires the determination of parameters that represent the physical phenomena taking place [START_REF] Damay | Thermal modeling of large prismatic LiFePO 4 /graphite battery. Coupled thermal and heat generation models for characterization and simulation[END_REF][START_REF] Mauracher | Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification[END_REF]. The most classical method for determining the parameters is the use of current pulses, also called GITT (galvanostatic intermittent titration technique) [START_REF] Schmidt | Pulse-fitting -A novel method for the evaluation of pulse measurements, demonstrated for the low frequency behavior of lithium-ion cells[END_REF][START_REF] Mihet-Popa | Charging and discharging tests for obtaining an accurate dynamic electrothermal model of high power lithium-ion pack system for hybrid and ev applications[END_REF]. This method uses the transient response of the battery to characterize the model. However, this method faces two difficulties which are the estimation of an OCV during the test, and the exploitation of the resulting overvoltage which may not correspond to the theory. To estimate the OCV, the classical approaches are to use a linear interpolation along the current pulses or to use a so-called "pseudo-OCV" curve that is measured independently by applying a very low current. While the pseudo-OCV gives an estimate with an accurate shape, it requires more tests than a linear estimate. Several tests may be needed to consider the shape modification of the OCV due to: a different electrode balancing [START_REF] Kleiner | Changes of the balancing between anode and cathode due to fatigue in commercial lithium-ion cells[END_REF][START_REF] Dubarry | State of health battery estimator enabling degradation diagnosis: Model and algorithm description[END_REF][START_REF] Mergo Mbeya | Off-line method to determine the electrode balancing of Li-ion batteries[END_REF]; an heterogeneous operation inside the cell [START_REF] Damay | Heterogeneous behavior modeling of a LiFePO 4 -graphite cell using an equivalent electrical circuit[END_REF] and the hysteresis phenomenon between charge and discharge [START_REF] García-Plaza | Hysteresis effect influence on electrochemical battery modeling[END_REF][START_REF] Konishi | Origin of hysteresis between charge and discharge processes in lithium-rich layer-structured cathode material for lithium-ion battery[END_REF][START_REF] Moskon | In depth discussion of selected phenomena associated with intrinsic battery hysteresis: Battery electrode versus rubber balloons[END_REF]. Determining pseudo-OCV curves for all these conditions can be long and may not be suited for certain applications.

In the case of a battery with a non-linear OCV, the use of simple methods such as linear interpolation does not allow satisfactory overvoltage estimates. Figure 1 illustrates this by presenting a voltage simulation and the overvoltage estimated by using a linear estimation of the OCV. The estimated OCV differs from the real overvoltage in shape, and cannot be used to determine electrical parameters. The difficulty of the overvoltage estimation lies initially in the estimation of the open circuit voltage of the battery. A small error in proportion, a few mV, on the OCV estimate can lead to a large error on the overvoltage, which has an order of magnitude of the mV. Especially, the shape modification of the overvoltage can greatly impact the extracted parameters, as illustrated in figure 1. Several phe-nomena make it complex to estimate precisely the OCV. It has been proved by experiments that the following parameters vary the open circuit voltage of a battery [START_REF] Farmann | A study on the dependency of the open-circuit voltage on temperature and actual aging state of lithium-ion batteries[END_REF]:

• the temperature;

• the immediately preceding operations of the battery, by hysteresis effect;

• age and state of health of the battery;

• history of the battery (all the operations of the battery during its life).

The estimation of the OCV during GITT is thus a difficult but important step toward a correct overvoltage and a correct model.

In this paper we present a new method to precisely estimate the OCV during a GITT. We also present a new method to estimate the overvoltage that allows the determination of the parameters of an electrical circuit model (ECM). The main interest is the modelling of the diffusion phenomena, which have the largest time constant among all the phenomena taking place in an operating battery, and which are particularly sensitive to the estimated overvoltage shape.

In section 2 we will present the proposed method for separating the OCV and the overvoltage during GITT tests. In section 3 we will present the application of the proposed method to two lithium-ion batteries, compare it to the linear OCV estimation and to the pseudo-OCV measurement and then discuss the results. In section 4 we compare the parameters obtained for our model based on the three OCV estimation methods and discuss the validity and consistency of those results.

Proposed method to separate OCV and overvoltage

Definition of an equivalent OCV

By construction, the ECM are often made with several components that represent the cell overvoltage and with a voltage source that represents the OCV of the battery, measured at the thermal and electrochemical equilibrium [START_REF] Mihet-Popa | Charging and discharging tests for obtaining an accurate dynamic electrothermal model of high power lithium-ion pack system for hybrid and ev applications[END_REF][START_REF] Gholizadeh | Estimation of state of charge, unknown nonlinearities, and state of health of a lithium-ion battery based on a comprehensive unobservable model[END_REF]. This definition of the OCV at equilibrium implies that any voltage variations from it must be included into the cell overvoltage.

We observed in our research team that this feature can lead to inconsistencies. For instance, the OCV shape is often less visible when current is getting higher, as if the OCV curve was smoothed. We found that this is potentially due to a heterogeneous operation of the cell [START_REF] Damay | Heterogeneous behavior modeling of a LiFePO 4 -graphite cell using an equivalent electrical circuit[END_REF]. We will demonstrate in this paper that this smoothing of the OCV shape should be modeled has a apparent change of the OCV curve itself, instead of a change of the overvoltage (thus a change of the impedance for an ECM). This apparent changes of the OCV curve lead to what we call an "equivalent OCV" OCV eq .

Definitions of transient and quasi-steady states overvoltages

The exploitation of current pulses to determine parameters aims at better understanding and characterizing the transient operation of a cell from a relaxed state to an operating state. Our measurements have shown that, given enough time, the voltage curve from current pulses joins the trace of a voltage curve from a constant current discharge at the same regime, as presented in figure 2a. On these figures are displayed the traces from a constant current discharge and from a current pulse discharge. Each pulse is 480 sec long at 1C, which is approximately 13% DoD. It can be seen that after 480 seconds the traces are the same.

The overvoltage is a quantity that is used on ECM to represent various electrochemical effects such as the charge transfer, the diffusion or the electrolyte concentration gradient. Those phenomena depends on the current, the temperature, the state of health and the state of charge [START_REF] Ovejas | State of charge dependency of the overvoltage generated in commercial Li-ion cells[END_REF]. The physical phenomena such as the double-layer capacity or the solid and/or liquid diffusion are dynamic with a time-constant that can be characterized by EIS. Thus we believe that after a certain duration the diffusion gradients are the same in the active particles and in the electrolyte [START_REF] Danilov | Mathematical modelling of ionic transport in the electrolyte of li-ion batteries[END_REF]. In this case, the overvoltages values of two discharges curves should be the same under the same experimental conditions (current, temperature, SoC, SoH): concentrations gradients are established. Some time is required for all the phenomena to reach this state, but if we consider a battery in this quasi-steady state then we can remove time from the expression of the overvoltage η GIT T (t, I, T, SoC), such as in Eq. 1:

η GIT T (t, I, T, SoC) → t→+∞ η GIT T,qss (I, T, SoC) (1)
This assumption is supported by the fact that the traces of the constant current and pulse current discharges are overlapping after a certain duration: the steady-state is reached. Thus the overvoltage at the end of each pulse η GIT T,qss is at the same level than the overvoltage from the constant current discharge (Eq. 2).

η GIT T,qss (I, T, SoC) = η CC (I, T, SoC) (2)
The value of this overvoltage is called here the quasisteady-state overvoltage. Let us note η qss this value. Recently, Chen et al. [START_REF] Chen | Li + concentration waves in a liquid electrolyte of Li-ion batteries with porous graphite-based electrodes[END_REF] also demonstrated that the concentration gradients in the electrolyte may change with the state of charge, leading to variations of the quasi-steadystate overvoltage with SOC. Given this, a constant current discharge is therefore a measure of the OCV and the quasisteady-state overvoltage generated by the application of a current I as presented in the equation 3 Provided that the OCV is the same for a constant current discharge and a discharge from a GITT over the same SoC range, we can also write the equation 4 for each GITT. They allow us to observe a transient regime towards the overvoltage in quasi-steady-state. We note the overvoltage during the GITT η transient,I . The transient overvoltage, which is usually called "overvoltage", allows us to characterize the behaviour of the cell from a relaxed state to a polarized state. The quasi-steady-state overvoltage, on the other hand, allows us to assess only the value of the overvoltage in the polarized state.

U CC,I = OCV + η qss,I (3)
U GIT T,I = OCV + η transient,I (4)

Theoretical development

It is then possible to define the transient overvoltage as a quantity that no longer requires the determination of an OCV, but by a difference between the voltage U CC and the voltage U GIT T thanks to the equation 5.

U GIT T -U CC = η transitory,I -η qss,I (5)
This we can rewrite to obtain the equation 6.

η transitory,I = U GIT T -(U CC -η qss,I ) (6)
In order to exploit this relation, we need to determine the quasi-steady-state overvoltage. Its value, η qss , is indirectly measured during the relaxation phase. Indeed, this phase represents the transition from the polarized state to a relaxed state. Independently of the dynamics of this relaxation, the voltage difference between these two phases gives us a value of the quasi-steady-state overvoltage for each of the GITT, noted η qss,meas . The dynamic of the relaxation was not studied because it is different from the polarization and requires a specific modeling [START_REF] Hentunen | Time-domain parameter extraction method for thévenin-equivalent circuit battery models[END_REF]. By interpolating these overvoltages measurements with respect to the SOC during the GITT, we can obtain an estimate of the overvoltage for any intermediate state of charge η qss,est . This interpolation is made with a polynomial of order 3 between each measured point to ensure a consistency between each measure. It is therefore possible to define the transient overvoltage thanks to these three quantities which are U CC , U GIT T and η qss,meas the quasisteady-state overvoltage measured during the relaxation phases.

In order to return to a more classical definition of the overvoltage, we propose to define an equivalent OCV as in eq. 7.

OCV eq,I = U CC -η qss,I [START_REF] Mergo Mbeya | Off-line method to determine the electrode balancing of Li-ion batteries[END_REF] which allows to rewrite the equation 6 into the equation 8. This term of equivalent OCV makes it possible to specify that what we estimate is not what we could define as a OCV, i.e. a measurement of the voltage in a relaxed and homogeneous state. This OCV is measured for a current regime, of value I, and corresponds to the voltage which allows us to find the values of U CC when we add the measured overvoltage η qss,mes . This equivalent OCV has a form that depends on the current, and the current regime is also indicated as a subscript to show the conditions for determining this OCV.

η transient,I = U GIT T -OCV eq,I (8)
In contrast to a linear estimation of the OCV, the use of the voltage U CC to estimate an equivalent OCV allows to keep an acceptable form of OCV. This method is also faster than a measurement of a pseudo OCV since the equivalent OCV estimation does not require to tend to a relaxed state at each instant. Thus, this method seems to be a good compromise between the use of a linear interpolation, which is fast but not very accurate, and the use of a pseudo-OCV which is long and is bound to be outdated when the OCV changes.

Application to a Li-ion cell

In order to test and compare the results of parameter extraction according to the OCV estimation method, we performed CC and GITT discharges on 2 types of batteries. Battery A is a power battery and battery B is more of an energy battery. These batteries have different chemistries, and therefore different OCV shapes. Battery A has a smooth OCV with little variation in slope, while battery B has an OCV with highly non-linear areas.

Experiments performed and experimental protocol

The experimental protocol consists of a charge at constant current and then at constant voltage (CC-CV) up to the maximum voltage given by the manufacturer, followed by a 2h relaxation period. A discharge at constant current until the minimum voltage given by the manufacturer is then carried out. After 2h of relaxation, the battery is recharged according to the same protocol as before. 2h of relaxation are again observed before carrying out partial discharges at constant current (GITT) for a time of 8 minutes, i.e., 480 seconds. This time was chosen to be able to observe overvoltage whose time constant is of the hundreds of seconds. It also makes it possible to highlight the phenomenon of variations of the steady-state overvoltage value. When the battery reaches its minimum voltage during a GITT, a period of relaxation of 2h is observed and the test is stopped. Figure 2 show the measurement from this protocol.

The pseudo OCV used in this article was measured at C/100 under constant current in a climatic chamber at 25 °C.

All the other tests are performed at 1C and 20 °C. They take place in a climatic chamber regulated in temperature, and the cell are tested one by one, to keep them at the regulated temperature as much as possible. Under the hypothesis of a low heating and a good thermal dissipation, which should be reasonable due to the experimental protocol presented in the this section, the temperature can be considered constant. The state of health will be considered identical because the tests (CC discharge and GITT discharge) are done consecutively. Based on this we considered an overvoltage that depends on current and SoC only in section 1.

Results

The estimate of the three different OCV estimations for cell B are shown in the figure 3. The three methods show globally the same shape, there are differences in areas where the OCV is non-linear. The linear estimation of the OCV can be very different in these areas. A zoom on the OCV shapes around the depth of discharge (DoD) 35 % is shown on figure 3b. It can be seen that the equivalent OCV -estimated thanks to the proposed methodis smoothed compared to the pseudo-OCV (see section 2). This smoothing is also visible at for higher DoD. We will see now that overvoltages estimated thanks to these three OCV can greatly differ.

Our estimates of overvoltages are presented in figures 4a and 4b for type A and B cells respectively. However, the results are not satisfactory, especially in terms of the shape of the overvoltages, which are different from what is expected in theory, despite our observations on the voltages measured. The overvoltages do not seem to converge towards a limit value but increase without any significant sign of a limit value. As the electrical model, presented later, used in this work has been used in several publications [START_REF] Von Srbik | Offer, A physically meaningful equivalent circuit network model of a lithium-ion battery accounting for local electrochemical and thermal behaviour, variable double layer capacitance and degradation[END_REF][START_REF] Berrueta | A comprehensive model for lithium-ion batteries: From the physical principles to an electrical model[END_REF][START_REF] Merla | An easy-to-parameterise physics-informed battery model and its application towards lithium-ion battery cell design, diagnosis, and degradation[END_REF], it is considered more robust than the overvoltage estimate. Therefore, these results do not question the model itself but our way of estimating the overvoltages.

Discussion

In figures 4a and 4b, the overvoltages were simply defined as in equation 8. However, as we saw when we constructed the equivalent OCVs, the quasi-steady-state overvoltage η qss,I is a quantity that has variations with the SoC. η qss,I has been reproduced on figures 4a and 4b to demonstrate that the transient overvoltage η transient,I tends toward η qss,I for cells A and B. This effect is particularly strong for cell B, because its OCV is very non-linear. From figure 4b, we can see some benefits of the proposed method for estimating the equivalent OCV. The overvoltage appears to tend smoothly towards the quasisteady-state, needing almost the same duration to reach it for each of the GITT pulses. To the contrary, the overvoltages estimated by the other methods (linear estimation and pseudo-OCV), appear to be less robust: they are correct for linear parts of the OCV, but they are skewed for cell B around DoD 35 %, 60 % and 80 % (see this cell OCV on figure 3). Finally, the overvoltage estimations after DoD 80 % are all bad. This can be explained by the complexity of the cell behavior in this operation range, mixing several effects due to heterogeneity and diffusive phenomena [START_REF] Damay | Heterogeneous behavior modeling of a LiFePO 4 -graphite cell using an equivalent electrical circuit[END_REF][START_REF] Kremer | 1D electrochemical model of lithium-ion battery for a sizing methodology of thermal power plant integrated storage system[END_REF].

To better observe the diffusion dynamic, we therefore propose to divide the transient overvoltage obtained by the estimated value of the quasi-steady-state overvoltage. This gives a dimensionless signal, whose value is between 0 and 1, and which is defined by the equation 9. In this equation the OCV term represents the chosen estimate of the OCV (linear, measured pseudo-OCV, equivalent OCV).

η adim = U GIT T -OCV η RP,est (9)
This method allows us to obtain the signals presented in figure 5a for a cell of type A and in figure 5b for a cell of type B. We observe a change in shape in both cases. The overvoltages have more regular responses and are closer to the expected overvoltage shape (see figure 1). The comparison of the overvoltages shows that the proposed method allows us to have overvoltages that can be modelled with the chosen electrical circuit, presented in the next section. In the case of a linear estimation of the open circuit voltage, the selected electrical circuit cannot model it, which leads to important errors. Although the final values of the overvoltages are comparable for the first overvoltages, the shape of makes it difficult to automatically optimize the circuit parameters. The direct consequence is either an error on the value of the overvoltage or an error on the dynamics of the battery response. For the type A cell, the contribution of this method is lighter but still leads to parameters without outliers, which is the case for the estimation based on a linearly estimated OCV.

Regarding the first GITT pulse, the proposed method cannot be applied because the GITT and CC discharge curves have approximately the same trace, thus our theoretical development gives for this peculiar case:

η adim = U GIT T -U CC + η qss,I η RP,est ≈ 0 + η qss,I η RP,est = 1 (10)

Extraction of parameters

Electrical model of a battery

A battery is an imperfect source of voltage that varies according to the current delivered. The simplest model of a battery is therefore a voltage source coupled to an impedance that models the effects of the current on the voltage. This impedance must model the overvoltage that will be seen at the terminals of the battery by the external system. Physical considerations [START_REF] Kuhn | Modeling diffusive phenomena using non integer derivatives -Application Ni-MH batteries[END_REF][START_REF] Kuhn | Modelling Ni-MH battery using Cauer and Foster structures[END_REF][START_REF] Andre | Characterization of high-power lithium-ion batteries by electrochemical impedance spectroscopy. II: Modelling[END_REF][START_REF] Von Srbik | Offer, A physically meaningful equivalent circuit network model of a lithium-ion battery accounting for local electrochemical and thermal behaviour, variable double layer capacitance and degradation[END_REF][START_REF] Berrueta | A comprehensive model for lithium-ion batteries: From the physical principles to an electrical model[END_REF][START_REF] Merla | An easy-to-parameterise physics-informed battery model and its application towards lithium-ion battery cell design, diagnosis, and degradation[END_REF] have led to equivalent circuit models of the U-R-RC-nRC type as shown in figure 6.

A voltage source U OCV represents the open-circuit voltage of the cell, as defined in section 2. The overvoltage of the cell is modeled by several components. A first resistor R s represents an instantaneous overvoltage that comes to model the contributions of the current collectors, the tabs, the electronic flow in the active material as well as the various connections from the battery to the system in which it operates. Then a surface resistance R surf associated to a surface capacitance C surf are linked to a fast dynamic. The surface resistance models the reactions at the electrode/electrolyte interfaces, as well as the possible contribution of a SEI-type layer on the electrode surface. Finally, on a slower dynamic, of the order of a hundred seconds, the solid and liquid diffusion phenomena are modelled by a number n of R d C d circuits, in theory infinite. These RC circuits approximate a specific impedance Z d related to these phenomena, which is in general a Warburg impedance [START_REF] Andre | Characterization of high-power lithium-ion batteries by electrochemical impedance spectroscopy. II: Modelling[END_REF]. In practice, to avoid large computation times, this number is limited. In this paper, which focuses on a new method of parameter extraction more than on the accuracy of the modelling, we have chosen to use only one RC circuit to model the effects of diffusion.

Such a model has a response to a current pulse that is constituted of several, 2 here, RC charges until a steadystate is reached. Studies [START_REF] Schönleber | A Consistent Derivation of the Impedance of a Lithium-Ion Battery Electrode and its Dependency on the State-of-Charge[END_REF][START_REF] Ovejas | State of charge dependency of the overvoltage generated in commercial Li-ion cells[END_REF] have shown that all the resistive phenomena of a cell depend on the state of charge. The steady-state value that is normally reached is then here a quasi-steady-state which vary with the SoC, as observed in our measurement presented in section 2.

In this paper we used an algorithm that seeks, for each overvoltage curve, the best set of parameters to minimize an error function between the estimated (via measurement) and simulated overvoltage. The simulated overvoltage comes from the chosen equivalent circuit model. Regarding the value of the series resistance, their values are low, and the optimization was struggling to determine them, or sometimes returned negatives values to have a better fit. To avoid this, we used an EIS to determine the values at different state of charges and fixed the value in the optimization algorithm. It appeared that the serial resistance value is almost invariant to a SoC change, which leads to the same value being used for all optimizations.

Results of ECM fitting

The diffusion results are the most influenced by shape variation, so we only present here the results for the resistance diffusion on the figure 7 and the time-constant diffusion on figure 8.

Analysis

Signal shape

For the two batteries studied, and for all overvoltages, the proposed method provides a signal shape can be modelled by the selected electrical circuit. Compared to a simple piecewise linear estimation of the OCV, we observe an improvement, especially in the case of non-linear OCV, such as that of battery A, or that of battery B during plateau transition. This phenomenon is particularly marked on the figure 5b for overvoltages 3 and 5, 6 & 7. Before a plateau transition, we see an increase in the equivalent resistance of the battery, and then a decrease after the plateau transition. Where an overvoltage will be poorly defined because of these variations in estimated OCV, the signal we propose to study has a form that can be used for parameter determination.

The applied shape correction requires to estimate the steady-state overvoltage for the range of SoC covered during the GITT. However, the variations obtained allow notable improvements, such as for example the slot n°7 of battery A (figure 4a) which becomes exploitable on figure 5a.

Value of the model parameters

We also observe that an electrical model applied to classical surges converges to inconsistent solutions, especially on the diffusion parameters with values several orders of magnitude above the values found with the pro-posed method or above a coherent value. The richness of the chosen electrical circuit allows us to propose a model even though it is physically incorrect. Although the RMS errors are comparable for the three methods, and of the order of percent, the value of the parameters allows us to discard the linear estimation method of the OCV.

The extremely large time constant values found for the battery A for a linear estimation of the OCV imply that the curves of the voltages U CC and U GIT T should not have the same trace after a few hundreds of seconds, which we observe. For battery A, even the time constant values obtained by using a pseudo-OCV are about 400 sec, which would imply that the traces displayed on figure 2a should not be the same after only 480 seconds. The proposed method allows us to obtain lower values, of the order of two hundred seconds for the diffusion time constant, which allow us to validate the concept of steady-state overvoltage and is coherent with the graphic observations. The values of the parameters obtained for an estimation of the OCV by a pseudo-OCV, and the proposed method give results which are globally similar. However, the use of a pseudo-OCV requires an accurate estimation of the OCV and faces problems of sustainability in time as said in the introduction. The open circuit voltage of a cell is a quantity which evolves with the ageing of the cell, and modifications of form can appear. This operation of determination of the OCV is long [START_REF] Damay | Heterogeneous behavior modeling of a LiFePO 4 -graphite cell using an equivalent electrical circuit[END_REF] and has to be repeated in a regular way. The proposed method makes it possible to avoid this operation by proposing an estimation at an identical state of health thanks to a discharge with constant current, while giving similar results.

Besides, we found that the diffusion time constant determined thanks to our approach does not depend on the SoC for the studied battery. It may be explained by the fact that this time constant represents the dynamic of the lithium concentration gradients creation inside the active materials and the electrolyte during operation [START_REF] Danilov | Mathematical modelling of ionic transport in the electrolyte of li-ion batteries[END_REF]. The creation of this gradients would thus have the same speed regardless of the SoC, but the effects of these gradients would be depending on SoC (namely the diffusion resistance whose shape varies in relation with the OCV shape).

Conclusions

We proposed a characterization method that allows the extraction of the diffusion resistance and time constant of a LIB battery. This method requires GITT and constant current tests.

The proposed method is based on a more realistic separation of the battery OCV and overvoltage. It has been tested on two different chemistries and gives reliable results, both on areas where the OCV is relatively linear and on areas where it is strongly non-linear. It brings results that seem more consistent with the electrochemical phenomena occurring inside a battery.

The benefits of the proposed approach are the characterization of the diffusion resistance as a nearly continuous function of the SoC and a more accurate determination of the diffusion time constant compared to the classical approach (that is linear interpolation of the OCV between GITT pulses).

Figure 1 :

 1 Figure 1: a) Simulation of the voltage response of a battery to a current pulse. b) Real and estimated overvoltage in the case of a linear estimation of the OCV.

Figure 2 :

 2 Figure 2: Traces of voltages measurements for discharges at constant current (CC) and by GITT

Figure 3 :

 3 Figure 3: a) Estimated OCV from a linear interpolation, a pseudo-OCV and the equivalent OCV obtained using proposed method. b) Zoom around 35% DoD.

Figure 4 :

 4 Figure 4: Overvoltages estimated with the different OCV estimation methods & estimated quasi-steady-state overvoltage for both cells

  Normalized overvoltages for cell A.

  Normalized overvoltages for cell B.

Figure 5 :

 5 Figure 5: Normalized overvoltages for the three methods.

Figure 6 :

 6 Figure 6: Equivalent circuit model of a battery that allows to model the main phenomenon.

  Diffusion resistances fitted for cell A.

  OCV Results with equivalent OCV RMS Error -pseudo OCV RMS Error -Equivalent OCV (b) Diffusion resistances fitted for cell B.

Figure 7 :

 7 Figure 7: Diffusion resistances fitted for normalized overvoltages with a pseudo OCV and the proposed method.

  Diffusion time constants fitted for cell A.

  Diffusion time constants fitted for cell B.

Figure 8 :

 8 Figure 8: Diffusion time constants fitted for normalized overvoltages with a pseudo OCV and the proposed method.

Acknowledgement

The authors would like to thank the ANRT for its financial support (No. 2017/0855), SNCF and Bogdan Vulturescu for their participation to the study.