Optimizing Power Transfer Efficiency in Biomedical Implants: A Comparative Analysis of SS and SP Inductive Link Topologies

Inductive link intended to deliver energy to implantable devices is recently becoming a popular method. To improve link efficiency, the compensated capacitors are adopted to resonate at the same frequency with the power transfer frequency on both sides. However, there is limited research on link efficiency and output power concerning quality factors, the selection between serial–parallel (SP) and serial–serial (SS) topologies, as well as coil design. This study delves into the efficiency of SP and SS topologies using rigorous mathematical derivations with reflected impedance models in terms of quality factors. It introduces a selection factor to help choose topology based on link efficiency. Results show that SS is better when the load quality factor (<inline-formula><tex-math notation="LaTeX">$Q_{L}$</tex-math></inline-formula>) is lower than the selection factor, and vice versa. The study also examines the performance limits of both topologies concerning frequency, inductance, and load impedance. It proposes a specific <inline-formula><tex-math notation="LaTeX">$Q_{L}$</tex-math></inline-formula> for maximum efficiency and a coil design procedure considering specific absorption rate limitations. By leveraging simulations and the design procedure, the study manufactures a pair of flexible coils for biomedical use. The designed link achieves a 62.99% efficiency under a 500 <inline-formula><tex-math notation="LaTeX">$\Omega$</tex-math></inline-formula> load, with an overall efficiency of 46.96% including a passive rectifier.


Optimizing Power Transfer Efficiency in Biomedical
Implants: A Comparative Analysis of SS and SP Inductive Link Topologies Fen Xia , Student Member, IEEE, Fangyu Mao , Member, IEEE, Yan Lu , Senior Member, IEEE, and Mohamad Sawan , Life Fellow, IEEE Abstract-Inductive link intended to deliver energy to implantable devices is recently becoming a popular method.To improve link efficiency, the compensated capacitors are adopted to resonate at the same frequency with the power transfer frequency on both sides.However, there is limited research on link efficiency and output power concerning quality factors, the selection between serial-parallel (SP) and serial-serial (SS) topologies, as well as coil design.This study delves into the efficiency of SP and SS topologies using rigorous mathematical derivations with reflected impedance models in terms of quality factors.It introduces a selection factor to help choose topology based on link efficiency.Results show that SS is better when the load quality factor (Q L ) is lower than the selection factor, and vice versa.The study also examines the performance limits of both topologies concerning frequency, inductance, and load impedance.It proposes a specific Q L for maximum efficiency and a coil design procedure considering specific absorption rate limitations.By leveraging simulations and the design procedure, the study manufactures a pair of flexible coils for biomedical use.The designed link achieves a 62.99% efficiency under a 500 Ω load, with an overall efficiency of 46.96% including a passive rectifier.Index Terms-Biomedical, coil design, compensated topology, implant, inductive link, link efficiency, specific absorption rate (SAR), selection factor.

I. INTRODUCTION
W IRELESS power transfer (WPT), based on inductive links, has gained popularity with advancements in modern power electronics, attributed to key advantages including Fig. 1.Architecture of a typical inductive coupling implantable biomedical system featuring coupled coils, a PMM, and a load.

TABLE I VARIOUS PARAMETERS OF AN INDUCTIVE LINK APPLIED IN IMPLANT
the absence of exposed electrical contacts, heightened reliability, and high levels of link efficiency (η Link ) [1].WPT-based biomedical implants, such as cardiac pacemakers [2], retinal prostheses [3], and hypoglossal nerve stimulators [4], must prioritize considerations to achieve low-power consumption, minimal footprint, and optimal power conversion efficiency.Fig. 1 illustrates a simplified architecture of a typical inductive coupling implantable system, including dual coupled coils, a power management module (PMM), and a load.PMM can be integrated on a chip to minimize invasive areas.An inductive link exhibits higher η Link with a larger coil size, yet it occupies a larger area.Achieving both high η Link and a minimally invasive area in the design of an inductive link poses a significant challenge.
The compensation method can enhance η Link by mitigating the reactance of the primary input impedance [10].Link efficiency is intricately related to the type of compensation topology, the size of the receiver coil, operation frequency, and load impedance [11], [12].However, there is a wide range of variations of these factors.As indicated in Table I, η Link spans from 0.12% to 90.3%, with serial-parallel (SP) topology chosen in [4], [5], [6], and [7] and serial-serial (SS) topology selected in [8] and [9].Besides, operating frequencies range from 2 [4] to 1000 MHz [6], conforming to industrial scientific and medical (ISM) standards [13].Similar variability exists in the size of the receiver coil and the load impedance defined by the nerve-electrode interface (NEI).Crucially, carefully selecting an appropriate compensation topology is paramount, given its substantial influence on other factors related to link efficiency.
The selection between SS and SP structures was discussed in [14]; however, the study only focused on the frequency boundary.Ishihara et al. [15] introduced a quasi-duality strategy for analyzing SS and SP topologies.This study employed approximate equivalent circuit structures rather than exact representations, and solely discussed the load impedance boundary.The reflected impedance model is equivalent to the conventional transformer model in an inductive link.This model has been utilized for the analysis of SS and SP topologies in [16] and [17].In [16], the η Link formula was explicitly expressed with quality factors.However, it assumed a real reflected impedance from the primary coil, complicating η Link calculations unnecessarily [17].Notably, Lu and Ki [17] did not derive η Link in terms of quality factors, and the selection analysis remains unexplored in both studies to date.Ibrahim and Kiani [18] investigated quality factors but was limited to SP topology.In addition, previous studies did not address the design of an inductive link for maximum efficiency at a specific load impedance or consider SAR limitations critical in biomedical applications.
In addition, optimizing coil design can significantly enhance η Link by tuning parameters such as turns, wire-width, and tracepitch [19], [20].It improves the self-inductance and/or reduces the parasitic impedance of the coil, consequently enhancing the quality factor and η Link [19].While integrating resonant coils into three or four-coil systems boosts η Link [21], [22], it increases invasiveness.A recent innovative approach, presented in [23], [24], and [25], focuses on enhancing the quality factor by introducing asymmetry in wire-width and trace-pitch within coil designs while maintaining a uniform outer radius.However, these studies were confined to single-layer coil design and did not consider factors, such as two-layer coil structure, load impedance, SAR limitations, and topology analysis between SS and SP.
The impedance control method enhances η Link by optimizing load impedance at a specific value [26], [27], [28], facilitated by a circuit block positioned between the receiver coil and the load (R L ).In a WPT-based biomedical system, the impedance observed from the receiver coil (R L,ref ) closely matches R L [29], as depicted in Fig. 1, obviating the need for this impedance controller.In strong coupling scenarios, where frequency splitting occurs, the frequency control technique stabilizes the voltage conversion ratio against load impedance variations [30], [31] by adjusting the operating frequency.Moreover, multiple harmonic analysis can be employed to evaluate the performance of WPT systems under conditions of strong coupling or heavy loads [32], [33], [34].In contrast, biomedical implants utilize a weak coupling inductive link and operate under milliamp-level load currents [1], [33], effectively mitigating frequency splitting.As a result, the coupling mode theory with fundamental harmonic frequency analysis suffices for achieving precision [33], [34].In biomedical applications, external and internal topologies resonate at the same frequency [4], [6], [7], [29] to maximize link efficiency (η max ) by tuning the external capacitor [8].In addition, the internal low dropout can deliver a constant voltage with considerable power conversion efficiency under maximum current load, rendering a frequency controller unnecessary in this study.
This article conducts an in-depth analysis of SS and SP topologies by evaluating η Link and output power in terms of quality factors.It employs a reflected impedance model alongside fundamental frequency analysis (see Section II).The derived formulas for η Link are then applied to two critical aspects (see Section III): SS versus SP topology selection, and the pursuit of η max .The analysis considers various factors, such as operating frequency, implantable coil inductance, and load impedance, providing insights into the boundary between SS and SP topologies and the conditions for achieving η max .Notably, the study provides a design procedure for an inductive link featuring a two-layer asymmetric receiver coil to achieve η max at a predefined load impedance.Section IV validates the selection and boundary analyses through simulations, examining fluctuations in η Link and output power under varying loads at 300 and 400 MHz frequencies.In addition, it presents the design and performance measurement of an inductive link operating at 13.56 MHz.Finally, Section V concludes this article.

II. RESONANT INDUCTIVE LINK
As the derivation of η Link and output power of SS topology is much easier than that of SP topology, this article omits the comprehensive procedure for SS.

A. SP Topology
The circuit structure of the SP resonant inductive link is shown in Fig. 2. L 1 and L 2 denote the primary inductance and secondary inductance, respectively, M is the mutual inductance, R 1 and R 2 refer to the parasitic resistances of the primary and secondary coils, respectively, and R L signifies the load.When analyzing η Link and output power for SS and SP topologies, the capacitance values of the compensation capacitors (C 1 and C 2 ) are defined to make sure that both sides are resonant at the same frequency with the power transfer frequency f 0 , as follows: The quality factors are defined as and the coupling coefficient k and turns ratio n are given The analysis of η Link and output power involves the following six steps.
1) Formulate the equations of V s and V 2 in Fig. 2(a): (4b) 2) Derive the expression for I 2 in terms of I 1 in SP topology: The total impedance of the secondary side is named as Z 2p .with the same resonance frequency provided in (1), Z 2p can be expressed in the form of quality factors as From (4b), I 2 can be deduced as 3) Determine the reflected impedance: Substitute I 2 in (4a) with the expression from (6), the reflected impedance Z eqp can be derived as 4) Compute the input current I 1 as a function of I o : Notably, the current I o that flows through the load is different from I 2 .Then, the input current I 1 can be derived based on (6) as 5) Derive η Link in terms of quality factors: Link efficiency is defined as where P in is the input power [35] given by P in = 1 2 V s I * 1 , and Z 1p is the total impedance of the primary side, which is a complex number, as given by By deriving the real part of Z 1p from the equation given in (10), the η Link for SP topology can be determined concerning the quality factors (11) 6) Calculate the output power in terms of input current I 1 : Based on (8), P o can be calculated related to I 1 as

B. SS Topology
Fig. 3 depicts the circuit of SS resonant topology and the corresponding reflected impedance model.The reflected (or equivalent) impedance Z eqs in the primary side is derived as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
In SS resonant topology, the η Link is derived as follows: The output power P o can be calculated related to I 1 as

III. APPLICATION-ORIENTED ANALYSIS
A selection factor based on quality factors is introduced to guide the choice between SS and SP topologies in terms of LE.Notably, the article explores the optimal load quality factor that results in η max .Considering SAR limitations in biomedical applications, a design procedure is proposed to achieve η max .

A. Proposed Selection Factor
The η Link equations for SS and SP topologies are articulated in terms of the quality factors, as denoted by ( 14) for SS and (11) for SP.Since Q 1 and Q 2 are determined by the structure of both coils, the analysis of Q L is essential under the condition of a specified inductive link.
The boundary to choose SS and SP should be calculated by making (11) equals (14).By solving this equation, Q L can be derived as where Q Lsf is denoted as the selection factor.Then verify that there is only one point of intersection of SS and SP topologies.By setting the derivative of η Link expressed in (14) for Q L to zero, η max for SS topology can be ascertained.This optimum value is achieved when Q L equals Considering it is evident that M 2 /L 2 2 is significantly less than 1, and R 2 /R 1 is smaller than 1 as well.In addition, Q 2 notably exceeds 1.Consequently, it is reasonable to derive that the quality factor Q ls is markedly smaller than 1.Furthermore, the analysis indicates that the selection factor in ( 16) is negligibly higher than 1.
When Q L is less than Q ls , the slope factor of the denominator in (14) concerning Q L is negative.This signifies that as Q L increases, the denominator decreases, increasing η Link .Conversely, when Q L exceeds Q ls , the slope factor of the denominator in ( 14) is positive.In this scenario, the denominator increases as Q L increases, leading to a decrease in η Link as Q L increases.
Meanwhile, the η max for SP topology can be determined, by setting the differentiation of η Link for Q L to zero.This occurs Since the value of ( 17) is less than 1, it is easy to deduce that the value of ( 19) exceeds 1, and the selection factor outlined in ( 16) hovers around 1.This phenomenon plays a crucial role in determining the singular point where the η Link of both SS and SP topologies intersect uniquely.
The selection between SS and SP topologies is summarized in Table II.When the value of load quality factor (Q L = R L /ωL 2 ) is lower than the selection factor Q Lsf specified in (16), SS demonstrates superior performance compared to SP, and the η Link for SS topology is computed using (14).Conversely, if the value of Q L exceeds the value of Q Lsf , SP exhibits better performance than SS, and the η Link for this topology is determined using (11).

B. Analysis on Load Quality Factor
The η Link for SS topology, expressed in (14), can be formulated as follows: The η Link for SP topology, as given in (11), can be expressed as follows: SS topology shows a better performance in this condition. 2) the η Link in SP topology reaches its maximum value, as indicated by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
When (11) can be simplified as As L can be neglected in the denominator term, then (24) can be further simplified as where This equation is the same as (4) provided in [18].Under the condition of the extremely high value of Q L , the impedance in the secondary side in SP resonant topology in ( 5) can be approximately expressed as In the same resonant condition, the impedance in the secondary side in SS topology can be expressed as Comparing ( 26) and ( 27), we can easily find that Z 2s is much higher than Z 2p , which means the reflected impedance Z eqs in SS topology in (13) is much lower than the reflected impedance Z eqp in SP topology in (7), and it means that the power received by the secondary coil in SS topology is much lower than in SP topology.It is easy to understand that the power absorbed by the reflected impedance is the power received on the secondary side. 3) When Q L is equal to Q Ls , the η Link in SS topology, as expressed in (14), attains its maximum value, denoted by , the η Link in SP topology [see (11)] can be simplified as

TABLE III SELECTION BASED ON LINK EFFICIENCY
Comparing (29) with (14), it is obvious that SS topology shows a better performance than SP topology, as the denominator in ( 29) is larger than in (14).
Link efficiency in terms of load quality factor can be depicted in Fig. 4.

C. Analyzing Boundaries
In the design of implantable devices utilizing inductive links, it is crucial to consider factors like operation frequency, load resistance, and implant size.A comprehensive summary of the choice between SS and SP topology for η Link , taking into account these factors, is presented in Table III.
Given the selection factor, the secondary inductance, and the load resistance, the boundary of operation frequency is determined as When the operating frequency surpasses the boundary frequency f b , as defined in (30), SS topology demonstrates higher η Link , and conversely, vice versa.When the quality factors of both coils are significantly higher than 1, Q Lsf is slightly greater than 1.The boundaries of the secondary inductance and load resistance are Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.determined as

D. Analyzing Maximum Link Efficiency
The η max in SS and SP topologies regarding operation frequency, secondary inductance, and load resistance are discussed.
1) Operation frequency: η max in SS topology occurs when operation frequency f equals f Ls , as indicated by and η max in SP topology occurs at f equals f Lp Based on the derivation in Section III-A, where it is shown that Q Ls is less than 1 and Q Lp is greater than 1, it becomes evident that f Lp is lower than f Ls .The link efficiency concerning operation frequency is illustrated in Fig. 5(a).

2) Secondary inductance (L 2 ):
In SS topology, η max is achieved when L 2 equals L 2s and η max in SP topology occurs when L 2 equals L 2p The trend of link efficiency for the value of L 2 is the same as that depicted in Fig. 5(a), which is not presented here.

3) Load resistance:
The η max in SS topology occurs when the load resistance equals R Ls , as indicated by and the peak link efficiency in SP topology occurs at the R L equals R Lp , as specified The diagram of link efficiency with load resistance is outlined in Fig. 5(b).The inductive link should be designed with a load resistance of around R Ls or R Lp .

E. SAR Consideration
The SAR is associated with tissue exposed to an electromagnetic field.The safety limits for a 1-g tissue, and the limit is 1.6 W/kg [36].The value of SAR [18] can be determined using the following equation: where σ denotes the conductivity of the muscle, ρ represents the density of the muscle, and E rms signifies the root-mean-square value of the induced electric field.The density of the muscle tissue is 1090 kg/m 3 [36].According to Maxwell's equations, the value of E rms is proportionate to the product of the operating frequency and current (f × I).Under a 1-A excitation current, the maximum SAR value is initially calculated through electromagnetic software simulation.As a result, the excitation current limit can be determined using the following equation [37]: Then, using this I 1,max , the peak power delivered to the load can be computed in (15) and (12).

F. Guidance for Designing an Inductive Link
There are six primary steps involved in the design of an inductive link, as illustrated in Fig. 6.Begin by establishing the requirements based on applications.For an implantable hypoglossal nerve stimulator example, if Rx is placed on top of the nerve to keep it small, the outer diameter of Rx is set to be 10 mm.This considers the nerve's diameter (around 2-4 mm) and accounts for electrodes and encapsulation [38].Given the implant's proximity to the mandible, where the skin depth is shallow, the depth is set between 8 and 14 mm.Considering the impedance of the hypoglossal NEI within a range of 0.5-2 kΩ [4], the output power of Rx is guaranteed to be at least 9 mW under a maximum stimulation current of 3 mA.The next step is the selection between SS and SP topology based on Table II.By considering an initial frequency and R L , the boundary of L 2 can be computed in (31), providing a framework for the subsequent Rx optimization.
The optimization of the implantable coil is based on the concept of maximizing quality factor (Q 2 ) as defined in (2).Three versions of the implantable coil (Rx1, Rx2, and Rx3) are considered.The optimization process for Rx1 encompasses varying wire-width (w 20 ), turns (n 2 ), and trace-pitch (p 20 ).Rx2 is a double-layer variant of Rx1, featuring symmetrical structures with consistent wire width for each turn and an identical trace pitch.
The self-resistance of a coil is composed of the skin-effect resistance R skin and the proximity-effect resistance R prox [23], as expressed by the equation The value of R skin is inversely proportional to the skin depth, which, in turn, is inversely proportional to the operating frequency f .The proximity-effect resistance R prox is directly correlated with the square of the magnetic field density, which has been studied through mathematical analysis, simulation, and measurement verification [23], [25], [39].The wire located in the innermost ring yields the highest proximity-effect resistance due to the presence of the largest magnetic field in this region.In [23], the asymmetric structure, characterized by a smaller innermost ring wire width and a larger outer ring wire width, demonstrates a quality factor enhancement of 19% compared to the conventional symmetric structure with the same width and trace-pitch.Rx3 optimization involves adjusting the wirewidth and trace-pitch asymmetrically while preserving the inner radius, outer radius, and number of turns consistent with Rx2.
Subsequently, verify whether the predefined load resistance aligns with this value using (37) or (38) If the load resistance significantly deviates from the specified value, return to step 2 by adjusting the operation frequency.Alternatively, if the load resistance is within an acceptable range, Tx optimization will be carried out to ensure that the quality factor Q 1 aligns with either (37) or (38).Taking advantage of the insight that Tx radius is approximately √ 2 times the distance between the two coils [40], the dimensions of the Tx are fine-tuned by varying the wire width, turns, and trace-pitch.
Finally, assess whether the design meets the SAR limitations.If not, revisit steps 2 or 5.If yes, output all the parameters of both Tx and Rx.An inductive link is designed following the procedure in the subsequent Section IV-B.

IV. SIMULATION AND MEASUREMENT RESULTS
In this section, two primary objectives are pursued.1) Validate theoretical formulations by examining link efficiency and output power fluctuations related to load quality in SS and SP topologies.This involves investigating the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE IV DIELECTRIC PROPERTIES OF MUSCLE UNDER DIFFERENT FREQUENCIES
selection factor to determine the boundary for operation frequency and load resistance.
2) Actively design an inductive link following the procedure to achieve η max and measurements are conducted to evaluate the performance of this link combined with a passive rectifier and a load resistor.The case in the first objective thoroughly analyzed an inductive link comprising an Rx coil (5 mm in diameter), a Tx coil (22 mm in diameter), and an operating frequency of 400 MHz, as depicted in Fig. 7.The choice of these particular sizes and frequency is informed by recent research, as outlined in [7], and aligns with the mid-range permissible frequencies of ISM standards [13].In addition, a comparative analysis was performed with an operating frequency of 300 MHz.
For the secondary objective, the chosen operating frequency is 13.56 MHz.This decision stems from the findings of the first case, where operating frequencies in the hundreds of MHz resulted in the η max at load impedances in the tens of kΩ.This load impedance range is significantly higher than what is typically encountered in the classical NEI.Lowering the operating frequency can shift the load impedance to a lower value at which η max occurs.

A. Verification of Simulations at Hundreds of MHz
Fig. 7 depicts the models of Tx and Rx, where Tx consists of a single turn on one layer with a wire width of 2 mm, and Rx comprises two turns on both layers.The separation between the coils is fixed at 10 mm.Simulations were conducted in Ansys electromagnetic software HFSS.The dielectric properties of muscle incorporated into the simulation model at 400 and 300 MHz are provided in Table IV, derived from online calculations [41].In addition, muscle properties at 13.56 MHz are included in this table for subsequent analysis.
Table V presents the electrical parameters obtained from HFSS simulations.For Tx coil, the simulated self-inductance is 38.69 nH at an operating frequency of 300 MHz, and the simulated L 1 is 40.63 nH at 400 MHz.Notably, there is a significant disparity in parasitic resistance between the two operating frequencies, with R 1 being 1.58 Ω at 300 MHz and R 2 being 4.51 Ω at 400 MHz.The parasitic capacitance C s1 is determined by frequency sweeping from 100 MHz to 30 GHz to obtain the self-resonant frequency (SRF), and analyze the inductance (L 1 ) at the operation frequency.Finally, utilize the SRF equation Typically, the operation frequency is set to be four times less than SRF [21].Utilizing the simulated electrical parameters for both coils, the variation of η Link with load quality in SS topology and SP topology is determined through ( 14) and (11), respectively.

TABLE V ELECTRICAL PARAMETERS OF BOTH COILS DERIVED FROM HFSS SIMULATION
1) Operation frequency (400 MHz): Fig. 8 illustrates the simulation results obtained under an operating frequency of 400 MHz.The PTE results reveal that SS topology outperforms SP topology when the load quality factor (Q L ) is below the selection factor (Q Lsf ).Conversely, when Q L exceeds Q Lsf , SP topology exhibits superior performance.The value of Q Lsf is nearly equal to 1, occurring when the load impedance is 300.7 Ω.Under SP topology, Q L must exceed 4.64 to achieve a η Link greater than 10%.As a result, R L should be greater than 1.4 kΩ.SS topology shows an output power that is inversely proportional to the load quality factor, while SP topology shows an opposite trend, as illustrated in Fig. 8(b).The maximum SAR is 6.85 W/kg at 1 A, indicating that the maximum excitation current should be 0.48 A, as determined by (40).In SP topology, the maximum power delivered to the load is 7.89 mW, which is the product of the squares of 0.48 and 1.8.Note that when the load resistance is 1 kΩ, the load quality is 3.32, and the power delivered to the load is only 46.2 mW under an excitation current of 0.48 A. 2) Operation frequency (300 MHz): In Fig. 9, simulation results obtained at 300 MHz are shown.It is noted that the η Link trends in both SS and SP topologies are in agreement with those observed at 400 MHz.The value of Q Lsf remains slightly higher than 1, occurring when the load impedance is 205.4Ω.Furthermore, to achieve a η Link greater than 10% under SP topology, Q L must surpass 2.56.This implies that R L should be greater than 0.52 kΩ, which is much less than the 1.4 kΩ resistance value observed at 400 MHz.The output power of SS topology is shown in Fig. 9(b).
The maximum SAR under an excitation current of 1 A is 4.17 W/kg, which means the maximum excitation current should be 0.62 A with (40).With it and R L of 1 kΩ (Q L = 2.43), the output power is 61.39 mW. 3) Boundary analysis Boundary of frequency: If the R L is less or around 300 Ω and the preference is to select SP topology, the operational frequency should be set lower than 400 MHz.The minimum resistance to select SP topology at 400 MHz is higher than at 300 MHz.Boundary of load resistance: SS topology performs better than SP topology at 400 MHz if the load impedance is less than 300.7 Ω.If the inductive link selects SS topology at 300 MHz, the load resistance should be less than 205 Ω.It is worth noting that with η max occurring, the load resistance is 16.72 kΩ at 400 MHz and 10.46 kΩ at 300 MHz), significantly surpassing the value of classical NEI.In addition, the load resistance needed to achieve η max at 300 MHz is 10.46 kΩ, which is lower than the value of 16.72 kΩ at 400 MHz.This indicates that opting for an operating frequency significantly lower than 300 MHz is preferable to attain η max , occurring at a load resistance of approximately 1 kΩ.Note that the measurement results for this inductive link at 400 and 300 MHz are not presented here, as previous research in [7] and [19] has shown that the measurement results match well with simulation results using the de-embedding calibration technique.

B. Application Design at 13.56 MHz
A pair of coils is designed following the procedure outlined in Section III-F.Fig. 10 illustrates the geometry of Rx.The simulated electrical parameters of Rx, the coupling factor between Tx and Rx, as well as the measured results, are presented in Fig. 11.Notably, L 2 in Rx3 (measured 1.15 μH, simulated 0.9 μH) surpasses that of Rx2 and is significantly greater than Rx1 in both simulated and measured results.Regarding resistance, R 2 in Rx3 (measured 1.46 Ω, simulated 0.98 Ω) is slightly smaller than Rx2 but higher than Rx1.The enhanced Rx3 exhibits the highest quality factor value.
The final geometry of Tx is shown in Fig. 12(a) and the outer radius is 14.5 mm with eight turns.The simulated inductance, parasitic impedance, and quality factor are compared with the measured results, as is shown In Fig. 12(b), we can see that the simulated L 1 (2.03 μH) is almost the same with the measured result (2.01 μH), and the measured R 1 is 1.5 Ω which is larger than the simulated 1.23 Ω, resulting the measured Q 1 of 114.1 is smaller than the simulated 140.5.Fig. 12(c) presents the SAR under 1 A excitation current with the maximum value of 2.79 W/kg.Based on (40), the maximum excitation current should be 0.76 A with the SAR limit.The simulated coupling factor remains nearly identical in the three versions of Rx (0.08),  slightly exceeding the measured value (0.065), as is shown in Fig. 11(d).
The measurement setup for the inductive link, featuring a passive rectifier and load, is depicted in Fig. 13, with labeled instruments.The GWINSTEK arbitrary function generator AFG3032 serves as an external power source denoted by 1 .The Keysight DSOX3034 T and Keysight MSOS604 A oscilloscopes, denoted by 2 and 3 , respectively, are tasked with monitoring the external voltage and internal voltage, respectively.External current analysis is performed by the Keysight CX3324 A, indicated by 4 .Note that the ground in the external coil differs from that in the implantable coil, an isolation transformer marked 5 is used to connect them.To obtain η Link , the overall efficiency of the setup is calculated and then divided by the power conversion efficiency of the rectifier.Fig. 14 presents the results of η Link and the output power variation with load quality for three versions of the implantable coil.The power delivered to the load is measured under an excitation current of 0.1 A, a value significantly below the maximum excitation current requirement of 0.85 A imposed by SAR limits.Compensated capacitors, C 1 and C 2 , are incorporated to achieve resonance with inductors L 1 and L 2 at 13.56 MHz.Specifically, C 1 is chosen as 68 pF, and C 2 in the Rx3 structure is set to 120 pF.In this experiment, beef was employed as a substitute to mimic muscle tissue.The depth of the implantable coil was set to 10 mm on a position stage, which was manually controlled by a rotating handle.
The results depicted in Fig. 14 indicate that SP topology exhibits significantly higher η Link compared to SS topology across all three versions of the receiver coil (Rx).The load resistance varies from 200 Ω to 2 kΩ, and the load quality is determined based on (2), incorporating the measured self-inductance from Fig. 11.
In Fig. 14(a), when R L is set at 200 Ω and a measured L 2 value of 0.23 μH, the resulting Q L is 10.2, significantly exceeding the selection factor.Rx1 exhibits higher η Link than Rx2 and Rx3 when R L is less than 400 Ω.However, with the increase in Q L , the η Link of Rx1 experiences a significant decline.With Rx3, when R L equals 1 kΩ (Q L = 10.2), the η Link reaches 71.45%, surpassing the η Link of Rx2 at 70.4%, and significantly outperforming Rx1, which records a η Link of 43.6%.This indicates that Rx3 improves efficiency by 27.85% compared to Rx1.Specifically, the highest η for Rx3 is reached when R L is 0.92 kΩ in the simulation result and 1.12 kΩ in the measurement result, which meets the design objective of η max occurring at R L of around 1 kΩ.Under an excitation current I 1 of 0.1 A, Rx1 achieves the maximum power delivery to the load (PDL max ) at 39.2 mW with Q L equal to 40.8 (R L = 800 Ω).In contrast, Rx3 demonstrates a PDL of 47.6 mW when Q L is 20 (R L = 1.96 kΩ).Significantly, Rx3 achieves the PDL max of 3.83 W, adhering to the SAR limit's maximum excitation current requirement of 0.76 A, corresponding to a Q L value of 59.2.
The simulated η Link and coupling coefficient concerning Rx3's misalignment in the X-direction and displacement in implant depth (Z-direction) are illustrated in Fig. 15   misalignment in the X-direction ranging from 0 to 5 mm, η Link exhibits a decrease from 0.78 to 0.75.In the case of displacement in the Z-direction spanning from -2 to 4 mm, η Link decreases from 0.8 to 0.67.Remarkably, the effect of displacement in the Z-direction on η Link is more significant than that in the X-direction.
In corresponding to the setup depicted in Fig. 13.Fig. 16 provides a photograph showcasing the same instruments with marked labels.The peak-to-peak value of the excitation voltage is 1.11 V, and the amplitude of the external current is 0.1 A. Under R L of 500 Ω, the measured voltage across the load is 3.61 V, with a peak-to-peak value between the implantable coil reaching 3.92 V, as is shown in Fig. 17.The total efficiency   VI illustrates a comparison of an inductive link aimed at implant applications with comparable outer diameters of Rx and frequencies in the MHz range.The measured η Link under beef tissue is 62.99% at R L of 0.5 kΩ, which is much higher than the value of 30.84% with the same operation frequency and same outer diameter of Rx provided in [43].Specifically, in [42], the η Link between two coils under an air medium is 76.3%, higher than our measured results of 72.5% due to the size of the Rx size being two times larger than ours.In terms of muscle variations, there is no significant influence on the mutual coupling coefficient, but they do impact the quality factor, as evidenced by studies in [19], [21], and [43].To investigate this further, simulations were conducted to analyze the inductive link under both air gap and muscle conditions.The results suggest that while the coupling coefficient remains relatively unchanged, there is a slight increase in the quality factor.Consequently, the link efficiency experiences a marginal increase of less than 1%.

V. CONCLUSION
In this work, formula expressions for link efficiency and output power are derived utilizing quality factors with reflected impedance model.Based on these expressions, the selection between SS and SP topologies, and maximum link efficiency are comprehensively analyzed with operation frequency, selfinductance, and load impedance.In addition, an inductive link design procedure aimed at achieving η max in inductive links is proposed.Besides, an asymmetric coil is designed to achieve a higher quality factor and enhance link efficiency in comparison to the conventional symmetric coil.The analysis is verified through simulation and measurement, along with SAR consideration.These contributions significantly advance WPT systems for biomedical applications, emphasizing efficiency enhancements and adherence to safety standards.

Fig. 2 .
Fig. 2. (a) Circuit configuration of coupled coils arranged in a SP resonant topology with parasitic resistance.(b) Equivalent reflected impedance model.

Fig. 3 .
Fig. 3. (a) Circuit configuration of coupled coils arranged in a series-serial resonant topology with parasitic resistance.(b) Equivalent reflected impedance model.

Fig. 7 .
Fig. 7. (a) Inductive coupling model under an operation frequency of 400 and 300 MHz.(b) Implantable coil with two-layer structure.(c) External coil with one-layer structure.

Fig. 8 .
Fig. 8. Simulation results under an operating frequency of 400 MHz.(a) Variation of link efficiency with load quality.(b) Variation of output power under an excitation current of 1 A with load quality.(c) Specific absorption rate observed under an excitation current of 1 A from a top-view perspective.

Fig. 9 .
Fig. 9. Simulation results under an operating frequency of 300 MHz.(a) Variation of link efficiency with load quality.(b) Variation of output power under an excitation current of 1 A with load quality.(c) SAR observed under an excitation current of 1 A from a top-view perspective.

Fig. 10 .
Fig. 10.Implantable coil at 13.56 MHz.(a) Rx1 is a single-layer design with a uniform wire width.(b) Rx3 is a two-layer design with asymmetric wire widths.Note: Rx2 is a two-layer design based on Rx1.

Fig. 12 .
Fig. 12.(a) External coil under an operation frequency of 13.56 MHz.(b) Parameters derived from HFSS simulation and measurement.(c) SAR observed from a lateral view under an excitation current of 1 A.

Fig. 13 .
Fig. 13.Measurement setup for SS or SP topology with a passive rectifier and load impedance.

Fig. 14 .
Fig. 14.Plotting of load quality factor with link efficiency (Y-axis on the left-hand side colored in red), and output power (Y-axis on the right-hand side colored in blue).Solid lines and dotted lines represent simulated and measured results, respectively.Different versions of receiver coils.(a) Rx1.(b) Rx2.(c) Rx3.
(a) and (b), respectively.With an increase in misalignment and displacement, there is a reduction in both coupling coefficient and η Link .For Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 15 .
Fig. 15.Simulated values of link efficiency and coupling coefficient plotted against Rx's (a) misalignment in the X-direction and (b) displacement in implant depth (Z-direction).

Fig. 16 .
Fig. 16.Photograph of measurement setup for SP topology with a passive rectifier and load.

Fig. 17 .
Fig. 17.Measurement voltages across the internal coil and the output voltage of the rectifier.

TABLE VI COMPARISON
OF INDUCTIVE LINK FOR IMPLANTABLE DEVICE (IMD) AT SEVERAL MHZ is calculated at 46.96%, comprising a η Link of 62.99% and a rectifier efficiency of 74.14%.Table