Differential-Evolution-Assisted Optimization of Classical Compensation Topologies for 1 W Current-Fed IMD Wireless Charging Systems

Abstract

Implantable medical devices (IMDs) necessitate a consistent energy supply, commonly sourced from an embedded battery. However, given the finite lifespan of batteries, periodic replacement becomes imperative. This paper addresses the challenge by introducing a wireless power transfer system designed specifically for implantable medical devices (IMDs). It begins with a detailed analysis of the four conventional topologies. Following this, the paper provides a thorough explanation for choosing the PS topology, highlighting its advantages and suitability for the intended application. The primary parallel capacitance necessitates power from current sources; thus, a Class-E amplifier was implemented. Additionally, the selected circuit was engineered to deliver 1 W at the biocompatible resonance frequency of 13.56 MHz. The delineation of the resonance parameters hinges on multifaceted solutions, encompassing bifurcation-free operation and the attainment of peak efficiency. To ensure the feasibility of the proposed solution, a Differential-Evolution-based algorithm was employed. The results obtained from simulation-based evaluations indicated that the system achieved an efficiency exceeding 86%. This efficiency level was maintained even in the face of frequency fluctuations and variations in the coupling between the coils, thereby ensuring stable operational performance. This aligns seamlessly with the specified application prerequisites, guaranteeing a feasible and reliable operation.

1. Introduction

Implantable medical devices (IMDs) are used for monitoring physiological signs, therapeutic functions, and the treatment of chronic diseases [1,2,3,4]. According to [3,4], in 2019, about 10-million people worldwide depended on implantable cardiac pacemakers, and this number has grown by about 15% per year. Most IMDs use built-in batteries for their operation, and for fully IMDs, the battery replacement is performed through a surgical procedure [1,2,3,4]. Battery replacement through a surgical procedure has the potential for infection and generates patient anxiety, and during battery replacement, the IMD must be turned off [1]. Based on these prerogatives, a wireless power transfer system (WPT) becomes very interesting for charging implantable devices; after all, this would significantly reduce the number of surgical procedures for replacing batteries, increasing the lifespan of the device and allowing operation during battery recharging [3,4,5]. Currently, wireless power transfer is being used in IMDs through a near-field resonant inductive coupling (NRIC) [4]. Some examples would be cochlear implants, whose function is to restore hearing in patients with hearing loss, ocular and retinal implants, whose function is to restore vision in patients with visual impairment, and neurostimulator implants, used to restore motor and sensory function [4,6].

Basically, in an application of WPT in IMDs, the primary transmitting coil is positioned close to the patient’s skin. From the circulation of a time-varying current, a magnetic field is induced in the receiving coil, implanted next to the load, and in this way, current is produced to charge the device [7]. However, applying WPT to IMDs brings with it some concerns. A serious restriction, related to biocompatibility, concerns the exposure of living beings to the electromagnetic field and the specific absorption rate (SAR) of tissues [4,8]. In addition to biological factors, it is essential to ensure operational stability and acceptable efficiency to make the application safe and viable.

In general terms, WPT classical compensation topologies are named series–series (SS), series–parallel (SP), parallel–series (PS), and parallel–parallel (PP). The main advantages of these topologies are the reduced number of elements and the consequent implementation simplicity. However, despite the reduced number of parameters, in order to achieve a good performance with classical topologies, a rigorous selection of the elements that will compose the circuits is essential. Based on this selection, the efficiency can be maximized and situations of instability can be avoided [9].

The inductive power transfer applied to battery charging has two stages, which are the constant current charging process followed by constant voltage charging. Charging starts in constant current mode while the battery voltage gradually increases; once the battery voltage reaches its maximum charging voltage value, charging switches to constant voltage mode, so the current decreases considerably. Charging ends when the current in the battery reaches a specific value. Since batteries are considered to have a variable load during the charging process, there is a significant fluctuation in the output voltage; therefore, a converter is needed to regulate its output precisely in order to implement charging [10,11,12].

In this paper, the authors intended to demonstrate that classical topologies with an input current source are more stable in the face of variations in the load, due to the charging process, as well as the coupling factor variation, and also vary little in terms of output power and input current. A first step is to carry out an in-depth review of the four classical topologies of inductive resonant coupling. After clarifying all the advantages and disadvantages of each, it is possible to justify the application of the PS topology in IMDs.

Commencing with the primary and secondary inductances, the choice of resonant components is influenced by multi-criteria decision-making processes. Consequently, a subsequent phase of this study involved the implementation of an algorithm based on Differential Evolution (as referenced in [13]). This algorithm is aimed at determining inductances and capacitances that are viable both technically and practically.

Presently, a diverse array of bio-inspired optimization techniques are utilized across various engineering disciplines, including power flow analysis. Notable among these are the Artificial Hummingbird [14], Prairie Dog [15], Marine Predators [16], and Slime Mold Algorithms [17], along with the Differential Evolution algorithm. Although each of these methods demonstrates commendable accuracy and convergence speed, the Differential Evolution (DE) algorithm has been specifically selected for this research. This choice is attributable to DE’s rapid convergence, high precision, straightforward implementation process, and proficient handling of constraints.

With a focus on experimental validation, other practical considerations must be taken into account. Topologies with series compensation on the primary have the great advantage of being able to operate from a voltage source at the input, avoiding the need for other stages, which could compromise the overall performance of the system. On the other hand, with parallel compensation in the primary, the circuit must be powered by a current source, which, in a way, makes the execution of the project difficult and can compromise the overall performance if an intermediate stage is necessary [18]. However, although parallel compensation in the primary brings with it the mentioned disadvantage, other technical advantages may justify the application of this topology.

In this paper, in order to circumvent the limitation of primary parallel compensation, it was decided to feed the chosen circuit through a Class-E amplifier. The Class-E amplifier stands out for its topological simplicity and for its good performance at MHz or GHz frequencies [19]. An aggravating factor of this application lies in the fact that the Class-E amplifier is highly sensitive to parametric variations, that is small variations in the resonant elements or in the load can impair its operation [20].

The results revealed that the Differential Evolution algorithm employed in this study precisely selected parameters that enabled the system to operate with an efficiency exceeding 94%. This efficiency level demonstrated notable stability against variations in frequency and coil coupling. Additionally, considering the incorporation of a Class-E power amplifier for generating alternating current, the system’s overall efficiency was maintained above 86%.

2. Classical Topologies

2.1. Fundamental Concepts

To facilitate the transmission of the desired active power in a wireless power transfer (WPT) system, a significant requirement arises for a substantial reactive power component. This necessity underscores the immediate need for compensation strategies, as elucidated in [9]. One of the most-straightforward approaches to achieve this compensation is by employing classical topologies, as discussed in [21]. In the context of the four topologies being examined in this study, the transmitting segment comprises a capacitance component (C₁) arranged either in series or parallel with a coil, characterized by its winding impedance, composed of an inductance (L₁) and a resistance (R₁). Similarly, the receiver section consists of a coil with a winding impedance that encompasses an inductance (L₂) and a resistance (R₂), with an associated capacitance (C₂) configured in series or parallel with the secondary coil. Regardless of the specific topology being considered, a load resistance (R_L) is positioned at the output of these circuits. The magnetic coupling between these coils is realized through mutual inductance (M). Visual representations of these four topologies are provided in Figure 1.

Figure 1 confirms that topologies with series compensation on the primary side are powered from voltage sources (Figure 1a,b), which considerably simplifies the implementation. Topologies whose primary compensation occurs through parallel capacitance (Figure 1c,d) are powered by current sources. This makes the implementation more laborious and, in most cases, leads to topological variations to achieve compatibility between a voltage source and the circuits in question [9].

At frequency ${\displaystyle {\omega }_r}$, the equivalent impedance (Z₁) must have its imaginary portion compensated for C₁; consequently, all reactive demand is eliminated. For the four classical topologies,

Table 1. Equations for the topologies, considering ${\displaystyle M=\kappa \sqrt{L_1{L}_2}}$ and ${\displaystyle C_2=\frac{1}{{\omega }_0^2{L}_2}}$.

	C1	${\displaystyle \mathit{\eta }}$	Z1
SS	${\displaystyle \frac{1}{{\omega }_r^2{L}_1}}$	${\displaystyle \frac{R_L}{R_1{\left(\frac{R_2{R}_L}{M{\omega }_r}\right)}^2+R_2+R_L}}$	${\displaystyle R_1+j(L_1{\omega }_r-\frac{1}{C_1{\omega }_r})+\frac{M^2{\omega }_r^2}{R_2+R_L+j(L_2{\omega }_r-\frac{1}{C_2{\omega }_r})}}$
SP	${\displaystyle \frac{L_2}{L_1{L}_2{\omega }_r^2-M^2{\omega }_r^2}}$	${\displaystyle \frac{R_L}{R_L+R_2+\frac{R_1{L_2}^2}{M^2}+\frac{R_1{R_2}^2}{{\left(M{\omega }_r\right)}^2}+R_2{\left(\frac{R_L}{{\omega }_r{L}_2}\right)}^2}}$	${\displaystyle R_1+j(L_1{\omega }_r-\frac{1}{C_1{\omega }_r})+\frac{M^2{\omega }_r^2(j{R}_L{\omega }_r{C}_2+1)}{(R_2+j{L}_2{\omega }_r)(j{R}_L{\omega }_r{C}_2+1)+R_L}}$
PS	${\displaystyle \frac{{\left(L_2{C}_2\right)}^2{R_L}^2{L}_1}{{R_L}^2{L_1}^2{L}_2{C}_2+M^4}}$	${\displaystyle \frac{R_L}{R_1{\left(\frac{R_2+R_L}{\omega M}\right)}^2+R_2+R_L}}$	${\displaystyle \frac{(R_1+\frac{M^2{\omega }_r^2}{R_2+R_L+j(L_2{\omega }_r-\frac{1}{C_2{\omega }_r})})+j{L}_1{\omega }_r}{(1-L_1{C}_1{\omega }_r^2)+j(R_1{C}_1{\omega }_r+\frac{M^2{\omega }_r^3{C}_1}{R_2+R_L+j(L_2{\omega }_r-\frac{1}{C_2{\omega }_r})})}}$
PP	${\displaystyle \frac{(L_1{L}_2-M^2)C_2{L_2}^2}{{(L_1{L}_2-M^2)}^2+\frac{{R_L}^2{C}_2{M}^4}{L_2}}}$	${\displaystyle \frac{R_L}{R_L+R_2+\frac{R_1{L_2}^2}{M^2}+\frac{R_1{R_2}^2}{{\left(M{\omega }_r\right)}^2}+R_2{\left(\frac{R_L}{{\omega }_r{L}_2}\right)}^2}}$	${\displaystyle \frac{\frac{M^2{R}_L}{L_2^2}+j{\omega }_r(L_1-\frac{M^2}{L_2^2})}{1-{{\omega }_r}^2{C}_1{L}_1+\frac{{{\omega }_r}^2{M}^2{C}_1}{L_2}+\frac{j{\omega }_r{C}_1{R}_L{M}^2}{L_2^2}}}$

summarizes the equations of C₁, the theoretical efficiency (${\displaystyle \eta }$), and the equivalent impedance (Z₁). It is interesting to note that, except for the SS topology, the primary compensation depends on mutual inductance. That is, considering that the inductances L₁ and L₂ do not change, C₁ also depends on the coupling factor ${\displaystyle \kappa }$. Another relevant fact is that the efficiency does not depend on the capacitive elements; however, for topologies with parallel resonance in the secondary, even for high frequencies, ${\displaystyle \eta }$ will be affected by L₂ and M.

2.2. Design for the Different Topologies

With a view toward low-power devices, a rated power (P_L) of 1 W was chosen as a design parameter [22]. Assuming that a rechargeable lithium-ion cell has a terminal voltage between 3.5 and 3.7 V [22], the load voltage (V_L) was defined as 3.6 V. For the resonance frequency, respecting biocompatibility restrictions, the frequency (f_r) of 13.56 MHz was chosen, and a value was estimated of 0.25 as the design coupling factor (${\displaystyle \kappa }$).

In order to identify the L₁ and L₂ pairs that correspond to the system’s maximum efficiency point, a range of inductance combinations from 0.01 ${\displaystyle \mathsf{\mu }}$H to 10 ${\displaystyle \mathsf{\mu }}$H was chosen, with intervals of 0.01 ${\displaystyle \mathsf{\mu }}$H. Based on these data, the efficiency behavior of the classical topologies is presented in Figure 2.

As depicted in Figure 2, the choice of L₂ is crucial to achieve good efficiency. On the other hand, the influence of L₁ is neglectable on the efficiency, and as long as it does not imply other constraints, it is reasonable to arbitrate its value as being identical to that of L₂. In order to obtain the system’s maximum efficiency point, according to the values obtained in Figure 2, the inductances L₁ and L₂ were selected. The values of C₁ and C₂ were calculated, as well as the mutual inductance and efficiency, based on the equations in Table 1.

Table 2. Design parameters.

	SS	SP	PS	PP
${\displaystyle L_1}$	610.0 nH	40.0 nH	610.0 nH	40.0 nH
${\displaystyle R_1}$	0.3381 ${\displaystyle \Omega }$	0.0222 ${\displaystyle \Omega }$	0.3381 ${\displaystyle \Omega }$	0.0222 ${\displaystyle \Omega }$
${\displaystyle L_2}$	610.0 nH	40.0 nH	610.0 nH	40.0 nH
${\displaystyle R_2}$	0.3381 ${\displaystyle \Omega }$	0.0222 ${\displaystyle \Omega }$	0.3381 ${\displaystyle \Omega }$	0.0222 ${\displaystyle \Omega }$
${\displaystyle C_1}$	225.8 pF	3.6736 nF	212.5 pF	3.4517 nF
${\displaystyle C_2}$	225.8 pF	3.4440 nF	225.8 pF	3.4440 nF
${\displaystyle M}$	152.5 nH	10.0 nH	152.5 nH	10.0 nH
${\displaystyle \eta }$	94.9%	94.8%	94.9%	94.8%

shows a summary of the parameters calculated for the different topologies. Loss resistances R₁ and R₂ were calculated as a proportionality rule in relation to the inductances.

2.3. Bifurcation Phenomenon

The system behavior was verified regarding deviations from a pre-established resonance, due to the practical difficulties of obtaining identical inductive and capacitive values to the designed ones. This type of analysis is necessary to evaluate the stability of the system through frequency variations. Thus, from the equations defined in Table 1 and the values presented in Table 2, the transmitter current behavior (I₁) was evaluated for frequency oscillations of ±30% in relation to the base frequency. The results obtained are shown in Figure 3.

Practical projects prove that the resonance frequency varies around 10% due to tolerances in the circuit parameters [9]. Taking that into consideration, as shown in Figure 3, for the SS topology, a reduction of 10% on the frequency of operation resulted in an input current 50% higher than the expected one for 13.56 MHz. A similar behavior was observed for the SP topology; the input current rose 40% for frequency variations of up to 10%. The obtained values were found to be impractical and could potentially jeopardize the integrity of the physical project. Conversely, both the PP and PS topologies demonstrated a significant reduction in the input current, achieving a decrease of approximately 70% even with oscillations reaching up to 10%.

The operation of a bifurcated system can be catastrophic for the project. However, respecting some criteria, the phenomenon can be eliminated, minimizing operational risks and ensuring stability in energy transfer [9]. As shown in

Table 3. Bifurcation criteria [9].

Topology	Condition
SS	${\displaystyle Q_p>\frac{4Q_s^3}{4Q_s^3-1}}$
SP and PP	${\displaystyle Q_p>Q_s+\frac{1}{Q_s}}$
PS	${\displaystyle Q_p>Q_s}$

, for stable operation, it is necessary to meet the criteria based on the primary and secondary quality factors Q_p and Q_s, respectively.

To address this undesired issue, it is necessary to select new design parameters. One viable strategy involves adjusting the L₂ values to mitigate instability, as outlined in

Table 4. New design parameters to minimize the bifurcation phenomenon.

	SS	SP	PS	PP
${\displaystyle L_1}$	610.0 nH	40.0 nH	610.0 nH	40.0 nH
${\displaystyle R_1}$	0.3381 ${\displaystyle \Omega }$	0.0222 ${\displaystyle \Omega }$	0.3381 ${\displaystyle \Omega }$	0.0222 ${\displaystyle \Omega }$
${\displaystyle L_2}$	300.0 nH	70.0 nH	430.0 nH	57.0 nH
${\displaystyle R_2}$	0.1663 ${\displaystyle \Omega }$	0.0388 ${\displaystyle \Omega }$	0.2385 ${\displaystyle \Omega }$	0.0316 ${\displaystyle \Omega }$
${\displaystyle C_1}$	225.8 pF	3.6736 nF	219.0 pF	3.5609 nF
${\displaystyle C_2}$	459.2 pF	1.9680 nF	320.2 pF	2.4168 nF
${\displaystyle M}$	107.0 nH	13.2 nH	128.1 nH	11.9 nH
${\displaystyle \eta }$	94.0%	93.8%	94.7%	93.6%

. While this approach may require operating outside the point of maximum efficiency, careful consideration can help minimize any significant impact on energy transfer. Table 4 provides a summary of the new parameters for the input current with bifurcation reduction.

From the new parameters, as presented in Figure 4, the SS (a) and SP (b) topologies became stable for frequency variations. In addition, the PS and PP topologies brought a peculiar behavior. For frequency variations of up to 10%, the input current was maintained practically constant, which is a very attractive feature for practical implementations. After the selection of the parameter values and confirming the minimization of the bifurcation phenomenon, a study was undertaken to evaluate the feasibility of the previously estimated coupling factor (${\displaystyle \kappa }$). This assessment was conducted using the student version of Ansys Maxwell^®. The inductances L₁ and L₂ were designed in a spiral–planar geometry, as depicted in Figure 5. This particular geometry was selected due to its associated beneficial characteristics, which include the ease of implementation, symmetrical behavior, and a robust coupling factor, especially in scenarios involving misalignment.

The study demonstrated that, for the coils designed, a coupling factor of 0.25 was achieved at a distance of 4 mm. This value aligns with the anticipated separation distance between the transmitting and receiving coils, specifically considering their application in implantable medical devices. Consequently, the obtained value of the coupling factor was validated and endorsed for this specific use.

2.4. Coupling Factor

One of the concerns in wireless transfer is to evaluate the behavior of the system through variations in the coupling factor (${\displaystyle \kappa }$). The coils are subjected to vertical and horizontal displacements in these applications. In relation to implantable devices, the situation is even more complex. In addition to the coils’ alignment unpredictability, the system is exposed to a medium of organic tissues and liquids of different compositions.

Figure 6 shows the input current behavior from the maximum misalignment (${\displaystyle \kappa }$ = 0) to situations where the coupling coefficient is about 20% above the nominal condition. For topologies with series compensation in the transmitter side, as the coupling factor ${\displaystyle \kappa }$ decreases, the input current increases considerably. Therefore, unless a strict primary current control is implemented, the system will be driven to incompatible current levels, making the implementation unfeasible. On the other hand, the PS and PP topologies are less sensitive to reductions in the coupling factor since the input current decreases with the distance between the coils. This characteristic makes them strong candidates for applications in which control simplicity and good stability are needed.

2.5. Input Impedance Behavior

As concluded before, the PS and PP topologies presented a convenient behavior in relation to the reduction of the coupling coefficient. However, the use of parallel compensation in the primary makes the use of voltage sources to supply the circuit unfeasible. In applications subject to biocompatibility restrictions, sources that oscillate in the order of MHz are not easily implemented by means of conventional current inverters. Therefore, a proposal that appears feasible is the use of Class-E amplifiers. The major drawback of using amplifiers as current sources is that they must be designed considering the load impedance. Thus, the evaluation of Z₁ becomes essential, as the parametric variations of the circuit can compromise the impedance matching between the amplifier and the wireless transfer circuit. Figure 7 and Figure 8 show, respectively, the magnitude and angle of the input impedance as a function of frequency. Figure 9 and Figure 10 show, respectively, the impedance modulus and angle as a function of the coupling coefficient.

Through Figure 7, it is notable that, at frequencies different from nominal, the impedance modulus tends to increase for the SS and SP compensations and to decrease for PS and PP. In Figure 8, one can conclude that, for the SS and SP topologies, the load tends to be inductive for frequencies lower than nominal and capacitive for higher. On the other hand, Figure 8c,d show that, for the PS and PP topologies, for frequencies lower than the design frequency, Z₁ presents an inductive behavior and, for higher frequencies, a capacitive behavior.

With respect to the behaviors of the topologies in relation to variations in the coupling coefficient, the data suggest that topologies exhibiting parallel resonance in the primary, as depicted in Figure 9c,d, exhibit an increase in their impedances under conditions where the coupling coefficient, ${\displaystyle \kappa }$, is diminished. Given that reductions in the coupling coefficient can be anticipated due to commonplace operational misalignment and offsets, this characteristic behavior confers a marked advantage to these particular topologies.

In Figure 10, it is noteworthy to highlight the SS topology’s consistent null angle throughout the entire range of ${\displaystyle \kappa }$. This conclusively demonstrates that fluctuations in mutual inductance do not introduce imaginary components to the equivalent impedance. Further, based on the observations from Figure 10b–d, it can be inferred that coil misalignments or displacements manifest as discernible inductive or capacitive segments in the input current.

3. Differential Evolution-Based Parameter Selection

Aiming at the selection of optimal parameters for the classical wireless energy transfer topologies, an algorithm developed from the Differential Evolution (DE) algorithm was used. DE is a meta-heuristic based on the process of species evolution, which consists of generating an initial random population, mutating, crossing characteristics between individuals, and selecting the best individuals at the end of each generation [13]. For the proposed application, some changes were made in the pseudocode of the presented algorithm.

First, an initial population is defined, composed of NP vectors of randomly chosen parameters, pop(i, 1) and pop(i, 2), in the interval between the minimum of 0.01 ${\displaystyle \mathsf{\mu }}$H and the maximum of 100 ${\displaystyle \mathsf{\mu }}$H, which correspond to the values of the primary and secondary inductances. The population is created from a uniform probability distribution and follows a natural evolution with the number of individuals fixed during the optimization process. It is noteworthy that the population must meet the non-bifurcation criteria. That is, if the generated population presents bifurcation, it will be discarded, and a new population will be formed. As shown in Figure 11, the generation of the initial population is illustrated.

In one generation, three individuals are selected, X(1), X(2), and X(3), obtained from a randomly established population, in an interval from 1 to NP. Another vector, named the target vector pop(i, j), is also selected randomly. In the first generation, therefore, the target vector and the vectors X(1), X(2) and X(3) are selected to be part of the mutation, crossover, and selection steps. The four individuals must be distinct from each other, meeting the condition that ${\displaystyle X\left(1\right)\ne X\left(2\right)\ne X\left(3\right)\ne pop(i,:)}$, as illustrated in Figure 12.

In the context of evolutionary computation, mutation can be defined as a change or perturbation with a random element [23]. From the random selection of three individuals of a generation G, X(1), X(2), and X(3), a vector of mutated parameters (V) is obtained. The differential mutation operation is performed according to (1).

$$V=X\left(3\right)+F\times \left(X\right(2)-X(1\left)\right)$$

(1)

In accordance with the operation, X(2) ${\displaystyle -}$X(1) is defined as the difference between the parameter vectors. Let F be a real constant ${\displaystyle \in }$ [0, 2], called the mutation factor. This factor is responsible for controlling the size of the step to be performed, that is orienting the vector amplitude. The weighted vector difference is added to the third individual X(3) as illustrated in Figure 13.

The mutated vector V is combined with the target vector, pop(i, j), i.e., an arbitrarily chosen target vector, and the combination results in the trial vector. This process is called crossover, increasing the diversity of mutated individuals. The algorithm performs the combination respecting the real crossover constant cr ${\displaystyle \in }$ [0, 1]. Let rand be a random scalar ${\displaystyle \in }$ [0, 1]; if rand is less than the cr, the muted vector parameters are selected; otherwise, the target vector is selected. The crossover operation is exemplified in Figure 14. It is noteworthy that, in this paper, the mutation and crossover operations were performed concurrently.

After the mutation and crossover of the individuals, the selection of the best individuals is carried out. Initially, the value of the characteristics was analyzed, and it was verified if the value was within the range of 0.01 ${\displaystyle \mathsf{\mu }}$H to 100 ${\displaystyle \mathsf{\mu }}$H. If it was within the range, the characteristic was maintained; otherwise, it was replaced by the characteristic of the previous individual. Subsequently, the bifurcation criterion was analyzed. If the values obtained presented the bifurcation phenomenon when implemented, they would be replaced by the characteristics of the individual of the previous generation; otherwise, they would be maintained.

Finally, the fitness function, in this case, the overall efficiency, was analyzed. Therefore, it was verified whether the value obtained with the new characteristics was higher or lower than the value obtained with the past characteristics. If it was superior, the new individual would keep its characteristics. Otherwise, the new individual would assume the characteristics of the individual of the previous generation. In this iterative framework, the mutation, crossover, and selection processes generate a new population in each cycle, continuing until the generation count aligns with the pre-set maximum or another stop criterion is achieved. The DE algorithm’s flowchart, as applied in the study, is showcased in Figure 15. Importantly, for a deeper, more-technical insight into the Differential Evolution algorithm proposed, the pseudocode is presented and detailed in Algorithm 1.

Algorithm 1 Differential Evolution algorithm.

1: Randomly generate the initial population following the bifurcation criteria.   2: for each generation k do   3:       for each individual i do   4:        Randomly select three different individuals X(1), X(2), X(3)   5:        for each dimension j do   6:              if rand(0,1) < cr then   7:                    Trial(j) = X(3) + F${\displaystyle \times }$(X(2) ${\displaystyle -}$ X(1))   8:              else   9:                    Trial(j) = pop(i, j) 10:              end if 11:        end for 12:        if f(Trial) > f(pop(i)) and the Trial meets the bifurcation criteria then 13:              pop(i) = Trial 14:        end if 15:       end for 16:       Perform replacements 17: end for

4. System Overview

The system implemented utilizes a parallel–series (PS) topology, powered by a current source. In this configuration, the correlation between the output power and the load is predetermined. Unlike most wireless power transfer (WPT) systems that are driven by inverters requiring at least four switches, our approach addresses the issue of significant switch losses in high-frequency applications. As an alternative to conventional inverters, a Class-E power amplifier (PA) was employed. This choice is particularly advantageous for low-power and very-high-frequency WPT applications. The Class-E PA demonstrates its efficacy even in dynamic charging scenarios, such as when there is misalignment between the transmitter and receiver coils [24,25].

This alternative exhibits notable characteristics, including a simplistic design, and conditions of zero voltage switching, as well as zero voltage derivative switching [25,26,27,28]. The Class-E power amplifier (PA) is capable of achieving efficiencies of approximately 90%, especially when operating at frequencies within the MHz range. However, it is important to note that the Class-E PA demonstrates sensitivity to variations in load [25].

In Figure 16, the Class-E power amplifier (PA) circuit is depicted on the left side, while the wireless power transfer (WPT) circuit is presented on the right side. The operating range of the Class-E PA was selected to be between 12 V and 14 V, thereby aligning with the predetermined voltage and power requirements at the load. The parameters of the circuit are concisely summarized in

Table 5. Circuit parameters.

Description	Value
DC supply voltage (${\displaystyle V_{CC}}$)	12.0 V
RE choke (${\displaystyle RFC}$)	1.0 mH
Choke resistance (${\displaystyle R_{FC}}$)	0.5 ${\displaystyle \Omega }$
Shunt capacitance (${\displaystyle C_s}$)	25.9 pF
Series inductance (${\displaystyle L_e}$)	12.44 ${\displaystyle \mathsf{\mu }}$H
Series resistance (${\displaystyle R_e}$)	6.8967 ${\displaystyle \Omega }$
Series capacitance (${\displaystyle C_e}$)	14.13 pF
Match capacitance (${\displaystyle C_m}$)	63.3 pF
Primary capacitance (${\displaystyle C_1}$)	219 pF
Primary coil resistance (${\displaystyle R_1}$)	0.3381 ${\displaystyle \Omega }$
Primary coil inductance (${\displaystyle L_1}$)	610.0 nH
Secondary capacitance (${\displaystyle C_2}$)	320.2 pF
Secondary coil resistance (${\displaystyle R_2}$)	0.2328 ${\displaystyle \Omega }$
Secondary coil inductance (${\displaystyle L_2}$)	430.2 nH
Capacitors’ ESR	0.050 ${\displaystyle \Omega }$
Switch RDS${\displaystyle {}_{on}}$ (${\displaystyle S}$)	0.025 ${\displaystyle \Omega }$
Load resistance (${\displaystyle R_L}$)	12.96 ${\displaystyle \Omega }$
Mutual inductance (${\displaystyle M}$)	128.1 nH

.

5. Simulation Results and Discussion

A Differential Evolution algorithm is proposed to obtain the best values for the primary and secondary inductances, aiming to maximize the efficiency of the system. In Figure 17c, the curve of the efficiency of the best individual in relation to the number of generations of the Differential Evolution is presented. Note that the algorithm showed rapid convergence to the global maximum of 94.79% for the PS WPT stage efficiency.

The initial population was generated within a predefined range. Figure 17a shows the distribution of the initial individuals. As expected, the position of the individuals was dispersed, since the population was randomly generated. Figure 17b shows the individual distribution after performing all iterative processes. Note that the position of all individuals converged to the global maximum, where L₂ was equal to 430.2 nH. Since the efficiency dependency of L₁ was neglected, the L₁ values were distributed throughout the defined range.

As mentioned before, the WPT circuit parameters were sized considering an output power of 1 W. However, aiming to evaluate the practical non-idealities and the effective losses in each stage depicted in Figure 16, a 1 W input power was imposed for the Class-E amplifier. Thus, the currents and voltages across the circuit elements, as well as the output power were expected to be lower than those previously calculated. Before analyzing the PS WPT circuit, it is reasonable to check the operation of the Class-E circuit. Looking at Figure 18, the input current I_IN is depicted in (a). As observed, its value was slightly superior to 80 mA, draining about 1 W from the DC power source. In Figure 18b,c, the I_e and C_m currents prove the resonance at 13.56 MHz; however, a small DC level was present, which denotes some mismatch caused by the values’ divergences and dissipative elements; this behavior did not make the circuit operation unfeasible; however, it increased the switch losses due to the current peaks, a consequence of the remanent voltage noticed in Figure 18e. Figure 18d shows the C_e voltage, and it is important to note its peak value, which can reach almost 150 V, being important data for a practical design.

Carrying out the analysis of the circuit for the WPT part, it can be concluded that the current in the primary inductor added to the current in the primary capacitor resulted in the input current I₁. Figure 19a,b illustrate the respective primary inductor and capacitor currents over time. The RMS currents in L₁ and C₁ were 311 mA and 306 mA, respectively. The current across all secondary elements (I₂) is presented in Figure 19c. Due to the losses, the RMS value for I₂ was slightly inferior to the expected one, about 256 mA, and the RMS voltage over R_L was 3.3 V (Figure 19d). Subsequently, the voltage stresses over the capacitances C₁ and C₂ can be noticed in Figure 19e,f, respectively. The expected RMS voltages for the respective primary and secondary capacitances were close to 16.4 V and 9.4 V.

Figure 20 shows the active power drained in different parts of the circuit. In Figure 20a, the DC source input power is presented. As can be noticed, its value corresponded to 1.0 W. Figure 20b shows the power delivered to the PS WPT stage, which resulted in being about 0.91 W. Finally, the output power is depicted in Figure 20c, reaching 0.86 W. Based on the obtained values, it is possible to conclude that the Class-E part operated with an efficiency of 91%; the PS WPT part reached about 94%; the overall efficiency was near 86%. It is worth mentioning the Class-E efficiency was sufficiently good in comparison with the other current source circuits, and the PS WPT efficiency was close to that expected in the preliminary results presented in Table 4. In summary, the overall expected efficiency was significantly satisfactory for this range of power.

Based on the comprehensive analysis of the simulation results, we can assertively conclude that the system’s overall performance met, if not exceeded, the anticipated standards set forth for this project. The recorded values of the currents and voltages across the inductances and capacitances played a pivotal role in the system’s behavior. It is imperative to understand that these measurements are not just mere numerical results; they offer valuable insights that are instrumental for the judicious selection of components. By ensuring the components align with these values, we can guarantee optimal performance and reliability, fortifying the system’s robustness against potential anomalies or discrepancies in real-world applications.

6. Conclusions and Future Works

The present study provided a comprehensive assessment of a Differential-Evolution-assisted optimization of classical compensation topologies for 1 W current-fed IMD wireless charging systems with a biocompatible operating frequency of 13.56 MHz. This innovative approach aimed to indicate that the parallel–series compensation topology fed through a Class-E power amplifier was more stable in the face of coupling factor variation and also tended to reduce the input current despite a frequency oscillation of up to 10%. The system was optimized using a Differential Evolution algorithm in order to select optimal parameters that must provide non-bifurcation criteria, along with higher efficiency.

A wide review of the four classical topologies of inductive resonant coupling was made by evaluating their behavior in terms of frequency and coupling factor variations. Once having clarified all the advantages and disadvantages of each, the PS topology was selected. The Differential Evolution algorithm showed rapid convergence to the global maximum of 94.79% for the PS WPT stage efficiency. The values of the primary and secondary inductances obtained for L₂ were equal to 430.2 nH; as the efficiency had no dependency on L₁, the L₁ value remained as previously selected, equal to 610 nH. The main results proved the good stability with an efficiency of around 94% of the PS WPT part. The Class-E PA developed achieved 91% efficiency, with the whole system standing above 86%.

The proposed system presents a potential path as a consistent energy supply for implantable medical devices, reducing the number of surgical procedures, also increasing the lifespan of the batteries and allowing operation during recharging. The results highlight the valuable role of the wireless charging systems in medical applications, considering biological factors so as to guarantee operational stability and appropriate efficiency, making the application safe and viable.

Future work will rely on evaluating the challenges of transitioning our idealized circuit to practical applications. It is paramount to consider the constraints posed by high-frequency operations. The high-frequency system necessitates sourcing suitable electronic components. GaN field-effect transistors emerge as a viable choice for Switch RDS_{${\displaystyle on}$}, due to their capability to handle high switching frequencies and compatibility with the project’s current requirements. Subsequently, a low-side ultra-fast gate driver optimized for switching GaN FETs in high-speed scenarios is indispensable. A clock oscillator with a frequency of 13.56 MHz will also be required. Designing inductors for high-frequency operation presents its unique challenges. The advantage of high-frequency operation is that it necessitates significantly lower inductance values. This makes air–core designs, which eliminate core losses at high frequencies, attractive [29]. Planar inductors on printed circuit boards (PCBs) stand out as they support higher operational frequencies and offer the added benefits of reduced weight and size—factors critical for biomedical applications [30]. High-frequency winding losses in PCB planar inductors are substantially influenced by skin and proximity effects, given the specific geometry of the PCB planar winding conductors [30]. This can lead to increased high-frequency losses. Therefore, an inductor design that holistically considers winding losses, overall performance, and thermal issues becomes crucial [31]. Furthermore, accurate temperature predictions and management of the coupling inductors are vital, not just for the electrical attributes, but also for the overall reliability and performance of the WPT system—integral for any IMD application.