# Ferrite Plate Multi-Objective Topology Optimization in a Wireless Power Transfer System for UAV Charging

Giulio Poggiana, Mohammed Terrah, Riccardo Torchio, Vincenzo Cirimele, Lionel Pichon, Mohamed Bensetti, Fabrizio Dughiero

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Giulio Poggiana, Mohammed Terrah, Riccardo Torchio, Vincenzo Cirimele, Lionel Pichon, et al. Ferrite Plate Multi-Objective Topology Optimization in a Wireless Power Transfer System for UAV Charging. IEEE Open Journal of Industry Applications, 2026, pp.1-12. 10.1109/ojia.2026.3665327. hal-05516578

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Submitted on 18 Feb 2026

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Ferrite Plate Multi-Objective Topology
Optimization in a Wireless Power
Transfer System for UAV Charging

Giulio Poggiana<fnref n="1" />,<fnref n="2" />, Mohammed Terrah<fnref n="2" />, Riccardo Torchio<fnref n="1" />,<fnref n="3" />, Member, IEEE,
Vincenzo Cirimele<fnref n="4" />, Senior Member, IEEE, Lionel Pichon<fnref n="2" />, Mohamed Bensetti<fnref n="2" />, and
Fabrizio Dughiero<fnref n="1" />

**ABSTRACT** Wireless power transfer (WPT) systems offer a promising solution for automating the charging process and extending the mission duration of unmanned aerial vehicles (UAVs). However, the weight of the WPT system's receiving part mounted on the UAV can significantly impact its energy consumption and overall performance. In this regard, the ferrite plate is a critical component. While it is necessary for improving magnetic coupling and reducing the stray magnetic field, it also increases weight. Topology optimization (TO) is an effective tool for addressing this challenge and identifying the most suitable shape for the ferrite plate. In this study, a multi-objective topology optimization (TO) approach based on the solid isotropic material with penalization (SIMP) method is employed to design an optimized ferrite plate for the receiver side of a WPT system for UAV charging. The multi-objective optimization, implemented through a weighted-sum formulation, identifies ferrite configurations that maximize mutual coupling while minimizing material usage. Three representative Pareto-optimal designs are fabricated and experimentally validated against a full-ferrite reference plate. The results show that the reference configuration is a dominated solution in the Pareto sense, as equivalent mutual coupling can be achieved with approximately 40% less ferrite material. The trade-off between the resulting reduction in charging efficiency and the decrease in the required flight power due to ferrite reduction is analyzed, further demonstrating the potential of TO to substantially reduce system weight without compromising overall system performance.

**INDEX TERMS** Wireless Power Transfer, Inductive Power Transfer, Topology Optimization, Unmanned Aerial Vehicles, Electromagnetic Modeling.

# I. INTRODUCTION

UNMANNED Aerial Vehicles (UAVs), more commonly referred to as drones, are increasingly being adopted in a wide range of fields such as environmental monitoring, precision agriculture, surveillance, and delivery services [1]–[4]. Their main advantage is the high level of automation. However, their operational time is limited by battery capacity. Traditional recharging requires the UAV to return to the starting point, reducing mission duration and operational range. To overcome this limitation, recent works proposed the integration of inductive wireless power transfer (WPT) systems into UAVs to automate the charging process [5]–[8]. In WPT systems, power transmission is achieved through

magnetic coupling between a transmitter (Tx) coil, installed in the charging base, and a receiver (Rx) coil, integrated into the UAV. This enables contactless charging that can easily be automated, making it a particularly flexible and promising solution for UAV applications.

However, several relevant open issues limit the spread of WPT systems in this type of application. In particular, the additional weight of the Rx components affects the UAV's energy consumption. Therefore, this weight must be minimized. Another important issue is the alignment of Tx and Rx coils to ensure efficient power transfer. Since perfect positioning is not always achievable when landing

<footnote n="1">¹Department of Industrial Engineering, Università degli Studi di Padova, 35131 Padova, Italy</footnote>

<footnote n="2">²Génie Électrique et Électronique de Paris (GeePs), Université Paris-Saclay, CNRS, CentraleSupélec, Sorbonne Université, 91192 Gif-sur-Yvette, France</footnote>

<footnote n="3">³Department of Information Engineering, Università degli Studi di Padova, 35131 Padova, Italy</footnote>

<footnote n="4">⁴Department of Electrical, Electronic, and Information Engineering, University of Bologna, 40136 Bologna, Italy</footnote>

<footnote>Corresponding author: R. Torchio (email: riccardo.torchio@unipd.it).</footnote>
the drone, the WPT system must be robust with respect to
coil misalignment.

To enhance magnetic coupling between the coils, improve
the efficiency of the charging system, and reduce hot spots
caused by induced currents in the UAV metallic frame, a
ferrite plate is commonly employed both on the Tx and Rx
side, even though this increases the UAV onboard weight [7]–
[10]. Recent studies have therefore addressed the design of
the Rx ferrite plate to achieve a lightweight configuration [11],
[12]. All these works adopt bars or tiles as basic units that
are then repeated according to fixed geometries or patterns.
This approach is based on qualitative assumptions about the
magnetic field distribution near the coils. The search for an
optimal configuration is therefore limited to adjusting a small
set of geometric variables, namely width, length, height, and
spacing pitch. Clearly, this strongly limits the design space
and excluding the exploration of unconventional shapes that
cannot be identified purely through empirical intuition or
prior experience.

Topology optimization (TO) can overcome this limitation
by enabling the identification of the optimal material dis-
tribution within a given design domain, based on defined
objective functions and constraints. In TO, the design domain
is divided into discrete elements, each of which is assigned
a design variable representing the presence or absence of
material. Unlike traditional approaches, TO therefore does not
require starting from shapes or distributions of the material
imposed a priori, allowing for the discovery of novel and
non-intuitive configurations tailored to specific performance
goals.

In many practical engineering problems, multiple con-
flicting objectives must be considered, and the solution is
often interpreted in the Pareto sense. TO approaches that
account for Pareto-optimal solutions have been implemented
in the literature using evolutionary algorithms [13]–[16], and
gradient-based methods by combining the objective functions
into a single weighted-sum formulation, which represents
the state-of-the-art method for solving multi-objective TO
problems thanks to its simple implementation strategies [17]–
[20].

While TO has been widely investigated across various
engineering fields, its application in electromagnetics has
only recently begun to attract significant attention [21]–[25].
Naturally, it has drawn increasing interest also for the design
of ferrite plates in WPT systems, where it has been applied
both in single-objective optimization problems [26]–[29] and
in multi-objective ones [30], [31]. Focusing on the multi-
objective TO, in [30], a genetic algorithm is employed to
optimize the ferrite distribution in the magnetic core of a WPT
system for electric vehicle applications. Similarly, in [31],
a genetic algorithm and a gradient-informed Pareto-based
algorithm are applied and compared for multi-objective TO
of the ferrite structure in a WPT system, with the aim of
maximizing magnetic coupling while minimizing the volume
of ferrite material. While these studies provide a thorough

analysis and comparison of different optimization algorithms
suitable for multi-objective TO in WPT applications, they
do not implement the optimized designs nor experimentally
validate the results. Furthermore, the WPT systems analyzed
in these works are not tailored to a specific target application
and, therefore, do not address the constraints associated with
a particular use case.

In contrast, the present work explores the application of
multi-objective TO to distribute ferrite within the Rx plate
of a WPT system specifically designed for UAV charging.
The goal is to identify Pareto-optimal solutions maximizing
the magnetic coupling between the Tx and Rx coils while
minimizing the amount of ferrite used, thereby achieving a
lightweight configuration. Additionally, a constraint on the
Tx self-inductance is incorporated to ensure its stability under
misalignment. This requirement arises because WPT systems
rely on the resonance with the elements of compensation
networks to enhance power transfer efficiency and reduce the
volt-ampere rating of the power electronics devices [32]. In
case of misalignment, ferrite can affect the magnetic field
distribution around the coils causing a variation in Tx and Rx
self-inductances and the potential detuning of the resonance
condition.

Some of the optimal ferrite configurations obtained through
the adopted TO approach are manufactured, and the resulting
characteristics are experimentally evaluated. To the best of
the authors’ knowledge, no previous work in the literature has
applied multi-objective TO to the design of the ferrite plate
in a WPT system for UAV applications, with subsequent
manufacturing of the optimized plates and experimental
validation of the numerical results.

The remainder of the paper is organized as follows. In
Section II, a general description of the WPT system for UAV
charging considered as the reference for this work is provided.
In Section III, the TO concept is briefly recalled, giving
particular attention to the approach implemented in the present
work. In Section IV, the electromagnetic model implemented
for this work and the numerical results from the performed
optimization are presented and discussed. Section V describes
the experimental validation of the proposed configurations.
The main outcomes of the work are finally summarized in
Section VI.

II. DEVICE UNDER STUDY

The analysis presented in this paper is based on a WPT
system specifically designed to operate around the nominal
frequency of 150 kHz and be integrated with a UAV model
that operates without additional landing gear [33]. This results
in the Tx and Rx coils of the WPT system being positioned in
very close proximity so achieving a better magnetic coupling.
A CAD image of the UAV model for which the WPT system
was designed is shown in Fig. 1. Further details on the system
design and performance can be found in previous works [5],
[34], [35].
The Tx coil, shown in Fig. 2, consists of a circular planar coil with six turns, constructed using litz wire with a diameter of 2.6 mm. The Rx coil also consists of six turns of litz wire, identical to that used in the Tx coil, and also adopts a circular geometry. However, to mitigate the effects of potential misalignment caused by inaccurate UAV positioning during landing, the Rx coil is designed with a diameter equal to half that of the Tx one [5], [36]. A ferrite plate is placed on top of the Rx coil to enhance the coupling between the two coils and serve as shielding to prevent electromagnetic interaction between the Rx coil and the drone landing gear [36]. This ferrite plate is fabricated by bonding 16 tiles, each measuring 25 mm × 25 mm and with a thickness of 2 mm. The ferrite is then attached to a printed circuit board (PCB) on the bottom side of the UAV using a thin adhesive layer. The remaining components of the WPT system, together with the electrical connections between the UAV's motors and their respective electronic drivers, are placed on the top layer of the PCB. Fig. 3 illustrates the Rx assembly, while the main electrical and design parameters of the two assemblies are summarized in Table 1.

The system, designed to transfer a rated power of 120 W, can be effectively represented by the equivalent circuit shown in Fig. 4, where Tx and Rx coils are modeled by the inductances $L_{Tx}$ and $L_{Rx}$, respectively, and their electromagnetic interaction is represented by the mutual inductance $M$. The parameters $R_{Tx}$ and $R_{Rx}$ represent the

<figure><img src="image_2.png" /><figcaption>FIGURE 1: Model of the UAV for which the WPT system is designed.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIGURE 2: Tx coil of the system under study.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIGURE 3: Rx coil and Rx ferrite glued on the PCB, taken as reference for the TO.</figcaption></figure>

<table><caption>TABLE 1: Design and electrical parameters of the reference Tx and Rx assemblies at nominal operating conditions.</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Tx coil external diameter</td><td>200 mm</td></tr><tr><td>Tx coil number of turns</td><td>6</td></tr><tr><td>Rx coil external diameter</td><td>100 mm</td></tr><tr><td>Rx coil number of turns</td><td>6</td></tr><tr><td>Litz wire number of strands</td><td>1000</td></tr><tr><td>Litz wire strands diameter</td><td>0.05 mm</td></tr><tr><td>Litz wire diameter</td><td>2.6 mm</td></tr><tr><td>Distance between Tx and Rx coils</td><td>11 mm</td></tr><tr><td>Rx ferrite plate width</td><td>105 mm × 100 mm</td></tr><tr><td>Rx ferrite plate thickness</td><td>2 mm</td></tr><tr><td>Ferrite relative permeability</td><td>2300</td></tr><tr><td>Operating frequency</td><td>150 kHz</td></tr><tr><td>Nominal power</td><td>120 W</td></tr><tr><td>Battery voltage</td><td>12.6 V</td></tr><tr><td>I<sub>Tx,RMS</sub></td><td>11 A</td></tr><tr><td>I<sub>Rx,RMS</sub></td><td>10.6 A</td></tr><tr><td>Tx coil self-inductance</td><td>13.08 μH</td></tr><tr><td>Rx coil self-inductance</td><td>7.66 μH</td></tr><tr><td>Mutual inductance</td><td>1.18 μH</td></tr><tr><td>Tx coil resistance</td><td>165 mΩ</td></tr><tr><td>Rx coil resistance</td><td>80 mΩ</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>FIGURE 4: Equivalent circuit representation of the WPT system under test.</figcaption></figure>

equivalent series resistances of the compensated coils, while the transmitter and receiver compensation capacitors are modeled through the capacitances $C_p$ and $C_s$, respectively.

Series–series compensation has been selected as the compensation topology for this system since the resonance
condition is not affected due to the variations in mutual inductance caused by coil misalignment [32], making it well-suited for UAV applications. However, the proximity of the Tx coil to the ferrite plate can still influence the Tx self-inductance under misalignment conditions, leading to mistuning of the compensation network. In particular, previous research [5], [34] has shown that a phase shift between voltage and current at the primary side occurs from 75 mm misalignment conditions, negatively affecting the volt-ampere rating of the source, and then the efficiency of the supplying converter. Reducing the amount of ferrite on the Rx plate can then contribute to reducing the weight, but can also help mitigate the variation of the Tx self-inductance under misalignment conditions. On the other hand, reducing the ferrite can negatively affect mutual coupling, which in turn reduces the overall efficiency of the system. Therefore, for this particular application, a multi-objective TO approach is particularly effective in identifying an optimal compromise design.

### III. ADOPTED TOPOLOGY OPTIMIZATION APPROACH

TO is a technique that enables the determination of the optimal distribution of a given material within a design domain $\Omega_d$, with the goal of minimizing or maximizing one or more objective functions $f_i$.

In TO problems, the design domain is discretized and a design variable $\theta$ (also called density) is assigned to each element of the mesh. Two main approaches can be defined, based on the mathematical space in which the design variables belong. In the binary approach, the design variable can only assume discrete values, representing the absence or presence of material:

$$ \theta_i \in \{0, 1\}, \forall i = 1 \cdots n, \quad (1) $$

where $\theta = 0$ indicates the absence of material and $\theta = 1$ indicates the presence of material.

In contrast, in the continuous approach, the design variable can take any value within the unit interval:

$$ 0 \le \theta_i \le 1, \forall i = 1 \cdots n, \quad (2) $$

and it is then correlated to the corresponding material property through a material interpolation scheme. From a manufacturing perspective, the use of binary design variables is advantageous, as it clearly delineates the regions where material is present or absent. However, the quality of the resulting design heavily depends on the mesh resolution used to discretize the design domain $\Omega_d$. In contrast, using continuous variables allows for higher-order interpolation functions, providing greater flexibility [31].

In this study, the Solid Isotropic Material with Penalization (SIMP) continuous approach [37] is adopted within $\Omega_d$, corresponding to the original Rx ferrite plate volume, such that the material property $\mu_{r,i}$ assigned to the i-th element is reconstructed through the following scheme:

$$ \mu_{r,i} = \mu_{r,air} + (\mu_{r,ferrite} - \mu_{r,air})\theta_i^\alpha, \quad (3) $$

<figure><img src="image_6.png" /><figcaption>FIGURE 5: Conceptual representation of the SIMP approach.</figcaption></figure>

where $\mu_{r,air}$ and $\mu_{r,ferrite}$ are the relative permeability of air and ferrite, respectively, and $\alpha$ is the penalization parameter. The correspondence between $\theta_i$ and the distributed material can be argued from Fig. 5. Similarly to the binary approach, when $\theta_i = 0$, the relative permeability of the element is 1, indicating the absence of ferrite in such element, while $\theta_i = 1$ when the equivalent permeability corresponds to $\mu_{r,ferrite}$, indicating the presence of ferrite.

The SIMP method is one of the most widely used approaches in TO for WPT systems [24], [28], [38] due to its computational efficiency and relative ease of implementation [39], as the material interpolation scheme (MIS) is based on the power law in (3). Consequently, most commercial TO software packages include an implementation of the SIMP technique.

When dealing with continuous TO, since the design variable $\theta_i$ is continuous, intermediate values of $\theta_i$ result in intermediate permeability values that do not strictly correspond to the presence or absence of material. One of the main challenges is then the presence of these intermediate material densities in the optimal design, since their physical interpretation in terms of real materials is not always straightforward. The SIMP method provides an important advantage in this context by introducing a penalization strategy through the choice of the $\alpha$ parameter in (3), which reduces intermediate densities and guides the solution toward designs that are more physically meaningful.

In this work, numerical techniques, such as the Helmholtz filter and the hyperbolic tangent projection, are also applied at each iteration to mitigate this so-called “gray-scale” phenomenon, as discussed in [21], [40]. Specifically, in the Helmholtz filtering stage, the filter radius was set equal to the maximum element size of the mesh in the design domain, while for the hyperbolic tangent projection, a projection slope of $\beta = 8$ and a projection point of $\theta_b = 0.5$ were used. Finally, a value of $\alpha = 7$ was adopted. Furthermore, at the end of the optimization, all elements with $\theta_i > 0.5$ are assigned to ferrite.
It is worth noting that, being the number of decision variables equal to the number of finite elements discretizing $\Omega_d$, TO falls into the class of large-scale optimization problems. As a result, TO is typically solved using gradient-based methods, as the high dimensionality of the design space can quickly lead to the curse of dimensionality [41]. Gradient-based optimization techniques applicable to such constrained problems often require transforming the multi-objective optimization problem into a single-objective formulation [42]. In this work, the two objective functions $f_1(\theta)$ representing the mutual inductance, and $f_2(\theta)$, representing the amount of ferrite material in the Rx plate, are combined into a single objective function $f(\theta)$ using the weighted sum approach through the weights $w_1$ and $w_2$. The resulting constrained optimization problem can be expressed as:

$$ \max \left\{ f(\theta) = w_1 f_1(\theta) - w_2 f_2(\theta) = w_1 \frac{M_0(\theta)}{M_{\text{full}}} - w_2 \frac{V(\theta)}{V_{\text{full}}} \right\} \\ \text{with } w_1 + w_2 = 1 \\ \text{subject to } 0.95 L_0(\theta) \le L_{75}(\theta) \le 1.05 L_0(\theta), \quad (4) $$

where $M_0(\theta)$ represents the mutual inductance subject to the TO material distribution at 0 mm misalignment and $M_{\text{full}}$ is the mutual inductance with the reference plate at 0 mm. $V(\theta)$ is the actual volume of material distributed in $\Omega_d$, while $V_{\text{full}}$ is the entire volume of $\Omega_d$. It is worth noting that maximizing $f(\theta)$ corresponds to maximizing the mutual inductance while simultaneously minimizing the amount of ferrite material within the design domain. Regarding the constraint, it ensures that the self-inductance of the Tx coil at 75 mm misalignment, indicated as $L_{75}$, remains within ±5% of its value at perfect alignment, indicated as $L_0$. This level of variation is considered acceptable, as it falls within the usual tolerance range of commercially available capacitors [43]. During the optimization process, $L_{75}$ and $L_0$ are evaluated at each iteration, verifying that $L_{75}$ remains within the bounds specified in Eq. 4. The constraint is handled through a penalization factor; therefore, solutions that do not satisfy it are not automatically discarded. As a result, certain solutions may not fully comply with the constraint.

The Pareto front representing the set of optimal solutions is obtained by performing multiple optimizations varying the weights in (4). Since the weights $w_1$ and $w_2$ must sum to one, the remainder of this article will refer only to $w_1$, assuming that $w_2 = 1 - w_1$. In particular, the Pareto front is approximated using 10 optimization points obtained by linearly sweeping $w_1$ from 0 to 1.

## IV. ELECTROMAGNETIC MODELLING AND NUMERICAL RESULTS

In this work, the TO process is performed using the finite element method (FEM). Specifically, the optimization module of the commercial software COMSOL Multiphysics is employed, where the SIMP approach is implemented. Since each optimization iteration requires evaluating the mutual

<figure><img src="image_7.png" /><figcaption>FIGURE 6: Geometry implemented to perform the TO.</figcaption></figure>

inductance at 0 mm misalignment as well as $L_{75}$ and $L_0$, the model was designed to account for both misalignment conditions within a single iteration by including two Tx coils in the geometry. These coils were alternatively excited in separate solver steps by prescribing the appropriate current density in the active coil while assigning air properties to the inactive one. This choice, combined with the large number of design variables in TO, makes the problem computationally demanding. Therefore, it is crucial to adopt strategies that reduce the overall computational effort. In the FEM model, a simplified equivalent representation of the Rx and Tx coils as homogeneous conductors with uniform current density is used to reduce complexity [44]. Additionally, geometric symmetry along one axis is exploited. The geometry of the model implemented to perform the optimization is shown in Fig. 6.

The problem is treated as a magneto quasi-static problem and solved via the magnetic vector potential A formulation. To further simplify the model, the conductive PCB located above the ferrite is treated as a perfectly electrically conductive (PEC) material by adopting a boundary condition that forces the magnetic flux density B to be tangential to its surface. This PEC assumption is considered appropriate for modeling the device, given the presence of the large ground plane that characterizes the PCB. This approach accounts for the PCB's influence on mutual coupling due to induced currents, without requiring a too fine mesh resolution to accurately capture the skin effect. Particular attention is also given to the discretization of the TO design domain, as it directly affects the optimization process. A structured quadrilateral mesh with 5 mm × 5 mm elements is first generated and converted into a triangular mesh by splitting each quadrilateral along its diagonal. The resulting mesh is then extruded through the thickness to obtain a hexahedral volume mesh. Finally, the magnetic behavior of the ferrite is assumed to be linear. This approximation is justified by preliminary analyses of the WPT system prior to optimization, which show that the maximum magnetic flux density induced in the ferrite plate under standard operating conditions is approximately 35 mT [5].
<figure><img src="image_8.png" /><figcaption>FIGURE 7: Pareto front obtained through the weighted sum approach. The solution with full material is a dominated solution. The black dots represent the manufactured solutions.</figcaption></figure>

<table><caption>TABLE 2: Simulated values of $M_0$, ferrite material percentage, Tx self-inductance under misalignment conditions and relative difference in $M_0$ for different weight values.</caption><thead><tr><th>w<sub>1</sub></th><th>Ferrite volume</th><th>M<sub>0</sub></th><th>ΔL<sub>Tx</sub></th><th>ΔM</th></tr></thead><tbody><tr><td>0</td><td>0%</td><td>0.26 μH</td><td>5%</td><td>-78%</td></tr><tr><td>0.1</td><td>0%</td><td>0.26 μH</td><td>5%</td><td>-78%</td></tr><tr><td>0.2</td><td>0%</td><td>0.26 μH</td><td>5%</td><td>-78%</td></tr><tr><td>0.3</td><td>18%</td><td>0.68 μH</td><td>4%</td><td>-44%</td></tr><tr><td>0.4</td><td>25%</td><td>0.88 μH</td><td>5%</td><td>-28%</td></tr><tr><td>0.5</td><td>32%</td><td>1.11 μH</td><td>5%</td><td>-10%</td></tr><tr><td>0.6</td><td>39%</td><td>1.19 μH</td><td>5%</td><td>-3%</td></tr><tr><td>0.7</td><td>50%</td><td>1.25 μH</td><td>6%</td><td>+1.5%</td></tr><tr><td>0.8</td><td>58%</td><td>1.28 μH</td><td>10%</td><td>+4%</td></tr><tr><td>0.9</td><td>71%</td><td>1.29 μH</td><td>11%</td><td>+5%</td></tr><tr><td>1</td><td>100%</td><td>1.23 μH</td><td>12%</td><td>-</td></tr></tbody></table>

The optimization is performed on a workstation equipped with an Intel Xeon processor (16 cores, 3.10 GHz) and 128 GB of RAM. The computation time for each set of weights is approximately 13 h.

The obtained Pareto front is shown in Fig. 7, while the corresponding values of mutual inductance at 0 mm misalignment conditions $M_0$ and ferrite material percentage are summarized in Table 2. Additionally, the table reports the relative difference in mutual inductance between each optimized solution and the reference configuration $ΔM = (M_{full} - M_0)/M_{full}$ (in percent), along with the relative variation in Tx self-inductance between the 75 mm and the 0 mm misalignment conditions for each optimized plate $ΔL_{Tx} = (L_{75} - L_0)/L_0$ (in percent).

It is particularly interesting to note that the full-material solution, i.e. the reference plate configuration, is a dominated

solution in the Pareto sense, as it presents a lower mutual inductance while using more ferrite material compared to other optimized solutions. Moreover, it is important to observe that the variation in the Tx self-inductance due to misalignment remains within the prescribed 5% range for solutions in which the ferrite usage remains below 39%, thereby satisfying the imposed constraint. For configurations with greater ferrite quantity, the constraint is no longer strictly satisfied.

The corresponding ferrite shapes resulting from the TO, which represent the points on the Pareto front for the different weights, are reported in Fig. 8. The configurations obtained with $w_1 = 0$ and $w_1 = 1$ are not included, as they correspond to trivial cases: the absence of ferrite and the fully filled reference plate, respectively. By analyzing the obtained optimized shapes, it can be observed that material removal from a given region indicates a limited contribution to magnetic flux guidance and, consequently, a negligible impact on system performance. This behavior is particularly evident in the central region, where ferrite is removed in all configurations, as the magnetic flux density is nearly zero and therefore does not influence the mutual inductance. This result is consistent with expectations and has also been reported in previous studies [29], [45].

It is evident that the final shapes generated by the TO are challenging to manufacture due to their irregular and complex geometries. Therefore, to ensure practical feasibility, the optimized shapes must be simplified prior to fabrication. Among the set of Pareto-optimal solutions, three representative configurations were selected and simplified for manufacturing and experimental validation. Specifically, the solution corresponding to $w_1 = 0.4$ was chosen for its ease of machinability. The configuration with $w_1 = 0.5$ was selected because it satisfies the constraint on the Tx self-inductance while achieving a significant reduction in ferrite volume, with only a 10% decrease in mutual inductance compared to the reference case. Finally, the solution corresponding to $w_1 = 0.8$ was included to demonstrate that a reduced amount of ferrite can yield a higher mutual inductance than the reference plate.

In the remainder of this paper, the simplified geometries associated with $w_1 = 0.4$, $w_1 = 0.5$, and $w_1 = 0.8$ are referred to as Case A, Case B, and Case C, respectively. The simplified geometries are shown alongside the original TO results in Fig. 9, Fig. 10, and Fig. 11 for Cases A, B, and C, respectively. The adopted simplification was driven by the need to obtain manufacturable geometries that favor straight cuts and avoid sharp corners and jagged edges, thereby enabling machining, especially in a laboratory-scale setting. A quantitative comparison of the simulated differences in mutual inductance, ferrite volume, and $ΔL_{Tx}$ between the original optimized designs and the corresponding simplified geometries is reported in Table 3. Moreover, the differences between the original and simplified shapes are highlighted in Fig. 7, where the simplified geometries are also reported.
<figure><img src="image_9.png" /><figcaption>FIGURE 8: Ferrite shapes resulting from the TO multi-objective optimization.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIGURE 9: Optimized ferrite plate for Case A: simplified shape (a) and original from TO (b). Units in mm</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>FIGURE 10: Optimized ferrite plate for Case B: simplified shape (a) and original from TO (b). Units in mm.</figcaption></figure>

<table><caption>TABLE 3: Simulated values of $M_0$, ferrite material percentage, and $\Delta L_{Tx}$ for the original optimized designs and the corresponding simplified geometries</caption><thead><tr><th>Case</th><th>Ferrite volume</th><th>M<sub>0</sub></th><th>ΔL<sub>Tx</sub></th></tr></thead><tbody><tr><td>A</td><td>30%</td><td>0.92 μH</td><td>5%</td></tr><tr><td>A original from TO</td><td>25%</td><td>0.88 μH</td><td>5%</td></tr><tr><td>B</td><td>36%</td><td>1.12 μH</td><td>4%</td></tr><tr><td>B original from TO</td><td>32%</td><td>1.11 μH</td><td>5%</td></tr><tr><td>C</td><td>62%</td><td>1.27 μH</td><td>11%</td></tr><tr><td>C original from TO</td><td>58%</td><td>1.28 μH</td><td>10%</td></tr></tbody></table>

## V. EXPERIMENTAL VALIDATION

The manufactured three simplified geometries derived from the TO results are shown in Fig. 12. Each ferrite plate has been manufactured by cutting the individual tiles using a diamond wire saw.

The self and mutual inductance values of the coils for each assembly are measured using a Wayne Kerr 4300

series LCR meter at the operating frequency and under the same injected current. The measured values are reported in Table 4 together with the corresponding simulation results. Table 4 also reports the measured values of mutual inductance, where $\Delta M$ denotes the relative difference between each optimized plate and the reference configuration, while Table 5 summarizes the measured relative variation in Tx self-inductance under misalignment conditions for each optimized
<figure><img src="image_12.png" /><figcaption>FIGURE 12: Manufactured ferrite plates.</figcaption></figure>

<table><caption>TABLE 4: Comparison between measured mutual inductance and simulated one. $\Delta M$ represents the relative variation with respect to the reference case.</caption><thead><tr><th>Case</th><th>A</th><th>B</th><th>C</th><th>Reference</th></tr></thead><tbody><tr><td>Ferrite volume</td><td>30%</td><td>36%</td><td>62%</td><td>100%</td></tr><tr><td>M<sub>0</sub> measured</td><td>0.90 µH</td><td>1.07 µH</td><td>1.18 µH</td><td>1.18 µH</td></tr><tr><td>M<sub>0</sub> simulated</td><td>0.92 µH</td><td>1.12 µH</td><td>1.27 µH</td><td>1.23 µH</td></tr><tr><td>Relative Error</td><td>2%</td><td>5%</td><td>7.5%</td><td>4%</td></tr><tr><td>ΔM</td><td>-23%</td><td>-9%</td><td>0%</td><td>-</td></tr></tbody></table>

<table><caption>TABLE 5: Comparison between measured Tx self-inductance variations and simulated ones.</caption><thead><tr><th>Case</th><th>A</th><th>B</th><th>C</th><th>Reference</th></tr></thead><tbody><tr><td>Ferrite volume</td><td>30%</td><td>36%</td><td>62%</td><td>100%</td></tr><tr><td>ΔL<sub>Tx</sub>measured</td><td>5%</td><td>4%</td><td>5%</td><td>9%</td></tr><tr><td>ΔL<sub>Tx</sub>simulated</td><td>5%</td><td>4%</td><td>11%</td><td>11%</td></tr><tr><td>Relative Error</td><td>-</td><td>-</td><td>6%</td><td>3%</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>FIGURE 13: System under test.</figcaption></figure>

plate. An example of the system employed for these tests is illustrated in Fig. 13.

As observed in both Table 4 and Table 5, the experimental results show good agreement with the numerical simulations. Minor discrepancies can be attributed to modeling approximations, particularly in the conductor representation, as well as to tolerances in coil placement during measurements and, most notably, machining tolerances in the manufacturing of the ferrite shapes. Furthermore, although Case C does not exhibit

<table><caption>TABLE 6: Measured values of $L_{Tx}$ and $L_{Rx}$, M and k under perfect alignment and 75 mm misalignment conditions.</caption><thead><tr><th>Case</th><th>A</th><th>B</th><th>C</th><th>Reference</th></tr></thead><tbody><tr><td>Ferrite volume</td><td>30%</td><td>36%</td><td>62%</td><td>100%</td></tr><tr><td>M<sub>0</sub> measured</td><td>0.90 µH</td><td>1.07 µH</td><td>1.18 µH</td><td>1.18 µH</td></tr><tr><td>L<sub>Tx,0</sub> measured</td><td>13 µH</td><td>13.07 µH</td><td>13.05 µH</td><td>13.08 µH</td></tr><tr><td>L<sub>Rx,0</sub> measured</td><td>5.42 µH</td><td>6.20 µH</td><td>6.61 µH</td><td>7.66 µH</td></tr><tr><td>k<sub>0</sub></td><td>0.11</td><td>0.12</td><td>0.13</td><td>0.12</td></tr><tr><td>M<sub>75</sub> measured</td><td>0.77 µH</td><td>0.89 µH</td><td>1.08 µH</td><td>1.1 µH</td></tr><tr><td>L<sub>Tx,75</sub> measured</td><td>13.63 µH</td><td>13.66 µH</td><td>14.01 µH</td><td>14.37 µH</td></tr><tr><td>L<sub>Rx,75</sub> measured</td><td>5.37 µH</td><td>6.22 µH</td><td>6.53 µH</td><td>7.77 µH</td></tr><tr><td>k<sub>75</sub></td><td>0.09</td><td>0.096</td><td>0.11</td><td>0.10</td></tr></tbody></table>

<table><caption>TABLE 7: Measured values of Rx resistance.</caption><thead><tr><th>Case</th><th>Ferrite volume</th><th>Rx resistance</th><th>ΔR<sub>Rx</sub></th></tr></thead><tbody><tr><td>A</td><td>30%</td><td>167 mΩ</td><td>108%</td></tr><tr><td>B</td><td>36%</td><td>130 mΩ</td><td>62%</td></tr><tr><td>C</td><td>62%</td><td>95 mΩ</td><td>19%</td></tr><tr><td>Reference</td><td>100%</td><td>80 mΩ</td><td>-</td></tr></tbody></table>

a mutual inductance value higher than that of the reference configuration, it is worth noting that the measured value of $M_0$ in that case is identical to that of the reference plate. This confirms that the reference case is a dominated solution in the Pareto sense, demonstrating that a significant reduction in ferrite material can be achieved without compromising magnetic coupling performance. Regarding the variation in Tx self-inductance, it is evident that all three tested cases successfully maintain the variation within the acceptable limit defined by the system constraints. To provide a more comprehensive characterization of the system behavior with the proposed ferrite plates, the measured values of the Tx and Rx self-inductances, together with the mutual inductance M and the coupling coefficient k, are reported for both perfect alignment and 75 mm misalignment conditions in Table 6. It is worth noting that Case C exhibits a higher coupling coefficient than the full-ferrite reference configuration under both alignment conditions.

When reducing the ferrite volume, it is also important to note that the current density distribution in the Rx coil is affected by the proximity effect with the conductive parts of the PCB. This phenomenon becomes more pronounced as the amount of ferrite decreases. Therefore, to provide a more comprehensive assessment of the impact of the optimized ferrite plates on the system, the Rx coil resistance is measured with an LCR meter at the working frequency for the different configurations and compared to the reference case. The measured resistance values are reported in Table 7, along with the corresponding relative increases compared to the reference configuration. As observed, reducing the ferrite volume results in a substantial increase in coil resistance, which negatively affects the overall system efficiency.
<table><caption>TABLE 8: Transmission efficiency and reduction in the required flight power.</caption><thead><tr><th>Case</th><th>η<sub>0</sub></th><th>ΔP<sub>f</sub></th></tr></thead><tbody><tr><td>A</td><td>60%</td><td>15%</td></tr><tr><td>B</td><td>69%</td><td>14%</td></tr><tr><td>C</td><td>75%</td><td>8%</td></tr><tr><td>Reference</td><td>77%</td><td>-</td></tr></tbody></table>

The DC/DC efficiency under perfect alignment, denoted as $η_0$, is reported in Table 8 together with the corresponding reduction in the required flight power, denoted as $ΔP_f$. The efficiency is evaluated under the same transferred power at the load for all cases. This is obtained by regulating the input DC voltage (i.e., the voltage $U_{dc}$ in Fig. 4).

$P_f$ is the power required for an UAV to remain in flight. As demonstrated in [46], this power is proportional to the total UAV mass $m$, according to the relationship $P_f \propto m^{3/2}$. The reduction in the required flight power, $ΔP_f$, resulting from the ferrite volume reduction achieved through the proposed TO is computed as:

$$ \Delta P_f = 1 - \left( \frac{m_{tk} + m_{fc}}{m_{tk} + m_f} \right)^{3/2}, \quad (5) $$

where $m_{tk} = 800$ g is the UAV takeoff mass without the WPT system [33], $m_f = 135$ g is the mass of the reference ferrite plate [34], and $m_{fc}$ represents the ferrite mass for Cases A, B, and C.

It can be appreciated that Case C, which achieves approximately a 40% reduction in ferrite volume, corresponds to an 8% reduction in the required flight power, while maintaining the same mutual coupling as the full-ferrite reference configuration. In addition, it presents a reduced variation of the Tx self-inductance under misalignment conditions by 5%, resulting in a higher coupling coefficient than the full-ferrite reference configuration under both alignment conditions. Despite an increase in the Rx coil resistance of less than 20%, which leads to an efficiency reduction of only 2%, this solution represents a well-balanced trade-off for the proposed application.

## VI. CONCLUSION

This paper presented the application of a dedicated topology optimization (TO) technique for the design of the ferrite plate in the receiver assembly of a wireless power transfer system intended for unmanned aerial vehicle (UAV) charging. In this study, the TO problem is formulated as a multi-objective optimization, with the goal of minimizing the amount of ferrite material used in the receiver side while maximizing the mutual coupling between the coils. These two objectives are combined into a single objective function using the weighted sum approach, and the optimal solutions are described in the Pareto sense. To make the proposed solutions feasible with the application requirements, a constraint on the variation of the transmitter coil self-inductance is also set. The main outcome

of this work is demonstrating the feasibility of employing multi-objective TO as a tool for the design of WPT systems for UAVs, enabling lighter designs while maintaining high performance.

Among the various configurations obtained, three are selected for manufacturing, and the simulation results are subsequently validated through experimental measurements. A key outcome of this work is also the confirmation that the full-ferrite solution can be a dominated configuration in the Pareto sense with respect to the two objective functions under consideration. In particular, for the case under study, the same level of magnetic coupling achieved by the full-ferrite configuration can be maintained while reducing the ferrite material by approximately 40%. This result is particularly relevant for UAV applications where reduction of weight is a key objective. It is clearly worth underlining that, although the proposed methodology is general, the optimized design is inherently tailored to the specific system under study and variation in geometrical parameters of the coil could result in a different geometry.

The experimental results on the optimized systems also revealed that reducing the ferrite material increases the receiver coil resistance, thereby affecting the overall system efficiency. In the present work, efficiency was not included among the objective functions, as the focus was on minimizing weight while preserving magnetic coupling. Clearly, the efficiency degradation associated with increased receiver resistance can be mitigated by adopting litz wires with smaller strand diameters. Nevertheless, for applications where efficiency is a priority, future research could focus on incorporating the resistance evaluation directly into the optimization process.

Finally, although thermal effects were not considered in this study, losses can lead to heating and potential thermal management challenges. These effects are another important parameter to investigate in future work, especially for higher-power applications.

Additionally, the proposed optimized designs consider only static misalignment conditions; however, practical WPT operation is also affected by dynamic variations of system parameters. Future work could investigate integrating recent recursive parameter identification techniques into WPT systems to track these variations and enhance the robustness of the optimized receiver designs under real-world operating conditions.

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<figure><img src="image_14.png" /></figure>

**Giulio Poggiana** received the M. Sc. in electrical energy engineering from the University of Padova, Padua, Italy, in 2023.

He is currently pursuing the Ph.D. degree in electrical engineering with the University of Padova. From February to July 2025, he was a visiting Ph.D. student at the Laboratoire de Génie Électrique de Paris (GeePs). His research interests include wireless power transfer, electromagnetic modeling and simulation, and induction heating.

<figure><img src="image_15.png" /></figure>

**Mohammed Terrah** received the M.Sc. degree in Electrical Engineering with a focus on the physics and engineering of energy from Paris-Saclay University, Paris, France, in 2022, and the Engineering degree in Electrical Engineering with a focus on electrical machines from the Higher School of Electrical and Energetic Engineering, Oran, Algeria, in 2021. He is currently pursuing the Ph.D. degree in Electrical Engineering, with research focused on wireless power transfer systems for unmanned aerial vehicle applications.

<figure><img src="image_16.png" /></figure>

**Riccardo Torchio** (Member, IEEE) was born in Padua, Italy, in 1992. He received the M.S. degree in electrical engineering from the University of Padova, Padua, in 2016, and the dual Ph.D. degree in electrical engineering from the University of Padova and Grenoble Electrical Engineering Laboratory (G2ELab), Università Grenoble Alpes, France, in December 2019.

He is currently with the University of Padova, as an Assistant Professor (RTDb). His research interests include numerical methods, optimizations, low-rank compression techniques, uncertainty quantifications, wireless power transfer applications, model order reduction, model predictive control, and the development of integral formulations for the study of low-and high-frequency electromagnetic devices.

<figure><img src="image_17.png" /></figure>

**Vincenzo Cirimele** (Senior Member, IEEE) received the M.Sc. (summa cum laude) and Ph.D. degrees in electrical engineering from the Politecnico di Torino, Turin, Italy, in 2013 and 2017, respectively, and the Ph.D.(Hons.) degree in electronics engineering from the Université Paris Saclay, Gir-sur-Yvette, France, in 2017.

Currently, he is an Associate Professor with the Department of Electrical, Electronic, and Information Engineering, Alma Mater Studiorum University of Bologna, Bologna, Italy. From 2020 to 2021, he was a Technical Responsible for the R&D and Innovation Group of the company Movyon s.p.a., Florence, Italy of Autostrade per l'Italia Group where he supervised projects related to energy sustainability and development of highway electric mobility. From 2017 to 2020, he held the position of an Assistant Professor with the Department of Energy, Politecnico di Torino. His research interests include technologies for electric mobility, inductive power transmission, electromagnetic modeling and simulation, and power electronics.

Prof. Cirimele acts as a Deputy Editor for IET Power Electronics and an Associate Editor for the journals IEEE Transaction on Transportation Electrification and Wireless Power Transfer.

<figure><img src="image_18.png" /></figure>

**Lionel Pichon** is Senior Researcher (Directeur de recherche) at the CNRS (French National Centre for Scientific Research) in GeePs (Group of electrical engineering of Paris). His research interests include computational electromagnetics, electromagnetic compatibility and wireless power transfer.

He received the Dip. Eng. from ESIEE-Paris (Ecole Supérieure d'Ingénieurs en Electronique et Electrotechnique) in 1984. In 1985 he joined the Laboratoire de Génie Electrique de Paris (LGEP, now GeePs) where he earned a PhD in electrical engineering in 1989. He got a position at the CNRS in 1989.

He is the author or coauthor of more than 130 journal articles in peer-reviewed journals. He is an Associate Editor of the European Physical Journal Applied Physics (EPJ AP) and of the Journal of Electromagnetic Waves and Applications.

<figure><img src="image_19.png" /></figure>

**Mohamed Bensetti** received his Master degree (DEA) in 2001 and his Ph.D. in Electrical Engineering in 2004 from the University of Paris-Sud, France.

From 2005 to 2007, he worked as researcher at Ecole Supérieure d'Electricité (SUPELEC) – Gif sur Yvette, France. In 2007, he joined Ecole Supérieure d'Ingénieur en Génie Electrique (ES-IGELEC), Rouen, where he was lecturer and researcher in the Research Institute for Electronic Embedded Systems (IRSEEM). In January 2013, he joined the energy department of SUPELEC. He received the "Habilitation à Diriger des Recherches" degree from the University of Paris-Sud in March 2014. He is now a Professor at CentraleSupelec. He is currently with GeePs (Group of electrical engineering, Paris).

His current research interests are Electromagnetic modeling, Electromagnetic Compatibility (EMC) and Power Electronics, Wireless Power Transmission including modeling, simulation and instrumentation.

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**Fabrizio Dughiero** is a Full Professor of electromagnetic processing of materials (EPM) with Università degli Studi di Padova, Padua, Italy, and the Chief Scientist with the Laboratory of Electroheat, Università degli Studi di Padova.

He has been the Founder of a spin-off company (InovaLab) and Unismart Padua Enterprise Ltd., Padua, a new in-house company aimed to develop Technology and Knowledge Transfer of Università degli Studi di Padova. He is currently the Director of the Department of Industrial Engineering, Uni-
versità degli Studi di Padova.

His current research activity, documented by more than 180 scientific publications, mainly deals with theoretical aspects and applications of EPM and electroheat technologies.