# Impact of Shift-Angle Topologies on the Performance of PCB-Embedded Solenoid Inductors

Mohamed Alkurdi, Zeeshan Umar, Christian Martin, Mounira Bouarroudj

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Mohamed Alkurdi, Zeeshan Umar, Christian Martin, Mounira Bouarroudj. Impact of Shift-Angle Topologies on the Performance of PCB-Embedded Solenoid Inductors. IEEE Transactions on Components, Packaging and Manufacturing Technology, 2025, 10.1109/TCPMT.2025.3635410. hal-05406975

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<figure><img src="image_1.png" /></figure>
# Impact of Shift-Angle Topologies on the Performance of PCB-Embedded Solenoid Inductors

Mohamed Alkurdi, Zeeshan Umar, Christian Martin, Mounira Bouarroudj

**Abstract**— The integration of magnetic components within printed circuit boards (PCBs) offers an innovative pathway to enhance electromagnetic performance while reducing size and improving manufacturability in power electronics. This study investigates the impact of shift-angle topologies in PCB-embedded solenoid inductors, focusing on minimizing electromagnetic losses to enhance high-frequency performance for applications such as compact DC/DC converters and power modules.

Four angular configurations were analyzed, assessing how different trace alignments influence flux distribution, eddy current generation, and inductance behavior. To isolate the effects of angle shift, the study employed Finite Element Method (FEM) simulations under direct current (DC) excitation, eliminating the influence of skin effect, proximity effect, core loss, and parasitic capacitance.

Results showed that modifying the shift angle significantly altered flux density—especially the perpendicular (Z) component responsible for eddy current formation. Among the DC-tested topologies, Configurations with two counter wise angles in the coils top-layer and bottom-layer traces demonstrated the lowest flux loss and minimal eddy current dissipation.

Experimental validation confirmed the simulation findings, with electromagnetic losses within a 6% margin.

Tests under alternating current (AC) conditions using an impedance analyzer verified the benefits of reverse angle configurations but revealed that the angle position affects the resonance frequency.

Future researches will explore advanced winding techniques, including litz wire implementations, alternative geometries like toroidal inductors, and integrating active components to enhance high-frequency performance and reliability.

**Index Terms**—PCB inductor, embedded magnetics, solenoid, finite-element modeling, shift angle, eddy currents.

## I. INTRODUCTION

RAPID advancement of embedded magnetic components has led to the development of smaller and more efficient electronic devices.

In printed circuit boards (PCBs), traditional magnetic components such as inductors and transformers often occupy significant large space compared to other electronic elements, making miniaturization the boards a key challenge in modern circuit design [1].

To address this, embedded PCB inductors have emerged as a promising alternative, where inductive components are

integrated directly into the PCB layers rather than being mounted as discrete elements [2].

Embedded PCB inductors offer several advantages, including reduced parasitic effects, improved electromagnetic interference (EMI) shielding, and enhanced mechanical stability [3].

Among the various core geometries used in embedded inductors, the solenoid core has demonstrated significant performance benefits due to its ability to generate a strong and uniform magnetic field along its axis. This structure enhances inductance values and efficiency, making solenoid inductors well-suited for applications requiring high magnetic field containment, such as testing and measuring magnetic field [4].

Moreover, the solenoid configuration is the simplest geometry to study and simulate, making it ideal for focusing on a specific physical phenomenon affecting inductor performance [5].

Its straightforward structure allows for more precise modeling of effects such as eddy current formation, the skin effect, the proximity effect, and Joule heating. This simplicity facilitates in-depth investigations into the thermal and electrical behavior of inductors, providing valuable insights for optimizing designs in high-frequency applications [6].

The challenges persist in the integration of inductors within PCBs are the increase in temperature and the reduction of inductance at higher operating frequencies. These issues arise due to the combined effects of electromagnetic losses and heat dissipation. Additionally, the winding orientation (angle alignment) and spacing play a critical role in determining the distribution of eddy currents within the coil, impacting both thermal performance and electrical efficiency. Optimizing the coil geometry and material selection is essential to mitigate these effects and ensure stable operation in high-frequency applications [7].

This study aims to isolate and examine the specific impact of coil alignment angle on eddy current formation in embedded PCB inductors [8]. By analyzing how different coil angle alignment configurations affect current flow and magnetic field
interactions, this research seeks to provide insights that could
contribute to the optimization of embedded inductor designs.
The findings of this work can help improve the performance of
compact electronic devices, particularly in applications
requiring high-efficiency power conversion and minimal
electromagnetic interference [9].

# I. METHODOLOGY

## A. Alignment Angle definition

The alignment angle shown in **Fig. 1** is defined as the angle
between the PCB trace and an imaginary line perpendicular to
the magnetic core [10]. In this study, it will be referred to as
Angle (A) for the traces on the bottom PCB layer and Angle (Q)
for the traces on the top PCB layer.

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Illustration for the top PCB layer angle (Q) and the bottom PCB layer angle (A)</figcaption></figure>

Since this study focuses exclusively on the effect of
alignment angle on the flux density and induced eddy currents,
several secondary factors will be neglected to isolate the
primary variable of interest. Specifically, the analysis will not
consider the following:

- Via Effects: Any additional flux disturbances or
resistive losses caused by vias will be excluded.

- Core Losses: Hysteresis and eddy current losses within
the magnetic core will not be factored into the
simulations.

- Permeability Variations: Changes in core permeability
due to frequency-dependent behavior will be ignored.

- Temperature Rise: Thermal effects, including resistive
heating and material property changes due to
temperature fluctuations, will not be accounted for.

By neglecting these influences, the study ensures that any
observed variations in flux density (Bz) and eddy currents are
solely attributed to the alignment angle [11].

Previous studies have demonstrated that the ideal alignment
angle is 0°, as this configuration minimizes losses and
optimizes performance. However, achieving a perfect zero-
degree alignment is practically impossible due to the
construction constraints of embedded inductors. In practice,
vias need to electrically connect the top and bottom layer traces;
however, these vias introduce a shift in trace alignment,
inevitably causing deviations from the ideal configuration.

This study investigating the impact of alignment angle on the
magnetic field responsible for eddy current formation in copper
coils concluded that an increase in the misalignment angle leads
to a rise in the harmful magnetic field. However, this increase
does not exceed 16.8% for an alignment angle of 20° as in **Fig.
2**.

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Comparison of the average maximum flux density
perpendicular to copper traces (top PCB/bottom PCB) based
on laboratory measurements and simulation results.</figcaption></figure>

## B. Investigated approaches

This study compares four different design approaches (Fig. 3) concerning the alignment angle, with the increase of perpendicular flux inside the coil. The goal is to of identifying the most practical configuration that minimizes the flux perpendicular to the copper coil. The objective is to determine the design that achieves those results. In other words, the least induced eddy currents, thereby enhancing overall efficiency and performance [12].

- First Approach (Config1 A = 0): The alignment angle is introduced only at the via location (trace end), where only the top layer traces (Q) have a nonzero value, while the bottom layer traces remain at 0°.

- Second Approach (Config2 A = -Q): The alignment
angle is also introduced at the via location, but both the
top (Q) and the bottom (A) layer traces have nonzero
values, ensuring that A = -Q to maintain symmetry.

- Third Approach (Config3 A = 0): The alignment angle is shifted to the middle of the trace instead of the via location, but follows the same rule as the first approach, where only Q has a nonzero value while A remains 0°.

- Fourth Approach (Config4 A = -Q): Similar to the
second approach, the alignment angle is applied at the
middle of the trace, with both Q and A having nonzero
values, ensuring that A = -Q.
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Illustration of the four different design approaches for alignment angle configurations in embedded PCB inductors.</figcaption></figure>

## II. FEM SIMULATION

A Finite Element Method (FEM) simulation tool, such as ANSYS Maxwell [13][14], will be used to analyze the four proposed design approaches under both DC and AC sources. To ensure a fair comparison, all models will share identical boundary conditions and simulation parameters as stated in Table I.

### A. DC Simulation

The primary objective is to evaluate the relationship between the alignment angle and the flux density (Bz) perpendicular to the copper traces. Since some configurations have A = 0°, while others have A = -Q, the comparison will be conducted using the total alignment angle defined as:

$$ \text{Total Angle} = Q + |A| \qquad (1) $$

By using this metric, the study maintains a consistent turn-to-turn clearance across all configurations, ensuring that differences in flux behavior are attributed solely to the alignment angle variations rather than trace spacing inconsistencies [15].

<table><caption>**TABLE I**
SIMULATION BOUNDARIES AND PARAMETERS</caption><thead><tr><th>Parameter</th><th>Value</th><th>unit</th></tr></thead><tbody><tr><td>Current</td><td>1.4</td><td>A</td></tr><tr><td>Trace width</td><td>2</td><td>mm</td></tr><tr><td>Core width</td><td>28</td><td>mm</td></tr><tr><td>Copper thickness</td><td>0.035</td><td>mm</td></tr><tr><td>Core thickness</td><td>2</td><td>mm</td></tr><tr><td>Core length</td><td>54</td><td>mm</td></tr><tr><td>Core permeability</td><td>26</td><td></td></tr><tr><td>Core PN#</td><td>00K9541B026</td><td></td></tr><tr><td>Clearance</td><td>8</td><td>mm</td></tr></tbody></table>

<table><tr><td>Number of turns</td><td>5</td></tr></table>

In order to reduce the simulation time, the mesh is fined only in the middle trace for both layers and the measurements are done in it, so that the maximum element length of the mesh is 60µm, while the current path and magnetic field directions are shown in **Fig. 4**.

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** (a) The model current path, (b) Front view of the magnetic field direction.</figcaption></figure>

To eliminate via effects and maintain the traces as a single continuous coil, the traces are represented as multiple windings connected in series through a matrix, as shown in **Fig. 5**. This series configuration ensures that the electrical behavior remains consistent with a single uninterrupted coil, while the segmented representation allows for precise modeling in the simulation environment.

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Simulation model of a continuous coil with series-connected windings to eliminate via effects.</figcaption></figure>

### B. DC Simulation Results

In the simulation software it is possible to observe the Flux (Ø) in the z direction inside the copper [16], [17]. This can show the impact on eddy current formulation in AC application according to the following relationships [18]:

$$ \varepsilon = \frac{d\Phi}{dt} \qquad (2) $$

$$ \Phi = \oint \beta \, ds \qquad (3) $$

$$ P_e = K_e B^2 f^2 d^2 \qquad (4) $$

$\varepsilon$ = Induced electromotive force (V), $Ø$ = Magnetic flux (Wb), $B$ = Magnetic flux density (T), $ds$ = Differential surface element, $P_e$ = Eddy current power loss (W/m³), $K_e$ = Eddy current coefficient (material & geometry dependent), $f$ = Frequency (Hz), $d$ = Conductor thickness (m).

The results in Fig. 6 are flux in the middle traces versus the angle increase in each configuration. They indicate that Config.2 A = -Q and Config.4 A = -Q achieve similar levels of flux reduction, both outperforming Config.1 A = 0 and Config3
A = 0. Among all configurations, Config.4 A = -Q proves to be the most effective in minimizing perpendicular flux density (Øz).

Specifically, for Config.4 A = -Q, the increase in (Øz) from 0° to 30° is only 3.7%, whereas for Config.1 A = 0, the increase reaches 14.8%, highlighting the significant advantage of the A = -Q configuration in reducing flux-induced losses.

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Ansys Maxwell simulation results showing Config.4 A = -Q with the lowest perpendicular flux density</figcaption></figure>

### C. AC Simulation

For the AC simulation, the same parameters from Table I are maintained, except for the current source, which is now an AC sinusoidal source with a sweep frequency ranging from 5 kHz to 10 MHz and an RMS current of 1.4 A.

Core loss and eddy currents in the magnetic material are neglected to focus the analysis on the effect of trace angles, while the core relative permeability is fixed at 26 to ensure that core properties remain unchanged across the frequency range. This simulation enables the study of resistance, inductance, and eddy current formation in the traces over a broad frequency spectrum.

### D. AC Simulation Results

First observation is the solid loss in the middle traces at 500kHz. In ANSYS Maxwell, when analyzing eddy currents, solid loss refers to the power dissipation due to eddy currents in a solid conductive material. This occurs when an AC magnetic field induces circulating currents (eddy currents) within the material, causing Joule heating (I²R losses).

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Solid loss variation with total angle for different configurations.</figcaption></figure>

Fig. 7 illustrates the impact of shift angle configurations on solid loss in a PCB-embedded solenoid inductor model. The Config4 A = -Q design proves to be the most effective in minimizing energy dissipation, aligning with DC simulation results that indicate counter-aligned angles (A = -Q) optimize performance.

At a shift angle of 30°, the loss in Config.2,4 A = -Q is 3.3%, whereas the loss in Config.1,3 A = 0 is notably higher at 7.3%.

The inductance value shown in Fig. 8 (a) demonstrates that Config1,2 A = 0 starts at approximately 342 nH at 0° and increases to around 370 nH at 30°, reflecting an inductance rise of 8.2%. In contrast, Config1,2 A = -Q exhibits a slight decrease of 0.6%, maintaining a more stable inductance value across the shift angle range.

<figure><img src="image_9.png" /></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 8. (a) Inductance variation with shift angle at 500 kHz, (b) Resistance variation with shift angle at 500 kHz</figcaption></figure>

Since Maxwell does not support resistance and inductance vs. frequency analysis, an HFSS model was created including the via connections between top and bottom layers, which contributed to a slightly higher inductance, as illustrated in Fig. 9.

<figure><img src="image_11.png" /><figcaption>Fig. 9. Ansys HFSS model for AC electrical characterization.</figcaption></figure>

All the boundaries are the same as in Maxwell simulation except for the mentioned in Table II:

<table><caption>TABLE II
HFSS BOUNDARIES</caption><thead><tr><td>PARAMETER</td><td>VALUE</td></tr></thead><tbody><tr><td>PORT TYPE</td><td>TERMINAL LUMPED PORT</td></tr><tr><td>FREQUENCY SWEEP</td><td>5kHz - 120MHz</td></tr><tr><td>SURROUNDING</td><td>VACUUM</td></tr><tr><td>VOLTAGE</td><td>1V</td></tr></tbody></table>

The Equations used to calculate Inductance and Resistance from the Y parameters are:

$$R(f) = \Re \left( \frac{1}{Y(f)} \right) \qquad (5)$$

$$L(f) = \frac{\Im\left(\frac{1}{Y(f)}\right)}{2\pi f} = \frac{\Im(Z(f))}{2\pi f} \qquad (6)$$

Where $\Im$ is the imaginary part, $\Re$ is the real part [1].

The simulation results of the four tested configurations (Config.1 A = 0, Config.2 A = -Q, Config.3 A = 0 and Config.4 A = -Q) shown in Fig. 10 (a) indicate that the inductance values are nearly identical across the entire frequency range (5kHz-120 MHz), following the same trend with a difference margin not exceeding 3-4%. This demonstrates that the shift-angle topologies have minimal impact on inductance within the studied frequency band.

The inductance increased to range from 410 nH to 460 nH due to the added segment of via, that was not included in Maxwell simulations.

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. (a) Inductance vs. Frequency at a total angle of 20°, (b) Resistance vs. Frequency at a total angle of 20°</figcaption></figure>

In contrast, the resistance increases consistently with frequency for all configurations as in Fig. 10 (b), mainly due to skin and proximity effects. While the resistance curves remain close, the maximum difference between configurations reaches 5-8% at higher frequencies. Configurations with counter angles (A = -Q) generally exhibit slightly lower resistance, suggesting improved current distribution and reduced eddy current effects.

Overall, the shift-angle configurations do not significantly affect the inductance but can slightly influence the AC resistance, particularly at higher frequencies where current redistribution effects are more pronounced.
### III. EXPERIMENTAL VALIDATION

#### A. PCB Design

The PCBs are designed to implement the proposed approaches similar to **Fig. 11**, each incorporating a 5-turn embedded solenoid inductor. The design consists of:

- 5 traces on the top PCB layer

- 5 traces on the bottom PCB layer, both connected in series to ensure electrical continuity.

The middle section serves as the test area, and the connector is placed 47mm away to minimize any potential interference from radiated emissions. The PCBs were designed using Altium Designer, with a focus on Design for Testing (DFT), and they include designated positions for future BNC connectors [19].

In total, 20 PCB designs were developed, corresponding to 4 configurations and 5 angles values in each.

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Examples of PCB Designs done in Altium SW</figcaption></figure>

#### B. Experiment setup for DC test

A customized test setup in **Fig. 12** has been developed to investigate the effect of angle shift on magnetic flux density. The setup includes a plastic support structure designed to hold the printed circuit boards (PCBs) in place by aligning them with a magnetic core [20].

The system consists of three key components positioned within the holder:

1. Top Layer PCB – Serves as the upper conductive layer.

2. Magnetic Core – Centrally positioned to influence the magnetic flux.

3. Bottom Layer PCB – Completes the structure, ensuring a stable configuration.

<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Experimental setup to validate the approaches measures</figcaption></figure>

A PCE-MFM 4000-ICA magnetometer with a resolution of 0.01 µT is employed for high-precision measurements. Due to the small variations in flux density caused by angle shifts, a high-precision probe is utilized instead of the general probe to enhance sensitivity and detect subtle changes [21].

The high-precision probe is positioned vertically along the trace edge to measure the maximum perpendicular flux density (Bzmax). This ensures that the measurement captures the peak flux component in the perpendicular direction to the copper traces [22].

To systematically evaluate different angle configurations, each PCB is designed with a specific angular arrangement. When a particular angle configuration is tested, the corresponding PCBs are placed in the holder, and measurements are recorded accordingly. This method illustrated in **Fig. 13** allows for a controlled investigation of the influence of PCB angle shifts on the observed magnetic flux density.

<figure><img src="image_16.png" /><figcaption>**Fig. 13.** (a) illustration of the experimental set structure, (b) The configuration changing mechanism</figcaption></figure>

Measuring the total flux (Øz) is highly complex, as it
requires sweeping the flux density (Bz) along the trace and performing an integral calculation over the trace surface. To simplify the process and obtain a more practical and reliable measurement, the approach focuses on measuring the maximum flux density (Bz max) instead.

For accuracy and consistency, flux density measurements are taken at four key points—positioned at the edges of the trace as shown in **Fig. 14**, where the magnetometer detects the highest Bz values. The average of these four measurements is then calculated to ensure a precise and representative result. The probe measures the perpendicular flux density (Bz) on the copper traces.

Due to the plastic casing of the probe, the internal sensor is positioned 4 mm above the edge of the copper trace. Therefore, for an accurate comparison with simulation readings, measurements should be taken at the same 4 mm height above the copper trace in the simulation, as illustrated in **Fig. 14**.

<figure><img src="image_17.png" /><figcaption>**Fig. 14.** Simulation model illustrating the measurement of perpendicular flux density (Bz) on the copper traces.</figcaption></figure>

Adding a 1-ohm series resistor in the circuit as in **Fig. 15** helps stabilize the current during the DC testing of the inductor. Since the inductor has a very low DC resistance, it can cause high current draw, potentially leading to power supply instability or current surges. The series resistor mitigates this issue by:

- Limiting Inrush Current – Prevents excessive current flow when the circuit is powered on.

- Improving Stability – Reduces fluctuations in the power supply, ensuring more consistent current.

- Enhancing Measurement Accuracy – Helps maintain controlled test conditions by preventing rapid variations in current.

<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Experimental setup circuit schematic</figcaption></figure>

### C. DC Experimental results

**Fig. 16** presents a comparison between experimental and simulated results, analyzing the impact of different angle configurations on the maximum flux density.

The results validate the simulation model, demonstrating that Config. 4 (A = -Q) achieves the best performance [23][24].

<figure><img src="image_19.png" /><figcaption>**Fig. 16.** Comparison of experimental and simulation results for different angle configurations</figcaption></figure>

Configurations with (A = -Q) show the lowest flux density values, confirming their effectiveness. The experimental data (solid lines) exhibit more variability than the simulation data (dashed lines). The average difference between experimental and simulated results is 2 – 4 µT, within the expected measurement error range of ±2 µT.

These results confirm that the simulation effectively represents real-world behavior, with Configurations of (A = -Q) emerging as the optimal approach for enhancing the uniformity and intensity of the magnetic flux.

### D. Experiment setup for AC test

AC testing introduces a complex interplay of physical phenomena, including skin effect, eddy current formation, parasitic capacitance and resistance, as well as core losses. While electromagnetic simulation tools allow for these effects to be selectively disabled—such as by omitting core loss models or disabling eddy current calculations—such isolation is not feasible in practical experiments. In physical measurements, these phenomena inherently interact and often dominate the observed behavior, making it challenging to isolate the contribution of individual parameters without computational support.

A potential measurement approach involves using an impedance analyzer to extract the frequency-dependent R and L parameters for different angular configurations. To ensure accurate characterization, the test model must be modified into a smaller section that isolates only the inductor segment.

After preparing the model, it could be placed in the impedance analyzer for measurement as shown in Fig. 17.
<figure><img src="image_20.png" /><figcaption>**Fig. 17.** Impedance characterization of a solenoid PCB inductor using the Keysight E4990A (20 Hz–120 MHz) Impedance Analyzer.</figcaption></figure>

The graph in Fig. 18 presents the frequency-dependent behavior of the inductance (L) in simulation and experiment for four configurations: Config.1 A = 0, Config.2 A = -Q, Config.3 A = 0 and Config.4 A = -Q. The x-axis represents the frequency in Hz on a logarithmic scale, ranging from 5kHz to 120MHz, while the y-axis represents inductance value in Henries (H).

Both the simulation and experiments showed a steady inductance between 400 - 500nH with a small change ranging between 1.6-4%.

<figure><img src="image_21.png" /><figcaption>**Fig. 18.** Inductance vs. Frequency for Four Configurations of the impedance analyzer results, sweeping from 5kHz to 120Mhz.</figcaption></figure>

The graph in Fig.19 illustrates the variation of normalized resistance (Rs) with respect to frequency for four different configurations.

<figure><img src="image_22.png" /><figcaption>**Fig. 19.** Normalized Resistance vs. Frequency for Four Different configurations of the impedance analyzer results, sweeping from 5kHz to 120Mhz.</figcaption></figure>

It's noticed that Config. 3 and 4 have earlier resonance point than Config. 1 and 2 causing an early rise of the resistance curve as in Fig.20.

<figure><img src="image_23.png" /><figcaption>**Fig. 20.** Resistance simulation vs. Frequency simulation for four different configurations, sweeping from 5 MHz to 1 GHz.</figcaption></figure>

The early resonance behavior in config.3 and config.4 is due to the capacitance change, as the copper traces overlapping area in these cases are bigger as shown in Fig. 21, causing an increase in the capacitance value of the inductor as in equation (7) and hence an early resonance point as shown from equation (8).

$$C = \varepsilon \frac{A}{d} = \frac{Q}{V} \qquad (7)$$

$$f_{res} = \frac{1}{2\pi\sqrt{LC}} \qquad (8)$$

Where C: Capacitance (in farads, F) — ability of a structure to store electric charge. Q: total charge on the conductor, V: the applied voltage. ε: Permittivity of the dielectric material between plates (in F/m). A: Area of one of the plates (in m²). d: Distance between the plates (in meters). fres: Resonance frequency (in Hz). L: Inductance (in Henries, H).

The capacitance difference is increased with frequency as this difference value could be noticed from Fig. 22.
<figure><img src="image_24.png" /><figcaption>Fig. 21. Traces overlapping area for all configurations.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Fig. 22. Capacitance vs. Frequency above 300 MHz to 700MHz.</figcaption></figure>

V. CONCLUSION

This study systematically investigated the influence of shift-
angle topologies on perpendicular magnetic flux distribution in
the coils of PCB-embedded solenoid inductors. Through both
DC and AC analyses—including FEM simulations and
experimental testing—it was demonstrated that alignment
configurations critically impact perpendicular flux behavior
and, consequently, electromagnetic losses.

Under DC excitation, Config.4 (A = -Q) proved most
effective in minimizing perpendicular flux density and eddy
current risks. However, AC testing revealed that, Config. 3 and
4 designs induce early resonance, limiting their high-frequency
performance. As a result, Config. 2 (A = -Q) was identified as
the optimal compromise for AC applications, offering both low
perpendicular flux and higher resonance frequency.

These findings confirm that both shift-angle value and
topological configuration are essential considerations in the
design of efficient PCB-integrated inductors, particularly for
applications where both DC and AC behaviors must be

balanced. Future research will focus on enhancing inductor
design efficiency through multiple approaches:

* Exploring Alternative Designs: Investigating different design methodologies, including the use of litz wire, to reduce eddy current losses and improve high-frequency performance.

* Expanding to Other Inductor Shapes: Evaluating the implementation of toroidal inductors, assessing their feasibility and performance under various design constraints.

* Integrating Active Components in Toroidal Inductors: Investigating the potential co-integration of active components to enhance functionality and optimize power efficiency.

* Assessing Thermal Behavior: Examining the heat dissipation characteristics of different configurations to ensure thermal stability and long-term reliability.

By addressing these aspects, the study aims to develop more
efficient, scalable, and high-performance embedded inductor
solutions for advanced static converters applications.

IV. ACKNOWLEDGMENT

This study is part of PhD program funded by ANR in the
frame of TECoCIP Project in coordination with CNRS,
SATIE lab in ENS Paris-Saclay and Ampere lab in Lyon.

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<figure><img src="image_26.png" /></figure>

**Zeeshan Umar**, is an Electrical Simulation Engineer at Siemens AG in Erlangen, Germany. He earned his M.Sc. in Computational Engineering from the University of Rostock in 2013 and completed his Ph.D. in Electrical Engineering at the University of Erlangen in 2024. His research focuses on the characterization of semiconductor devices and the 3D electromagnetic modeling and simulation of power electronic devices.

<figure><img src="image_27.png" /></figure>

**Christian Martin**, is a full Professor at Claude Bernard Lyon 1 University and leads his research with the AMPERE Laboratory. He received the Engineer degree from the Institut National Polytechnique de Grenoble in 2002. Then, he received the M.Sc and Ph.D degrees from the Joseph Fourier University, in Grenoble, France in 2002 and 2005 respectively. His research interests include magnetic material characterization and modelling, design and integration of magnetic component for power electronics converters.

<figure><img src="image_28.png" /></figure>

**Mounira Bouarroudj**, is a Director of Research at Université Gustave Eiffel, affiliated with the SATIE Laboratory. She received the M.Sc. degree from Université Paul Sabatier, Toulouse, in 2005, and the Ph.D. degree in electrical engineering from ENS Cachan, in 2008. Her research interests include semiconductor device characterization, power electronics reliability, and Multiphysics modeling.

<figure><img src="image_29.png" /></figure>

**Mohamed Alkurdi** (CNRS Researcher, SATIE) received the B.Sc. degree in electrical engineering in 2017 and the M.Sc. degree in electrical control engineering in 2023 from Aswan University, Egypt. He is currently pursuing the Ph.D. degree in power electronics with Université Paris-Saclay, Paris, France, in collaboration with the French National Centre for Scientific Research (CNRS). His research focuses on embedded magnetics in printed circuit boards for improved performance and reliability in power electronics. He previously worked with Excel Codes, Cerebrus Co., and Siemens EDA in hardware and PCB design related roles.