# Investigating the Impact of Alignment Angle Between Copper Turns and Magnetic Core on Magnetic Field Coupling in PCB Integrated Inductors

Mohamed Alkurdi, Christian Martin, Mounira Bouarroudj, Mickael Petit

► To cite this version:

Mohamed Alkurdi, Christian Martin, Mounira Bouarroudj, Mickael Petit. Investigating the Impact of Alignment Angle Between Copper Turns and Magnetic Core on Magnetic Field Coupling in PCB Integrated Inductors. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0182 . hal-05106019

HAL Id: hal-05106019

https://utc.hal.science/hal-05106019v1

Submitted on 10 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Investigating the Impact of Alignment Angle Between Copper Turns and Magnetic Core on Magnetic Field Coupling in PCB Integrated Inductors

Mohamed ALKURDI¹,², Christian MARTIN³, Mounira BOUARROUDJ¹,⁴ and Mickael PETIT¹,⁵

¹SATIE, ²CNRS, Université Paris-Saclay, ³Ampère, Université Claude Bernard Lyon 1,

⁴Universite Paris Est Créteil, ⁵CNAM

4 Av. des Sciences, 91190 Gif-sur-Yvette, 1 Av. de la Terrasse, 91190 Gif-sur-Yvette, 3 rue Joliot Curie, Breguet Building, 91190 Gif-sur-Yvette, 36 Avenue Guy de Collongue 69134

Ecully cedex, 61 Av. du Général de Gaulle, 94000 Créteil, 2 Rue Conté, 75003 Paris, France

Tel.: +33 (0)1 81 87 55 01, +33 1 44 96 46 06, +33 04 72 43 16 10, +33 01 45 17 10 00

E-Mail: Mohamed.Alkurdi@ens-paris-saclay.fr, christian.martin@univ-lyon1.fr, mounira.bouarroudj@ens-paris-saclay.fr, mickael.petit@ens-paris-saclay.fr

URL: https://satie.ens-paris-saclay.fr/fr, https://www.iledefrance-gif.cnrs.fr/fr, http://www.ampere-lab.fr, https://www.u-pec.fr/, https://iim.cnam.fr/

## Acknowledgements

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

## Keywords

Embedded inductor, PCB magnetics, Leakage flux measurement, Angular-shift effect, PCB eddy currents.

## Abstract

This study investigates the influence of copper trace alignment relative to the magnetic core in PCB-embedded inductors, focusing on the effects of the alignment angle on perpendicular magnetic flux density that shall cause eddy currents in AC applications. Although aligning traces at right angles is common in traditional inductor winding, it may not be feasible for PCB designs due to manufacturing constraints. The study proposes an investigation of the angular-shift effects on magnetic flux while maintaining practical PCB manufacturability in potential applications like static power converters. By evaluating how the angular shift impacts magnetic flux behavior in copper traces, this research is part of a study that aims to address efficiency losses and heating caused by eddy currents. Through Finite Element Method (FEM) simulations using Ansys Maxwell software and experimental validation, the results demonstrate that smaller alignment angles, particularly at 0°, reduce perpendicular flux what shall mitigate eddy current formation in AC applications, improving inductor performance.

## Introduction

Embedding active and passive components as shown in Fig. 1 participates in reducing PCBs size, power losses and increasing signal integrity [1].

<figure><img src="image_2.png" /><figcaption>Fig. 1 Integration of power dies and integrated circuits together in a common "pre-package", which is then embedded, together with large magnetic components, in a PCB to form a complete converter.</figcaption></figure>

While embedding passive components such as resistors and capacitors is well-established and offers the advantage of reducing PCB dimensions, this study focuses on the challenges associated with embedding magnetic components.

Recent researches discussed the potential of embedding toroidal magnetic inductors inside PCB layers [2],[3], though the study highlighted problems regarding the temperature rise of the inductor, the drop of inductance and the fringing effect in the magnetic core gap[4].

This study starts to investigate one of the potential causes for the embedded toroidal inductor drop of inductance and heating, that is the effect of the alignment angle of the coil turns with the core centerline.
## Problem formulation

A key issue in this technology is inductor heating, which can arise from various factors, including eddy currents within the inductor turns. These eddy currents are caused by time-varying magnetic flux through the copper traces as in Faraday's law [5].

$$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \quad \text{Eq. 1} $$

In its integral form:

$$ \oint_C \mathbf{E} \cdot dl = - \frac{d}{dt} \int_S \mathbf{B} \cdot \mathbf{n} da \quad \text{Eq. 2} $$

Where E is the electromotive force in Volt, B is the flux density in Tesla, n is number of turns and a is cross section area.

From Eq.2, not only the flux density (B) value is affecting but also the change rate (frequency) and the area of copper exposed to the flux, hence, one critical factor influencing this flux is the alignment angle of the copper traces relative to the core centerline, as illustrated in Fig. 2. This angle is necessary due to the PCB manufacturing process for creating the inductor coil [6]. Therefore, this study investigates the potential effect of this alignment angle on flux behavior and aims to characterize how flux changes as the angle varies.

<figure><img src="image_3.png" /><figcaption>Fig. 2 Illustrative image of Shift angle (Θ) between core center line and coil traces created in Ansys Maxwell 3D software</figcaption></figure>

The magnetic field is a function of the current and the magnetic path length [7].

$$ H = \frac{l}{l} \quad \text{Eq. 3} $$

$$ B = \mu \cdot H \quad \text{Eq. 4} $$

$$ \mu = \mu_r \mu^\circ \quad \text{Eq. 5} $$

$$ \emptyset = B \cdot a \quad \text{Eq. 6} $$

$$ E = - \frac{d\emptyset}{dt} \quad \text{Eq. 7} $$

Where H is the magnetic field is A/m, μ is magnetic permeability and Φ is the magnetic flux in Weber.

Because the behavior of the magnetic field in the PCB trace in not homogeneous as in cylindrical conductor the integral form of magnetic field has been considered. As illustration, a comparison between the flux in toroidal and cylindrical copper conductors have been performed and obtained results are depicted in Fig. 3.

$$ \oint_C H \cdot dl = \int_S J_f \cdot da \quad \text{Eq. 8} $$

$$ \emptyset = \int_S B \cdot da \quad \text{Eq. 9} $$

$$ \oint_C \mathbf{E} \cdot dl = - \frac{d}{dt} \int_S \mathbf{B}_z \cdot da \quad \text{Eq. 10} $$

Bz is the flux density in Z dimension, the perpendicular direction to the copper trace most contributing the eddy current formation in AC.

<figure><img src="image_4.png" /><figcaption>Fig. 3 Magnetic field distribution in the same volume of copper for PCB trace of 0.7mm thickness in (a) and cylindrical conductor of 0.3mm diameter in (b) supplied with 0.7A DC current</figcaption></figure>

Angle (Θ) is the variable affecting the area what reflects to (Ø) value, then Eq.10 can be modified as:

$$ \oint_C \mathbf{E} \cdot dl = - \frac{d}{dt} \int_S \mathbf{B}_z \cdot d(\mathbf{W} \cdot \mathbf{L} \cdot f(\Theta)) \quad \text{Eq. 11} $$

W is the trace width and L is the trace length at Θ = 0°.

The term f(Θ) can vary based on the inductor core shape and coil trace shape, for example in the toroidal shaped core the inner and outer radius are factors in f(Θ), and the behavior of conductor surface area would remain incremental.

In the presence of an angle (Θ), the field exhibits an asymmetric distribution, with a greater concentration toward the North on one side and toward the South on the opposite side. This result is depicted in Fig. 4b and is in contrast to the uniform distribution observed along the sides when no angle is applied as illustrated in Fig. 4.
<figure><img src="image_5.png" /><figcaption>Fig. 4 Example of Flux density in Z direction inside PCB inductor at (Θ = 0°) in (a) and (Θ > 0°) in (b)</figcaption></figure>

In this case, there is a point along the trace edge where Bz = Bzmax (North) and another point on the opposite side where Bz = Bzmax (South) in negative value, to get the average value for both top layer and bottom layer traces, we shall have four points of measurement.

$$B_{zmaxavg} = \frac{B_{zmaxNt} + B_{zmaxNb} + B_{zmaxSt} + B_{zmaxSb}}{\text{Measurement points number}} \quad \text{Eq. 12}$$

## Simulation analysis

Though toroidal shapes PCB inductors have proven to be the least radiative for magnetic field [8], only a part of the inductor represented in Fig. 2 has been modeled by using a I/solenoid shape inductor. This approach allows computing flux density in copper layer and simplifying the design and simulation model with the same concept as illustrated in **Error! Reference source not found.**

<figure><img src="image_6.png" /><figcaption>Fig. 5 Illustrative image of a solenoid shaped model created in Ansys Maxwell 3D software</figcaption></figure>

The study is neglecting the effect of vias and core losses, investigating only the effect of the angle shift (Θ) [9].

The simulation employs Finite Element Method (FEM) through the ANSYS Maxwell eddycurrents solver. Due to the angular shift, which causes displacement of the magnetic field in three dimensions (X, Y, and Z), the magnetostatic solver was deemed unsuitable, as it

treated the barriers (Simulation area limits) as magnetic resistance as shown in Fig. 6. In real application the magnetic field is distributing in infinity space. By using the eddycurrents solver and adjusting the frequency to its lowest possible value, the results closely approximated those of a DC simulation, with an error margin of less than 2.5%, this allows using the radiation boundaries which represent infinity space.

<figure><img src="image_7.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(b)</figcaption></figure>

Fig. 6 (a) Model flux path, (b) the equivalent magnetic circuit in Magnetostatic simulation. The simulation model consists of an I shaped magnetic core inside the PCB that has 5 turns of inductor traces as illustrated in Fig. 7 and the dimensions as in Table I.

<figure><img src="image_9.png" /><figcaption>Fig. 7 Simulation model created in Ansys Maxwell 3D software</figcaption></figure>

<table><caption>**Table I: Simulation model dimensions**</caption><thead><tr><td>Parameter</td><td>Dimension</td></tr></thead><tbody><tr><td>Model length (Y direction)</td><td>54mm</td></tr><tr><td>Trace length (X direction)</td><td>14mm</td></tr><tr><td>Trace width</td><td>2mm</td></tr><tr><td>Trace thickness</td><td>0.07mm</td></tr><tr><td>Trace clearance</td><td>8mm</td></tr><tr><td>Core thickness</td><td>2mm</td></tr><tr><td>Prepreg layer thickness</td><td>1.5mm</td></tr></tbody></table>
To reduce the simulation time, only the middle trace on both the top and bottom layers is set to a very fine mesh as shown in Fig. 8, where the measurements are taking place, the simulation boundaries are shown in Table II.

<figure><img src="image_10.png" /><figcaption>Fig. 8 (a) Model mesh visualization in Ansys Maxwell 3D software, (b) Close view of the middle trace mesh formation in Ansys Maxwell 3D software</figcaption></figure>

<table><caption>Table II: Simulation boundaries</caption><thead><tr><th>Parameter</th><th>Description</th></tr></thead><tbody><tr><td>Current source</td><td>Standard 0.7 A</td></tr><tr><td>Frequency</td><td>10<sup>-8</sup>Hz</td></tr><tr><td>BH curve</td><td>Smooth</td></tr><tr><td>Nonlinear residual</td><td>0.01</td></tr><tr><td>Region</td><td>Air</td></tr><tr><td>Boundaries</td><td>Radiation</td></tr><tr><td>Eddy current effect</td><td>Non</td></tr><tr><td>Core material</td><td>00K9541B026</td></tr><tr><td>Core permeability</td><td>26</td></tr><tr><td>Angle rotation position</td><td>The middle of the inductor</td></tr><tr><td>Middle traces mesh</td><td>Length based with maximum element length 0.06mm</td></tr><tr><td>Core mesh</td><td>Length based with maximum element length 1.2 mm</td></tr></tbody></table>

The simulation sweeps along angle (Θ) for values (0°,5°,10°,15°,20°,30°), so the calculation of Bz inside the software calculator is as in Eq.12 and Øz as in Eq.9.

The maximum flux density value (Bzmax) increase is more related to the increase of the trace length along the angle increase (marked area in Fig. 9), while the flux (Øz) is integral of the flux density along the trace area which increase with a triangular function of angle (Θ) along the copper trace angle (Θ) increase.

<figure><img src="image_11.png" /><figcaption>Fig. 9 Top view of the simulation model in Ansys Maxwell 3D showing the change in copper trace area above the magnetic core from angle 0 to angle 30</figcaption></figure>

The simulation results showed an incremental behavior for both the flux density (Bzmaxavg) and the flux (Øz) inside the copper along the increase of the shift angle (Θ) as illustrated in Fig. 10 and Fig. 11.

<figure><img src="image_12.png" /><figcaption>Fig. 10 The behavior of Bzmaxavg along the change of angle (Θ) positioned in the middle of the trace</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11 The behavior of Øz inside copper middle traces along the change of angle (Θ)</figcaption></figure>

The curve in Fig. 10 looks linear while in Fig. 11 looks more polynomial. The cause of this is the increase rate in Bzmax value and the trace area. As the measurement sensor shall be placed directly on the top of the middle trace and sweep of the maximum flux density point (Bzmax), there shall be a small distance around 0.5 mm from each side due to the sensor plastic cover, so the measurement level shall be 0.5mm above the trace as marked in Fig. 12.
<figure><img src="image_14.png" /><figcaption>Fig. 12 Measurement location 0.5mm over the middle trace due to the sensor package (plastic cover)</figcaption></figure>

The simulation calculations are done at that point as well to mimic the field going through the copper and passing by that point above the copper with 0.5 mm, the results shown in Fig. 13.

<figure><img src="image_15.png" /><figcaption>Fig. 13 The behavior of Bzmaxavg along the change of angle (Θ) positioned 0.5mm above the middle of the trace vs inside the trace</figcaption></figure>

## Experimental validation

The test model built is designed to be flexible as each angle is represented by an individual PCB, so that a top PCB is representing the top traces and prepreg layer and the bottom PCB is representing the bottom traces and prepreg layer, in-between there is the magnetic core and everything is hold together by a plastic platform as illustrated in Fig. 14.

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

<figure><img src="image_17.png" /><figcaption>Fig. 14 Experimental setup (a) shows six PCBs of different angles, (b) shows the setup placing with the bottom PCB and the magnetic core place on it, (c) shows the final form after placing the top PCB and connecting it to the bottom PCB.</figcaption></figure>

When it's required to get the measurement of a different angle, the plastic holder is opened to remove the top PCB of angle (Q°) and replace it with another top PCB of angle (Q+5°), then the measurements are registered and the process is repeated.

The PCBs are designed to have the angle segment where the measurements are taken to be in the middle section as in Fig. 15.

<figure><img src="image_18.png" /><figcaption>Fig. 15 PCB Designs in Altium Designer, (a) is showing the bottom PCB and (b) is the top PCB</figcaption></figure>

A 1 Ω 50 W resistor is added in series to help stabilizing the supplier voltage as the inductor appears as short circuit in DC as shown in the circuit representation in Fig. 16 [10].
<figure><img src="image_19.png" /><figcaption>Fig. 16 Experimental setup circuit representation</figcaption></figure>

The magnetic flux is measured by using PCE magnetometer PCE-MFM 4000-ICA which has a resolution of 0.01μT [11]. Its general prop can detect the DC field polarity by measuring the perpendicular field to the prop as in Fig. 17 [12].

<figure><img src="image_20.png" /><figcaption>Fig. 17 PCE-MFM 4000-ICA general prop directional measuring illustration</figcaption></figure>

The experiment boundaries are stated in **Error! Reference source not found.** and the measurements are taken along the angle.

<table><caption>**Table III: Experimental boundaries**</caption><thead><tr><th>Parameter</th><th>Description</th></tr></thead><tbody><tr><td>Current source</td><td>0.7 A DC</td></tr><tr><td>Frequency</td><td>0 Hz</td></tr><tr><td>Core thickness</td><td>2mm</td></tr><tr><td>Core permeability</td><td>26</td></tr><tr><td>Region</td><td>Air</td></tr><tr><td>Trace length in shifted area</td><td>17mm</td></tr><tr><td>Trace width</td><td>2mm</td></tr><tr><td>Trace clearance</td><td>8mm</td></tr><tr><td>Angle rotation position</td><td>The middle of the inductor</td></tr><tr><td>Copper Weight</td><td>1oz</td></tr><tr><td>PCB thickness</td><td>1.5mm</td></tr></tbody></table>

## Results comparison

Comparing the measurements with simulation values from Fig. 13 shows an incremental similar to the simulation ensuring the effect of the angle (Θ) on the increase of perpendicular flux in the copper trace as illustrated in Fig. 18.

### Measurements vs Simulation

<figure><img src="image_21.png" /><figcaption>Fig. 18 The average of maximum flux density perpendicular to the middle traces (top PCB/bottom PCB) in lab measurements compared to simulation results</figcaption></figure>

While the simulation showed 5.8% increase rate in flux between angle 0° and angle 20°, the experimental results showed 16.8% increase for the same angle value, this difference can be resulting due to the size difference between the simulation and the experimental sample as the experimental sample is bigger and has extended parts for connectors, though the incremental behavior is still valid.

## Conclusion

This study demonstrates the significant impact of the angle between copper traces and the magnetic core on the performance of PCB-embedded inductors. By evaluating the effect of this angular shift on the perpendicular magnetic flux density within copper traces, it was found that the orientation of the copper traces plays a crucial role in mitigating the perpendicular flux to copper traces in PCB embedded inductors.

The simulation results, validated through experimental testing, show an incremental relationship between the angle (Θ) and the magnetic flux density perpendicular to the copper traces. A lower angle, particularly at (Θ = 0°), minimizes perpendicular flux and reduces the induced eddy currents, improving inductor performance. However, while the 0° alignment
may be optimal from a flux reduction perspective, it is not always practical in industrial applications due to PCB design constraints.

By optimizing the angle to a practical value, it is possible to achieve a balance between minimizing unwanted magnetic flux and maintaining efficient PCB designs.

Future investigations shall be applied to validate the formed eddy currents inside the copper traces in AC applications based on this study result and their role in heating up the inductor coil.

This optimization leads to enhanced performance of embedded magnetic components, reducing losses, improving thermal management, and increasing overall reliability, particularly in applications like static power converters.

Thus, the continuous effort shall include defining the other potential elements affecting the PCB integrated inductor performance such as core selection and shaping plus performing some reliability tests to investigate the mechanical and thermal issue. A future model study shall include integrating this technology in industrial applications such as static converters as part of investing the research results in industry.

## References

[1] G. WEIS, I. SALKOVIC, E G. WEIDINGER, "PCB EMBEDDING OF MAGNETIC MATERIAL FOR INDUCTOR-BASED APPLICATIONS", EM IMAPS DPC 2020, FOUTAIN HILLS, ARIZONA: IMAPS DPC 2020, MAR. 2020.

[2] R. CAILLAUD ET AL., "EVALUATION OF THE PCB-EMBEDDING TECHNOLOGY FOR A 3.3 KW CONVERTER", EM 2019 IEEE INTERNATIONAL WORKSHOP ON INTEGRATED POWER PACKAGING (IWIPP), IEEE, ABR. 2019, P. 1-6. DOI: 10.1109/IWIPP.2019.8799080.

[3] ZEESHAN UMAR, "DESIGN OF AN EMBEDDED PCB INDUCTOR WITH A FERRITE MAGNETIC CORE FOR POWER APPLICATIONS DESIGN", NUREMBERG, ABR. 2024. ACESSADO: 27 DE JANEIRO DE 2025. [ONLINE]. DISPONÍVEL EM: HTTPS://D-NB.INFO/134898421X/34

[4] G. WEIDINGER ET AL., "NEW EMBEDDED INDUCTORS FOR POWER CONVERTER APPLICATIONS", EM MINAPAD 2018, GRENOBLE; FRANCE, MAIO 2018. [ONLINE]. DISPONÍVEL EM: WWW.ATS.NET

[5] W. G. HURLEY E W. H. WÖLFLE, TRANSFORMERS AND INDUCTORS FOR POWER ELECTRONICS, FIRST. WILEY, 2013. DOI: 10.1002/9781118544648.

[6] R. R. TUMMALA, M. SWAMINATHAN, S. SURESH, E H. GARMESTANI, "MODELING, DESIGN AND FABRICATION OF SUBSTRATE-EMBEDDED INDUCTORS WITH HIGH INDUCTANCE DENSITY AND LOW DC RESISTANCE FOR INTEGRATED VOLTAGE REGULATORS", TEXT, GEORGIA INSTITUTE OF TECHNOLOGY, GEORGIA, 2020.

[7] C. WM. T. MCLYMAN, TRANSFORMER AND INDUCTOR DESIGN HANDBOOK. CRC PRESS, 2017. DOI: 10.1201/B10865.

[8] BY NEEL MAYUR SHAH E B. PARKHIDEH BADRUL CHOWDHURY JONATHAN BIRD, "INVESTIGATION ON TOROIDAL PCB EMBEDDED INDUCTOR FOR POWER ELECTRONICS APPLICATIONS", THE UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE, 2015.

[9] A. MSELLATI, M. JAHNES, E M. PREINDL, "I-CORE PCB PLANAR INDUCTOR DESIGN FOR HIGH FREQUENCY AND HIGH POWER CONVERTERS", EM 2022 IEEE TRANSPORTATION ELECTRIFICATION CONFERENCE & EXPO (ITEC), IEEE, JUN. 2022, P. 377-382. DOI: 10.1109/ITEC53557.2022.9813880.

[10] R. CAILLAUD ET AL., "DESIGN, MANUFACTURING AND CHARACTERIZATION OF PRINTED CIRCUIT BOARD EMBEDDED INDUCTORS FOR POWER APPLICATIONS", EM 2018 IEEE INTERNATIONAL CONFERENCE ON INDUSTRIAL TECHNOLOGY (ICIT), IEEE, FEV. 2018, P. 694-699. DOI: 10.1109/ICIT.2018.8352262.

[11] MUNIMA HAQUE, QUAMRUZZAMAN MOHAMMAD, E SHAHINA HAQUE, "MEASUREMENT OF MAGNETIC FIELD EMITTED FROM ELECTRICAL APPLIANCES IN CSE LABS AND CLASSROOMS OF SOUTHEAST UNIVERSITY, BANGLADESH", INT J SCI ENG RES, VOL. 7, NO 10, P. 295-304, OUT. 2016.

[12] TUMANSKI S, "MODERN MAGNETIC FIELD SENSORS - A REVIEW", ELECTROTECHNICAL REVIEW, VOL. 89, NO 10, P. 1-12, 2013.