# Inductive, Capacitive and Resistive Aspects Modeling of Planar Windings Applied to a 1 MHz LLC Converter

Oussama Dihane, Tanguy Phulpin, Eric Laboure, Alain Gascher, Lingeshwaren Sobrayen, Daniel Sadarnac

► To cite this version:

Oussama Dihane, Tanguy Phulpin, Eric Laboure, Alain Gascher, Lingeshwaren Sobrayen, et al. Inductive, Capacitive and Resistive Aspects Modeling of Planar Windings Applied to a 1 MHz LLC Converter. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0199. hal-05101057

HAL Id: hal-05101057

https://utc.hal.science/hal-05101057v1

Submitted on 6 Jun 2025

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# Inductive, Capacitive and Resistive Aspects
Modeling of Planar Windings Applied to a 1 MHz
LLC Converter

Oussama Dihane¹,², Tanguy Phulpin¹, Eric Laboure¹, Alain Gascher², Lingeshwaren Sobrayen³, Daniel Sadarnac¹

1) Paris-Saclay University, CentraleSupelec, Groupe of Electrical Engineering of Paris. 2) Renault S.A.S.

3) EnerSys S.A.R.L.

GeePs : 11 Joliot Curie Street, 91192 Gif-sur-Yvette. Renault S.A.S : 1 Golf Av, 78280 Guyancourt.

Enersys: EnerSys S.A.R.L., Rue A. Fleming, Z.I. EST, CS40962, 62033 Arras, France.

Gif-sur-Yvette, France.

(+33) 07 53 64 30 51

oussama.dihane@ampere.cars

**Index Terms** – Planar, Modeling, Capacitance, Losses, Impedances, Simulations, Optimisation.

**Abstract** – Planar transformers and inductors are subject to parasitic capacitances that can alter the operation of the converter using them. This article integrates capacitive aspects into conventional inductive and resistive modeling developed by Dowell [1]. This leads to a relatively simple software program that enables fast simulation of planar components' frequency behavior, predicts conductors' current distribution, and deducts the resulting losses. Finally, an application example is proposed consisting of optimizing the design of a transformer by adjusting its windings, the nature of its materials, and the position of an air gap, to minimize current oscillations in the converter that could result from excessive parasitic capacitors.

## I. INTRODUCTION

The planar winding concept was introduced in the 1980s to facilitate the design of transformers and inductors suitable for high-frequency operation. In addition to simplifying the winding and making the resulting characteristics reproducible on a large scale, this concept aligns itself perfectly with the concept of interleaving transformer windings, intending to minimize proximity losses and optimize magnetic coupling.

However, it is clear that a complicated windings interleaving results in an undesirable capacitive coupling relative to the ElectroMagnetic Compatibility (EMC) [2]. Moreover, the windings interleaving in a planar transformer implies the use of parallel windings to ensure the desired transformer's ratio. Precautions must be taken to guarantee a homogeneous current sharing between the parallel windings to avoid hot spots [3].

For the reasons mentioned, different windings interleaving techniques have been developed and continue to emerge. These techniques rely on logical qualitative reasoning that aims to achieve the best compromise for the given application [4], [5], [6].

However, the estimation and the validation of the desired characteristics are not obvious, such as the current distribution inside the conductor and between the windings, the generated losses, the leakage inductance, the parasitic capacitances, and the impact of frequency. Thus, the first part of the article introduces the analytical modeling of the characteristics mentioned above. The second part introduces the resulting algorithm which can be implemented in Matlab or Octave software. This algorithm is developed in a general application context of transformers associated with galvanically isolated converters such as resonance converters or Dual Active Bridge (DAB) converters. This article completes [7].

## II. ANALYTICAL MODELING

Fig 1 describes the modeled structure. Three windings designated by their respective colors are shown as an example. The number of parallel turns that make up each winding, and the number of windings per layer can be taken as desired.

<figure><img src="image_2.png" /><figcaption>Fig. 1: The considered configurations of windings and airgaps.</figcaption></figure>

To increase the field of the study, the copper turns are split into
half-turns that can be connected in any manner between them. The magnetic circuit is considered as parallel ferrite plates, between these plates, ferrite pillars are nested along with air gaps. The number of pillars and air gaps can be random. The air gaps could be located at the central leg, at the two or the three external legs.

## INDUCTANCE AND RESISTANCE MODELING

The concept of half-turn has no meaning in electromagnetism. However, it will be used to present a real object in which the voltage can be measured. Fig 2 highlights an encountered problem, which is that the voltage difference measured depends on the mode of operation.

To clear out any ambiguity, the model presented will use the electrical potential rather than the voltage difference. This electric potential is relative to an external reference.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Half-turn ambiguity(a) The use of a potential reference (b).</figcaption></figure>

Fig 3 illustrates the magnetic fluxes inside the system defined in Fig 1. The flux flows in the sides of each conductor, within its width, and through each air gap.

To ensure a fast estimation of the physical variables characterizing a given transformer, simplifying hypotheses are considered :

- The permeability of the ferrite pillars is considered infinite. The permeability of the plates corresponds to the complex permeability specified by the constructor. By taking it into account, the estimation of the damping of certain resonance phenomena is possible.

- The flux lines through the windings are straight, perpendicular to the central core, and with the same length, even if the winding extends beyond the magnetic circuit.

- The flux lines going through the air gaps do not deviate and do not reach the conductors.

- The magnetic circuit's resistivity is neglected to estimate the parasitic capacitance between the first and last copper layers.

Fig 4 represents an example of a half-turn enumerated k among the half-turns presented in Fig 3. The conductor

in question is divided into layers of a width of $\Delta x$. Each layer carries a fraction $\Delta i$ of the total current $i_k$.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Magnetic flux.</figcaption></figure>

The layer identified by its x-axis is located between $x$ and $x + \Delta x$ forms a turn around the top leg and it is subject to the flux $\Psi_k - \phi(x)$, where $\Psi_k$ is the flux through the winding coming from the upper leg, and $\phi(x)$ is the leakage flux through the conductor up to the x coordinates. The layer at the $x + \Delta x$ coordinates is subject to a different flux of $\Psi_k - \phi(x + \Delta x)$. The difference corresponds to the flux through the layer $\phi_s$, these two layers do not carry the same induced voltage.

$$ \Delta u = u(x) - u(x + \Delta x) \quad (1) $$

$$ \Delta u = \frac{d[\phi(x + \Delta x) - \phi(x)]}{dt} \quad (2) $$

$$ \Delta u = \frac{d\phi_s}{dt} \quad (3) $$

Since the applied electrical potentials $U_k$ and $V_k$ are the same, the currents $\Delta i$ are different. The difference between these currents is only limited by the layer's resistance. Fig 5 represents an equivalent schematic to illustrate the current $i_k$ distribution inside the conductor.

<figure><img src="image_5.png" /><figcaption>Fig. 4: A single conductor divided into elementary layers of a width of $\Delta x$.</figcaption></figure>

Fig 6 represents a layer in the k-th half turn, the layer can be decomposed into smaller conductor portions connected in series. One of these portions, illustrated in the right corner
of Fig 6 is itself composed of parallel elementary conductors. One of these conductors is represented in black in Fig 6. Its length is $ΔL$, its width is $dy$ and its thickness is $Δx$ equal to the layer's thickness.

<figure><img src="image_6.png" /><figcaption>Fig. 5: Half-turn equivalent schematic.</figcaption></figure>

The equation (4) is deduced from Fig 5

$$ \Delta i(x + \Delta x) - \Delta i(x) = \frac{\Delta u}{r} = \frac{1}{r} \cdot \frac{d\phi_s}{dt} \quad (4) $$

It is subject to a flux density $B$, a magnetic field $H$, and is traversed by a flux $Δφ$ within its thickness, which is a fraction of the total flux $φ_s$ in a layer.

<figure><img src="image_7.png" /><figcaption>Fig. 6: A portion of an elementary conductor inside a layer.</figcaption></figure>

$J$ denotes the magnetic field circulation calculated at the height $h_c$. $J$ is shown in Fig 4. By definition, it is written as :

$$ J = \int_{h_c} H \cdot dy = \int_{h_c} \frac{B}{\mu_0} \cdot dy \quad (5) $$

$$ J = \int_{h_c} \frac{\Delta\phi}{\mu_0 \cdot \Delta L \cdot \Delta x} \cdot dy = \frac{\Delta\phi}{\mu_0 \cdot \Delta x} \cdot \int_{h_c} \frac{dy}{\Delta L} \quad (6) $$

The elementary conductor is characterized by a resistance or admittance that depends on its length $ΔL$ and its cross-section $Δx \cdot dy$. The integral of the admittances calculated throughout the width gives out the admittance of the layer in question and estimates its resistance $Δr$ as shown in the equation (7).

$$ \frac{1}{\Delta r} = \int_{h_c} \frac{1}{\rho \cdot \frac{\Delta L}{\Delta x}} \cdot dy = \frac{\Delta x}{\rho} \cdot \int_{h_c} \frac{dy}{\Delta L} \quad (7) $$

The flux $φ_s$ in the width of the layer corresponds to the sum of $Δφ$ calculated throughout its length. In the same way, the total resistance $r$ of a layer corresponds to the sum of $Δr$ calculated throughout the same length.

$$ \phi_s = \sum \Delta \phi = \sum J \cdot \frac{\mu_0 \cdot \Delta x}{\int_{h_c} \frac{dy}{\Delta L}} \quad (8) $$

$$ \phi_s = \sum J \cdot \frac{\mu_0 \cdot \Delta r \cdot \Delta x^2}{\rho} = J \cdot \frac{\mu_0 \cdot \Delta x^2}{\rho} \cdot r \quad (9) $$

$$ \Rightarrow \Delta i(x + \Delta x) - \Delta i(x) = \frac{\mu_0 \cdot \Delta x^2}{\rho} \cdot \frac{dJ}{dt} \quad (10) $$

Using Ampere's law :

$$ \Delta i(x) = J(x + \Delta x) - J(x) = \Delta J(x) \quad (11) $$

$$ \frac{\Delta i(x - \Delta x) - \Delta i(x)}{\Delta x^2} = \frac{\Delta J(x - \Delta x) - \Delta J(x)}{\Delta x^2} \quad (12) $$

$$ \frac{\Delta i(x - \Delta x) - \Delta i(x)}{\Delta x^2} = \frac{\mu_0}{\rho} \cdot \frac{dJ}{dt} \quad (13) $$

considering a harmonic system, equation (13) becomes :

$$ \frac{\mu_0}{\rho} \cdot j\omega \cdot J = \frac{d^2 J}{dx^2} \quad (14) $$

$$ \Rightarrow J = J_{Ak} \cdot e^{\alpha x} + J_{Bk} \cdot e^{-\alpha x} \quad (15) $$

Where $\alpha = \frac{1+j}{x_0}$ and $x_0 = \sqrt{\frac{2\rho}{\mu_0 \cdot \omega}}$

It is possible to calculate the coefficients $J_A$ and $J_B$ relative to the k-th conductor as a function of the particular values $J_0$ and $J_e$ at the $x = 0$ and $x = e$ coordinates as shown in the Fig 4

$$ J_{Ak} = \frac{-J_{0k} \cdot e^{-\alpha e} + J_{ek}}{e^{\alpha e} - e^{-\alpha e}} \quad (16) $$

$$ J_{Bk} = \frac{J_{0k} \cdot e^{\alpha e} - J_{ek}}{e^{\alpha e} - e^{-\alpha e}} \quad (17) $$

The potential difference $U_k - V_k$ at the terminals of the k-th turn can be calculated at any given x coordinate. For example, the layer at the $x = 0$ coordinates form a turn subject to the flux $Ψ_k$. It is therefore subject to an induced of $j\omega \cdot Ψ_k$, which leads to writing:

$$ U_k - V_k = j\omega \cdot \Psi_k + r\Delta i(0) \quad (18) $$

$$ \frac{\Delta i}{\Delta x} = \frac{\Delta J}{\Delta x} = \frac{dJ}{dx} = \alpha \cdot (J_A - J_B) \quad \text{if } x = 0 \quad (19) $$

$$ \Rightarrow U_k - V_k = j\omega \cdot \Psi_k + r \cdot \Delta x \cdot \alpha (J_{Ak} - J_{Bk}) \quad (20) $$

If the windings were made of ribbons of copper wrapped around themselves, the resistance $r$ of one layer would be calculated as a function of its length. for a planar-type winding, the average length $L_c$ of the inductor presented in Fig 6 is chosen. It can be possible to introduce a correction factor. This simplification leads to :

$$ r = \frac{\rho \cdot L_c}{\Delta x \cdot h_c} \quad (21) $$

$$ U_k - V_k = j\omega\Psi_k + \frac{\alpha \cdot \rho \cdot L_c}{h_c} (J_{ak} - J_{Bk}) \quad (22) $$

For each k-th half-turn, the flux $Ψ_k$ is calculated with the sum of the previous fluxes as suggested in Fig 7.
<figure><img src="image_8.png" /><figcaption>Fig. 7: The different magnetic fluxes at play.</figcaption></figure>

These fluxes correspond to the fluxes $\phi_i$ within the thickness $e$ of the conductor, to the fluxes $\phi'_i$ and $\phi_i$ within both thickness $e'_i$ and $e_i$ of the isolation layers to the left and the right of the conductors and the flux $\phi_F$ in the core at the right of the core:

$$ \phi_i = \int_0^e \frac{\mu_0 \cdot J}{h_{ci}} \cdot dx \qquad (23) $$

$$ \phi_i = \frac{\mu_0 \cdot L_{Ci}}{\alpha \cdot h_{ci}} \cdot \frac{e^{\alpha e} + e^{-\alpha e} - 2}{e^{\alpha e} - e^{-\alpha e}} \cdot (J_{0i} + J_{ei}) \quad (24) $$

$$ \phi'_j \text{ or } \phi_j = \frac{\mu_0 \cdot L_{Cj}}{h_{Cj}} \cdot (e'_i \text{ or } e_i)(J_{0i} \text{ or } J_{ei}) \quad (25) $$

The determination of the flux $\phi_F$ depends on the magnetic circuit's structure, taking into account the magnetic field's circulations $J_e$ at the rightmost corner under the two external legs, the ferrite's reluctance and the air gaps at that same location. These calculations were developed in [7].

## CAPACITANCE MODELING

The proximity of the copper layers between them in a PCB is the source of parasitic capacitances. In the example presented in Fig 8a, several capacitances are represented since numerous windings are engraved in the same layer. In reality, it represents the capacitances distributed along the length and the width of the conductors. The hypothesis was considered and validated that it is possible to discretize the conductors to locate the capacitances. The discretization step was taken as a half-turn, which appeared to be enough and it corresponds to the inductance model previously presented. Moreover, this step allows to differentiate between the frequency behavior of the transformer depending on whether its terminals are superposed or diametrically opposed.

Figure 8b illustrates the model adopted for the example of half-turn n°3: the parasitic capacitance between 3 and 1 is split to form two capacitances located at the two ends of the half-turn. These capacitors are subject to potential differences $U_3 - U_1$ and $V_3 - V_1$. Their values correspond to half the calculable capacitance between 3 and 1 (for ¼ turn) [8]. The current $i_3$ and the electrical potentials $U_3$ and $V_3$ correspond to the quantities $i_k$, $U_k$, and $V_k$ considered in the inductive model (with $k = 3$).

<figure><img src="image_9.png" /><figcaption>Fig. 8: Parasitic capacitance between two conductors (a). The half-turn model (b).</figcaption></figure>

$$ C_{13} = \frac{1}{2} \cdot \epsilon \cdot \frac{\text{Surface between 1 and 3}}{\text{distance between 1 and 3}} \quad (26) $$

## III. NUMERICAL SIMULATION

The purpose of the simulation program used to analyze a given inductor or transformer is threefold:

- Determine the losses in each half-turn and the overall losses as a function of the constraints imposed (voltages, currents, frequency).

- Determine the evolution of various impedances with frequency: no-load or short-circuit seen from the primary, between primary and secondary...

- Determine the steady-state waveforms in a given converter, including a transformer.

## CALCULATING LOSSES AND IMPEDANCES

The magnetic component is considered alone and subjected to given currents and/or voltages. First of all, it's necessary to determine the current $i_k$ and the potentials $U_k$ and $V_k$ for each half-turn. This is not straightforward for several reasons: the currents are not necessarily shared equally between turns connected in parallel, the primary and secondary currents are not in a simple relationship in the presence of an air gap or tertiary winding, and because of the parasitic capacitances. Determining the current $i_k$ in each half-turn involves determining all the circulations $J_{0k}$ and $J_{ek}$: $i_k = J_{ek} - J_{0k}$ according to Ampère's law. For each half-turn, there are 4 parameters to determine: $J_{0k}$, $J_{ek}$, $U_k$, and $V_k$. resulting in a total of 4 unknowns $N_{DS}$ for $N_{DS}$ half-turns.

The electrical potentials of the external terminals and the currents flowing through them must also be determined to derive impedances. This represents 4 additional $N_E$ unknowns for $N_E$ windings (primary, secondary...).

There is therefore a total of $4N_{DS} + 4N_E$ unknowns that can only be determined by writing as many equations. These equations are:

- The equations describing the electrical quantities imposed on the windings. Fig 9 shows two examples. The imposed quantities are shown in red. $2N_E$ equations must be written for $N_E$ windings. These equations describe the
known voltage potentials, voltage differences, currents, or the relationship between the currents.

<figure><img src="image_10.png" /><figcaption>Fig. 9: Imposed quantities example.</figcaption></figure>

- The equations describing the connections between half-turns. Fig 10 shows two examples of windings. The connections between half-turns are shown in blue. There are $N_B \cdot (N_{DS} - 1)$ for a winding with $N_B$ windings in parallel and $N_{DS}$ half-turns per winding. It is, therefore, possible to write as many equations relating the equality of the potentials of the connected points and as many node laws involving the complex impedances of the parasitic capacitances, i.e. a total of $2N_B \cdot (N_{DS} - 1)$ equations for a winding.

<figure><img src="image_11.png" /><figcaption>Fig. 10: Example of connections between half-turns to form a winding.</figcaption></figure>

- The equations describing the external connections. These connections are shown in red in Fig 10, and there are $2N_B$ per winding. It is possible to write any number of electrical potential equalities, but only 2 node laws, i.e. a total of $2n_B + 2$ equations per winding. Added to the equations describing the connections between half-turns, we obtain a total of $2N_{DS} + 2N_E$ equations for all windings.

- The equations describing the electrical potential differences across the half-turns. Their number is $N_{DS}$. Each is obtained from equations (22) by using the expressions (16)(17)(24) and (25) of $J_{Ak}$, $J_{Bk}$, $\Psi_k$, $\varphi_i$, $\phi'_j$, $\phi_j$, we obtain an equation between the electrical potential difference $U_k - V_k$ at the terminals of half-turn n°k and all $J_0$ and $J_e$. Together, these equations form a linear system whose complex coefficients are calculable and depend, among other things, on the operating frequency (by $\alpha$).

- The equations describing the conservation of flux between the half-turns of the same layer. In the example shown in

Fig 7, the flux $\varphi_1$ is equal to the flux $\varphi_2$. We can thus deduce from the equation (24) :

$$ \frac{L_{C1}}{h_{C1}} \cdot (J_{01} + J_{e1}) = \frac{L_{C2}}{h_{C2}} \cdot (J_{02} + J_{e2}) \quad (27) $$

The number of equations with this type is $N_{DS} - N_c$ for $N_{DS}$ half-turns and $N_c$ copper layers.

- The equations describing the relationships between the current flows in the insulators of a single PCB. In the example shown in Fig 7: $J_{04} + J_{05} = J_{e3}$.

- The equations describing the relationships between the currents' flow on either side of the air gaps (except for the last air gap, which is already taken into account when calculating $\varphi_F$). In the example shown in Fig 7: $J_{e1} + J_{e2} = J_{03}$ + the J of the air gaps. The $J_s$ of the air gaps can be calculated from the downstream flows and are therefore expressed as a function of the downstream $J_{0s}$ and $J_{es}$. The number of relations of this type added to the previous number for the insulators is equal to the number $N_C$ of copper layers. Overall, a total of $4N_{DS} + 4N_E$ equations is obtained, corresponding to as many unknowns to determine. The calculation program resulting from this analysis essentially consists in automatically writing a square matrix ($4N_{DS} + 4N_E$) x ($4N_{DS} + 4N_E$) before inverting it in a single instruction to find out all the currents, all the voltages and finally all the relevant information (current distribution in each half-spire, losses, impedances...).

## WAVEFORM PREDICTION

The transformer is now included in a converter. In its current form (which can evolve), the proposed generic diagram is given in Fig 11: it is a converter with NE inputs/outputs and using $N_E$ active bridges (e.g. a DAB, a CLLLC...). In this case, $N_E$ is equal to 2, but could be any number. The commands of all bridge arms can be phase-shifted and are compatible with ZVS-type switching. Each bridge is assumed to be equivalent to two trapezoidal voltage sources with 1/2 duty cycle and edges of adjustable duration. The NE-winding transformer is supplied via resonant circuits between points a and A of each channel, and via wiring inductors between points b and B. The impedances of the resonant circuits can be any, zero or purely inductive. A “parasitic” capacitance between point m and the potential reference (ground, case...) allows common-mode currents to close.

Compared to what was presented above for the transformer considered alone, three new potentials must be taken into account for each path: at points a, b and m. Knowing these potentials enables us to deduce the resonant current and the common-mode current (marked in red in Fig 11). These three new unknowns per path imply writing $5N_E$ equations instead of $2N_E$ to describe the electrical quantities imposed on the NE windings: 3 Ohm's laws for the 3 impedances + 2 imposed voltages. Finally, we need to write and solve $4N_{DS} + 7N_E$ equations.
The program has been validated on the basis of various projects in very different fields (aeronautics, industry). We will illustrate the interest of this program by taking the example of a 1 kW designed in the thesis [9]. We will not go into the details of the transformer optimization, which has already been described in detail in the thesis and various publications. We will simply use the main steps to illustrate the scope of the program. The converter including the transformer under consideration is of the LLC hybrid bridge type. An image of the first transformer produced is shown in Fig 12. Table I presents the transformer specifications. Fig 13a illustrates the construction of the two windings by drawing the successive copper layers from the bottom of the circuit to the air gap, which is indicated by a red line. The 11 primary turns are connected in series. 4 secondary turns are connected in parallel.

<figure><img src="image_12.png" /><figcaption>Fig. 11: A converter's generic model.</figcaption></figure>

## IV. APPLICATION

<figure><img src="image_13.png" /><figcaption>Fig. 12: First transformer prototype.</figcaption></figure>

Table I presents the transformer specifications.

<table><caption>TABLE I: The studied transformer's specifications.</caption><tr><td>Core type</td><td>E43/10/28 + PLT</td></tr><tr><td>Air gap</td><td>0.9 mm</td></tr><tr><td>Ferrite</td><td>3F4</td></tr><tr><td>PCB</td><td>8 layers of 2.4 µm</td></tr><tr><td>Copper</td><td>105 µm</td></tr><tr><td>N<sub>primary</sub></td><td>11</td></tr><tr><td>N<sub>secondary</sub></td><td>1</td></tr></table>

Based on essential numerical data (maximum power currents, potentials, frequency (1 MHz), thicknesses, permittivities, etc.), the program determines the profiles of induction, current density and losses in each half-turn of each copper layer located under the same leg (Fig 14).

<figure><img src="image_14.png" /><figcaption>Fig. 13: The considered winding types.</figcaption></figure>

Calculations are made considering only the fundamental components of voltages and currents. The clear blue bars refer to an 'average' current density defined by the current/section ratio. The bars of unequal height reflect, in particular, a poor distribution of currents between turns connected in parallel. The Comparison of the blue curves with these bars reveals the impact of induced currents. The global losses shown have been experimentally validated under sinusoidal conditions.

<figure><img src="image_15.png" /><figcaption>Fig. 14: Calculated profiles.</figcaption></figure>
The algorithm can also determine any impedance. Fig 15, for example, shows the impedance as seen from the primary, with the secondary in an open circuit. It also shows an experimental reading under the same conditions, which validates the calculations.

Fig 16 shows an experimental measurement carried out under the same conditions, which validates the calculations.

<figure><img src="image_16.png" /><figcaption>Fig. 15: Calculated impedance as seen from the primary.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 16: Measured impedance as seen from the primary.</figcaption></figure>

The resonance at 2.8 MHz and subsequent resonances demonstrate the presence of parasitic capacitances. These capacitances are always revealed by further readings or impedance determinations. Fig 17, for example, is obtained with the program by calculating the impedance between the two connected terminals of the primary and the two connected terminals of the secondary. It shows a capacitance close to 1.75 nF at low frequencies.

As shown in Fig 18, these parasitic capacitances have a strong influence on the converter's behavior through current oscillations, which cause additional losses, malfunctions in ZVS switching and electromagnetic interference. When sizing the converter, it would be interesting to use numerical simulation software to assess the significance of these oscillations. For this purpose, it is possible to approximate the operation by

enriching the classic equivalent diagram of the transformer with 6 capacitors between its 4 terminals to take account of parasitic capacitances [10], [11].

The values of the 6 capacitors can be obtained by taking 6 suitable readings. This method is well known, but a prototype is needed! The program allows these 6 capacitances to be determined, since it is designed to calculate any impedance (between any terminals, no-load, short-circuit...).

<figure><img src="image_18.png" /><figcaption>Fig. 17: Calculated impedance between primary and secondary.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 18: Transformer's current (blue) and the applied voltage by the primary bridge (red).</figcaption></figure>

Another option offered by the program is to plot waveforms directly in specific contexts (DAB, CLLC...). The predetermined waveforms on the primary and secondary sides are shown in Figure 19.

<figure><img src="image_20.png" /><figcaption>Fig. 19: Transformer currents (blue), bridge voltages (red), common-mode currents (green).</figcaption></figure>

However, this program is not exactly adapted to the hybrid
bridge that delivers the 4-level voltage shown in Fig 18 (a structure that can be switched between half-bridge and full-bridge). The primary voltage in Fig 19 has only 2 levels, but this is sufficient to demonstrate the current oscillations observed experimentally. The program also highlights other aspects: the sensitivity of the operating point as a function of wiring inductances and transformer characteristics (air gap, insulation, etc.), the intensity and frequency of common-mode currents... With the harmonic composition of currents known for a given transformer environment, the program deduces overall winding losses and highlights the influence of certain harmonic ranks: Fig 20 shows that the presence of the current oscillations observed in Fig 18 results in a significant increase in losses.

<figure><img src="image_21.png" /><figcaption>Fig. 20: Integral of winding losses as a function of the harmonic rank of currents</figcaption></figure>

To minimize the primary to secondary capacitance, a variant is to implement a “two-pack interleaving” (Fig 13b): the number of primary-secondary interfaces is thus reduced from 7 to 4. The latter can also be enlarged by increasing the PCB thickness to 3.2 mm. This interleaving also makes it possible to limit current imbalances between parallel coils [7], thus avoiding the “hot point” in the second copper layer shown in Fig 14. The resonant frequency in the primary winding impedance is thus increased from 2.8 to 4.3 MHz, and the primary-secondary capacitance is reduced from 1.75 to 0.5 nF. To minimize PCB temperature rise, two PCBs can be connected in parallel, each under an E-shaped circuit (Figure 13c). Overall losses are then divided by 2, but the primary-secondary capacitance is multiplied by 2.

This capacitance can be reduced by interleaving in packs of four (Figure 13d). The number of interfaces is reduced from 8 to 4, and these interfaces can be enlarged by bringing the primary layers closer together and the secondary layers closer together. The primary to secondary capacitance then drops to 230 pF. On the other hand, the current distribution between the secondary turns is not ideal (Fig 21): four turns grouped together are as thick as a turn four times thicker.

To remedy this, the Litz wire principle can be adapted to planar windings (Fig 13e) to better balance the currents in the secondary conductors: a coil is formed by two half-coils on two different layers. On the primary side, the two coils are placed symmetrically relative to the air gap to balance the currents. The overall result is improved as shown in Fig 22. In addition, the two PCBs are identical.

In order to reduce the magnetic field through the copper layers close to the air gap, a simple idea is to segment the magnetic circuit more finely into several air gaps. Fig 23 illustrates this principle: instead of a single 0.9 mm air gap, three air gaps of 0.225, 0.45 and 0.225 mm are used. This reduces overall losses.

<figure><img src="image_22.png" /><figcaption>Fig. 21: Interleaving by packs of 4.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 22: Litz principle and symmetrization.</figcaption></figure>

The current distribution in the thickness of the copper layers is still far from homogeneous. It seems appropriate to test for a lower copper thickness. Fig 24 is plotted by replacing the thickness of 105 µm by 70 µm. Overall losses have been further reduced.

<figure><img src="image_24.png" /><figcaption>Fig. 23: Segmentation to 3 air gaps.</figcaption></figure>

A second transformer prototype based on the previous reasoning was built. Given the industrial considerations of the time, not everything was implemented, but the choice of materials was optimized. Fig 25 shows the calculated impedance as seen from the primary, with the secondary unloaded.

The experimental measurements under the same conditions are shown in Fig 26. It validates the principle of the calculation
and justifies the technological choices made, with the resonance frequency increasing from 2.8 MHz to 6.5 MHz.

<figure><img src="image_25.png" /><figcaption>Fig. 24: Copper thickness reduction.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 25: Calculated impedance seen from the primary.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Fig. 26: Measured impedance seen from the primary.</figcaption></figure>

The precalculated waveforms in Fig 27 confirm the effective reduction in parasitic capacitance, since the residual oscillations have been reduced and are of higher frequency. Common-mode currents are therefore easier to filter.

<figure><img src="image_28.png" /><figcaption>Fig. 27: Transformer currents.</figcaption></figure>

## CONCLUSION

This paper has proposed an analytical model of a planar transformer, taking into account inductive, resistive and capacitive aspects. Based on this model, a program was developed to calculate various characteristics of a given transformer (current distribution, losses, impedances) and predetermine electrical waveforms in a converter. The logic of this program can easily be exploited by a designer to rewrite a customized script and adapt an optimized solution amongst a list of magnetic materials according to his applications. The validity of the model and calculation logic is demonstrated by a number of very different real-life examples. This script, usable with Matlab or Octave, can be supplied free of charge on request.

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