# Design and Optimization of an Unbalanced-flux Transformer for 1 kW LLC Transformer

Sobhi Barg, Adane Hailu, Souhaib Barg, Kent Bertilsson

► To cite this version:

Sobhi Barg, Adane Hailu, Souhaib Barg, Kent Bertilsson. Design and Optimization of an Unbalanced-flux Transformer for 1 kW LLC Transformer. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0149. hal-05105998

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# Design and Optimization of an Unbalanced-flux Transformer for 1 kW LLC Transformer

Sobhi Barg¹, Adane Hailu¹, Souhaib Barg², Kent Bertilsson¹

¹Mid Sweden University, Sundsvall, Sweden

²Central University of Tunis, Tunis, Tunisia

E-Mails: sobhi.barg@miun.se, admu1900@student.miun.se, souhaib1.barg1@gmail.com, kent.bertilsson@miun.se

## Keywords

«Unbalanced-flux magnetics», «LLC transformer», «optimization», «MHz approach», «balanced-flux magnetics».

## Abstract

Existing works to increase the power density of LLC transformer are mainly based on the MHz approach. This paper proposes a new LLC transformer design based on the recent introduced concept called the unbalanced-flux magnetics. The transformer was optimized using Genetic algorithm and iterative optimization approach. The unbalanced-flux transformer was designed and experimentally tested in 1 kW 400/48V LLC converter. The LLC converter is built, and the transformer is successfully up-to 750 W without heatsink and 1 kW with heatsink. The realized realizing power density is 224W/cm³ which is approximately 3x higher than the results in the state-of-the-art.

## Introduction

The recent years has witnessed a raising awareness for resonant converters in both academia and industry. This converter topology becomes a dominant solution for (400V-12V and 400V-48V) telecommunication systems and it is remarkably pervading all other power electronics applications such as EV chargers. Among the many advantages this topology possesses, their natural-soft-switching feature is to be a major one. The switching loss, which was long considered to be a serious obstacle that hinders the efficiency and the power density of the converter, was significantly minimized which solves the issues related to the switching devices. The recent emerging of the Wide-band-Gap devices such as GaN and SiC helps to relatively improve the efficiency and the power density of the magnetic components issued by their

capability to increase the switching frequency to the MHz range. In recent years, the MHz operation range is regarded as the key solution and the most adopted approach to overcome the limitations of magnetics. Pr. Fred from Virginia Tech and his students are the principal pioneers of this approach [1-2]. Numerous research, which have been carried out, argued that the power density and the efficiency has improved because of increasing the switching frequency to MHz range [1-3]. Undoubtedly, the overall power density and efficiency of the converter has increased. Nevertheless, despite this improvement, the existing works lack the scientific evidence, which prove that the choice of the MHz is what contributes to the increase of the power density and the efficiency of magnetics. Recently, the study in [4] has shown a new concept of magnetics that can enable to increase the power density by 250% compared to existing approach. This concept is called the unbalanced-flux magnetics (UBFM) which allows to operate the magnetic core at high magnetic flux density without any thermal issues as found in existing designs. This idea enables to increase significantly the power density and achieve high power density without the need to increase the switching frequency to MHz range. Increase the frequency can allow to shrink the size of magnetic core, however it leads to several issues such as thermal and EMI issues resulting from the high core loss and high switching frequency. The UBFM is a radical magnetic concept and an alternative approach to overcome the limitations of the MHz approach.

The main contribution of this work is to validate the concept of the UBFM concept, presented in [4], in a 400 V-48V, 1 kW LLC converter.

This paper is structured as follows. A review of the existing methods, with a detailed examination and discussion of their principles, their advantages and shortcomings, is presented in section II. The proposed UBFM concept and the
design of the transformer is developed in section III. A design example and a comparison with existing methods is presented in section IV. Finally, the conclusion of the work is presented in section V.

## Review of the existing methods

LLC converter is a key solution for applications requiring high power density and high efficiency. Thanks to its natural soft switching feature, the switching loss are negligible allowing the frequency to be increased to MHz range which in turns helps to reduce the size of the magnetic components and therefore improving the power density and the overall efficiency.

Minimizing the switching loss, which was known to be the main issue in DC-DC converter, is not something to deal with in LLC converter. However, another remaining and important issue which is more severe at high frequency is the minimization of the magnetic components' losses. Increasing the frequency is a double-edge solution. From one side, it allows to shrink the size of magnetics, and from another it increases significantly the losses hindering achieving high power density. Designing magnetics at MHz range has several challenges such as minimization of the winding losses due to the proximity and skin effect, the thermal constraints and minimization of core loss due to eddy currents [1].

Several techniques were proposed in the literature to overcome these issues. Magnetics integration, matrix transformers, flux cancellation and fractional turns, integrated magnetic shunt to integrate the resonant inductor are technical solution employed in several works to improve the efficiency and the power density [5-12].

In [12], the realized power density is 70.4 W/cm³ using a split phase and fractional turn techniques. A 45 W/cm³ was achieved in [9] using half turn solution. A quarter design solution was proposed in [11] leading to increase the power density, calculated on the box-volume base, to 70.82 W/cm³. 78 W/cm³ was achieved in [10] using half turn design solution and flux cancellation technique in UI core shape. In [3], the power density was increased to 66 W/cm³, calculated on the box-volume base, through improving the shielding technique of solution in [2]. In general, the realized power density using the MHz approach varies approximately from 45 to 70 W/cm³ for 400/12 V LLC converter [92]. An approximate 100 W/cm³ power density was

achieved for 3 kW 400V/48V three phase LLC transformer.

Recently, a new concept called the unbalanced-flux magnetics was introduced in [4]. Based on the magnetic flux density balance propriety, transformer can be classified into two types. the first are called the balanced-flux-magnetics (BFM) as it was invented by Faraday and conventionally used up to nowadays. The second are called the unbalanced-magnetic-flux magnetics (UBFM) as proposed in [4] and in which, the magnetic flux density is unbalanced. In an ideal case, where we assume perfect homogeneity in the magnetic core and negligible edge and eddy current effects, a BFM is characterized by a constant magnetic flux density throughout the entire core paths. As illustrated by the FEA simulation in Fig.1-a, the intensity of B remains uniform along the core length in a BFM if we neglect the corner and edge effects. This is because the cross section-to-flux ratio is constant. In contrast, a UBFM exhibits variable magnetic flux density, as depicted in the accompanying Fig.1-b. This variation is achieved by altering the core's cross-sectional area.

The theoretical results developed in [4] shows that this concept can have a significant impact to increase the power density. The results shows that the core loss can be decreased by 70% compared to an equivalent BFM design. This significant improvement in loss minimization has a great advantage to improve the thermal performance and increase the power capability of the magnetic core by several times. The registered maximum specific power density and the maximum power capability approximately increase by 2.7x and 7x compared to a conventional BFM design.

<figure><img src="image_2.png" /><figcaption>Fig.1. Principle of BFM (a) and UBFM (b)</figcaption></figure>

In the following section, we present the design method of the unbalanced-flux magnetics for a 1kW 400/48V LLC converter.
# Principle and Steps of the Optimization Method

The design of the unbalanced-flux magnetic core was carried out through two main steps. The first step, the optimal core was designed using Genetic algorithms and Pareto front concept. In the second step, the PCB integrated winding is optimized accordingly using an iterative approach.

To achieve high power density and high efficiency, designing custom cores is necessary because commercialized cores cannot allow to achieve high power density and high efficiency. Choosing the suitable variables to optimize the core is a challenging task for the designer and requires a good understanding of the optimization problem because of the interdependence between these variables between each other's and the nonlinear dependence of the objective functions on these variables.

The selection of the required variables should also be minimum to increase the results accuracy and reduce the convergence time of the optimization algorithm.

In our case, we opt for the following. The physical core variables that we aim to optimize are the diameter of the core central leg $D_c$, the core window-height $W_h$ and the core window-width $W_w$. From these three variables, we can derive the cross section, the window area, the core length and obviously the core volume. based on the results in [4], the unbalanced-flux condition of which the cross section of the outer parts is imposed to be twice the cross section of the center legs.

the objective functions can be expressed through these variables and through the intermediate equations. In our problem, we have optimized the transformer and the resonant inductor volume and loss. GOSET algorithm was used and its principle is described in Fig.1 [17].

<figure><img src="image_3.png" /><figcaption>Fig.2. Structure of GOSET algorithm [17]</figcaption></figure>

The necessary optimization variables are summarized in Tab. I.

<table><caption>Tab. I. Optimization variables</caption><thead><tr><th>Variables</th><th>Unit</th><th>type</th><th>interval</th></tr></thead><tbody><tr><td>D<sub>c</sub></td><td>mm</td><td>Real</td><td>[1-10]</td></tr><tr><td>W<sub>h</sub></td><td>mm</td><td>Real</td><td>[1-5]</td></tr><tr><td>W<sub>w</sub></td><td>mm</td><td>Real</td><td>[1-20]</td></tr><tr><td>D<sub>ci</sub></td><td>mm</td><td>Real</td><td>[1-10]</td></tr><tr><td>W<sub>hi</sub></td><td>mm</td><td>Real</td><td>[1-5]</td></tr><tr><td>W<sub>wi</sub></td><td>mm</td><td>real</td><td>[1-10]</td></tr><tr><td>N<sub>tp</sub></td><td>U</td><td>integer</td><td>4</td></tr><tr><td>N<sub>tpi</sub></td><td>U</td><td>integer</td><td>[1-8]</td></tr><tr><td>N<sub>pcb</sub></td><td>U</td><td>integer</td><td>[1-4]</td></tr><tr><td>l<sub>g</sub></td><td>mm</td><td>real</td><td>[]</td></tr><tr><td>T</td><td>C</td><td>real</td><td>[40-100]</td></tr><tr><td>C<sub>r</sub></td><td>nF</td><td>real</td><td>[1-100]</td></tr><tr><td>l<sub>gi</sub></td><td>mm</td><td>real</td><td>[0-2]</td></tr></tbody></table>

As stated above, the objective functions of the optimization problem are the transformer and resonant inductor loss and volume. The core loss model of the unbalanced-flux magnetic core is given by [14]:

$$P_{c\_unb} = k f^{\alpha} B_{m}^{\beta} V_{cc} + k f^{\alpha} B_{mo}^{\beta} V_{o\_unb} \quad (1)$$

$V_{cc}$ and $V_{o\_unb}$ are the volume of the central legs and the remaining parts respectively. $B_m$ is the magnetic flux density of the central leg and $B_{mo}$ is the magnetic flux density in the outer parts, and they are related by the following relationship.

$$\frac{B_{mo}}{B_m} = \frac{\frac{1}{2} A_c}{A_o} \qquad (2)$$

The core volume, which is the second objective function, is the product of the core cross section and the core volume calculated from the optimization variables.

The PCB winding designed was refined using an iterative approach and based on the designed magnetic core. The cross section, and the window area are given data and what remain is to optimize the switching frequency and the winding which includes the number of turns, the trace width and the PCB layer thickness taking into account the constraints imposed by the dimensions of magnetic core.

The algorithm starts by sweeping the switching frequency $f$ within the interval [200:1:400] kHz. This choice is made based on the optimal frequency obtained in the previous step. The primary number of turns is firstly calculated using Faraday's law as follows:

$$N_p = \frac{V_{in} D}{2 f A_c B_{max}} \qquad (3)$$

$B_{max}$ is the maximum magnetic flux density equal to 0.25 T and $A_c$ is the cross section of the center
legs. $B_{max}$ is fixed to 0.25 T because we predict that the designed magnetic core can operate at this range of B and f.

The secondary number of turns is known from the turn ratio relationship defined by the input/output voltage ratio.

$$n = \frac{N_p}{N_s} \qquad (4)$$

As the number of layers is unknown, the primary number of turns per layer is derived from the window width constraints as follows (5):

$$N_{t_{p}l_{p}} = \text{round} \left( \frac{W_{max}}{W_{p}} \right) \qquad (5)$$

Where $W_{max}$ is the maximum trace width given by the window width of the core and $W_p$ is the primary trace width which is derived by (6):

$$W_p = \frac{A_{wp}}{l_{th}} \qquad (6)$$

Where $A_{wp}$ is the cross-sectional area of the primary wire determined based on the maximum primary current and $l_{th}$ is the thickness of the PCB layer. This later is an optimization variable determined by iteration. In other terms, the optimization is performed for 4, 5 and 6 oz PCB-type.

For the secondary, the trace width is intuitively equal to $W_{max}$ and the total thickness of the secondary turn $l_s$ should be determined first to determine how many layers will be conducting in parallel:

$$l_s = \frac{A_{ws}}{W_{max}} \qquad (7)$$

Where $A_{ws}$ is the cross-sectional area of the equivalent secondary wire determined from the maximum secondary current. The number of layers for one turn can be calculated as follows:

$$N_{t_{s}l_{s}} = \text{round} \left( \frac{l_{s}}{l_{th}} \right) \qquad (8)$$

Where $N_{t_{s}l_{s}}$ is the number of layers for one turn. The number of layers for the primary and secondary can be calculated by the following two equations.

$$N_{lp} = \frac{N_p}{N_{t_{p}l_{p}}} \qquad (9)$$

$$N_{ls} = \frac{N_s}{N_{t_{s}l_{s}}} \qquad (10)$$

The total number of layers is the sum of the primary and secondary layers number and then the number of PCB can be determined.

Based on the previous data, the winding loss can be optimized for each frequency and for each PCB-type using iterative techniques.

The design constraints are the following:

* The thickness of the PCB must be less than the height of the core window, which is limited to 4 mm.

* PCB height < 4mm.

* The maximum width of the trace ≤3mm.

* The total number of layers ≤16.

* The thickness of one layer varies from 4 to 6 oz. The area of the trace.

In the following, we limit to present the results of the transformer as it is the one verified in experiments and because the inductor is not fabricated, and another commercialized one was used instead.

The shape and the dimension of the optimized transformer core is depicted in Fig.3. It has the following dimensions: core volume ($V_c$) is 2.17 cm³, box volume is 3.36 cm³ and footprint is 420 mm². It is in fact two integrated EE cores to make two transformers, where the primaries are connected in series and the secondaries in parallel in the secondary.

<figure><img src="image_4.png" /><figcaption>Fig.3. Prototype of customized transformer</figcaption></figure>

## Design Example

The LLC converter circuit is shown in Fig. 4 and its specifications are summarized in Tab. II.

<figure><img src="image_5.png" /><figcaption>Fig. 4 Studied LLC converter circuit</figcaption></figure>

<table><caption>Tab. II. LLC converter specifications</caption><tr><td>Input voltage</td><td>370-430 V</td></tr><tr><td>Output voltage</td><td>48 V</td></tr><tr><td>Output power</td><td>1 kW</td></tr><tr><td>frequency</td><td>[200:10:400] kHz</td></tr><tr><td>PCB layer thickness</td><td>[4 5 6] oz</td></tr></table>
<figure><img src="image_6.png" /><figcaption>Top layer: sec</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Second layer: pri</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Third layer: pri</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Bottom layer: sec</figcaption></figure>

<caption>Fig. 5 Transformer PCB winding design</caption>

The secondary of each transformer is connected to a full-bridge fast-recovery diode rectifier. Each transformer is composed by 8 turns in the primary and 1 turn in the secondary. The primary of the transformers are connected in series while the secondaries are connected in parallel. The winding is made of two 4-layers-4oz PCB. Fig.5 shows the winding design of the top-side PCB.

# Experiments and Verification

## Transformer test

The transformer was tested before the implementation in the LLC transformer. Two fundamental tests were performed. The open circuit test to measure the core loss and the short circuit test to measure the winding loss.

Fig.6 depicts the temperature variation as a function of B for different frequencies. As it can be noticed the transformer can operate at 0.25 T and f=350 kHz, without facing any thermal issue. the registered temperature at these operations conditions is 78°C. This demonstrates the advantage of using unbalanced-flux magnetics in reducing loss and enhancing thermal performance.

Fig.7 shows the experimental results of the short circuit test. The registered core temperature for a magnetizing current of 5 A is 65°C.

<figure><img src="image_10.png" /><figcaption>Fig. 6. Core temperature variation with respect to flux density B.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 7. primary current and voltage waveform in short-circuit test.</figcaption></figure>

Applying the superposition technique as understood from eq (11), the total temperature rise ($T_r$) of the transformer is the sum of the temperature rise due to core loss in no-load test and the temperature rise due to the winding loss in short-circuit test.

Considering an ambient temperature of 25°C, the maximum temperature of the core is calculated to be 94°C at 400 V input voltage and 5 A input current.

$$T_r = R_{th}(P_{core} + P_w) \quad (11)$$

The performance of the transformer can be improved using forced cooling. A fan delivering an air flow of 1.51 m³/min has reduced the transformer temperature from 78°C to 45°C in open-circuit test.
## LLC Converter test

The transformer was built in 1 kW 400/48V LLC transformer as shown in Fig.8. The half-bridge inverter is realized GaN switches (GS66508T) [18]. The full-bridge rectifier is realized using Schottkey diode devices (SDT30B100D1-13) [19]. The resonant inductor is 4.2 uH and the resonant capacitor is 50 nF.

<figure><img src="image_12.png" /><figcaption>Fig. 8. A prototype of the realized 1kW LLC converter.</figcaption></figure>

The converter was successfully tested to 750 W without heatsink. Fig.9-a shows the resonant currents and the primary voltage at different power levels. As it can be noticed, the converter is operating under soft switching at turn-on and turn-off.

<figure><img src="image_13.png" /><figcaption>Fig. 9. Primary voltage and resonant currents waveforms at different power levels: (a) transformer with air cooling, (b) transformer with heatsink and air cooling</figcaption></figure>

The transformer was also tested using a heatsink and the converter was run up to 1 kW. Fig.9-b shows the resonant currents and the primary voltage at different power levels.

Fig. 10 shows the thermal images at different output powers. The converter was operated continuously for 54 min: 15 min at 503 W, 10 min at 554 W, 8 min at 603.7 W, 5 min at 656 W and 8 min at 754 W. This test is repeated two times. The temperature at the different output powers were registered after verifying that the thermal equilibrium of the transformer is attained. A forced air cooling using a fan was used to cool down the transformer temperature.

Fig. 11 shows the thermal image of the converter at 1 kW using a heatsink for the transformer. The maximum temperature of the output rectifier is 93 °C registered in the diode-rectifier.

<figure><img src="image_14.png" /><figcaption>Fig. 10. Thermal images at: (a) 503 W, (b) 554 W, (c) 603.7 W, (d) 653.1 W, (e) 706.9 W, (f) 753.4 W.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 11. Thermal image of the converter at 1 kW, transformer with heatsink.</figcaption></figure>
Fig.12 shows the measured efficiency of the converter. The maximum registered efficiency without using a heatsink is 94.27% measured at 603.7 W. The peak efficiency is 94.6%, registered at 900 W. The efficiency is almost flat from 600 W to 1 kW.

<figure><img src="image_16.png" /><figcaption>Fig. 12. Measured efficiency up to 1 kW.</figcaption></figure>

Conclusion

This paper introduces a new design methodology based on the principle of unbalanced-flux distribution to enhance the power density and efficiency of LLC transformers. Theoretical analysis and experimental validation were conducted on a prototype transformer, achieving a power density of 224 W/cm³—3.5 times higher than the current average. There is significant potential to further increase power density. Additionally, this work provides a comprehensive review of existing studies, addressing key issues such as efficiency measurement for LLC transformers and high-frequency magnetic component design methods. The Unbalanced-Flux Magnetic (UBFM) concept represents a groundbreaking innovation in the design of high-frequency magnetic components. This revolutionary approach challenges the long-standing notion that magnetic components are inherently bulky in power electronics. For the first time, it enables power densities that exceed the average of other components within the system. Moreover, the UBFM concept is poised to introduce novel shapes to the magnetic components market, potentially reshaping existing standards and redefining industry benchmarks. Its transformative potential positions it as a key driver of progress in power electronics design.

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