A New Optimization Method for Gapped and Distributed Core Magnetics in LLC Converter

LLC converter design optimization remains a challenging task for varying loads such as in battery chargers. There are numerous L-L-C combinations to choose from a design space that can satisfy the required voltage gains of the application. An accurate magnetic model is essential to optimally size the passive components according to the application needs. This paper provides a new design tool for gapped core magnetics to optimize the transformer and resonant inductor in LLC converters. Unlike conventional design algorithms, the proposed algorithm considers multiple distributed cores and selects the optimal magnetic flux density by minimizing a penalty function that includes power loss, cost and volume of the magnetic components using Big-Bang Big-Crunch Algorithm. The gapped core transformer and inductor design equations have been verified in Ansys Maxwell and Simplorer co-simulation environment for a 3700 W 48 V LLC and calculated power loss have been compared with experimental results. For the given <inline-formula> <tex-math notation="LaTeX">$L_{m}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$L_{r}$ </tex-math></inline-formula> pair of <inline-formula> <tex-math notation="LaTeX">$37.52~\mathrm {\mu H}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$9.38~\mathrm {\mu H}$ </tex-math></inline-formula>, the proposed magnetic model designed a transformer with two-distributed cores, each one exhibiting a magnetizing inductance of <inline-formula> <tex-math notation="LaTeX">$17.73~\mathrm {\mu H}$ </tex-math></inline-formula> and leakage inductance of <inline-formula> <tex-math notation="LaTeX">$0.7~\mathrm {\mu H}$ </tex-math></inline-formula> on a EE422120 with 3F36 material. The total power loss of the transformers are measured as 12.44 W on a 3700 W prototype switched at 350 kHz.


I. INTRODUCTION
Light Electrical Vehicles (LEVs) cover a wide segment of the electric vehicle (EV) market including e-scooters, e-bikes, e-rickshaws, e-forklifts, e-motorbikes, etc. Due to the low power requirement, LEVs can be charged directly from the mains (i.e. any standard wall outlet) up to 3700W due to the low battery capacity and charging power requirement. Conventional LEV chargers generally have a two-stage power structure consisting of a power factor correction (PFC) and an The associate editor coordinating the review of this manuscript and approving it for publication was Jiann-Jong Chen .
isolated DC/DC power conversion stages [1]. In the DC/DC conversion stage, LLC resonant topology is widely used for converting high voltage to isolated and controlled low voltage in LEV battery chargers [2], [3]. The primary-side power switches achieve zero-voltage switching (ZVS) under entire load range when LLC converter is properly designed [4]. However, variable switching frequency control makes the selection and design of the magnetic components a difficult task, particularly for varying loads such as LEV batteries [5], [6]. Different magnetizing inductance (L m ), resonant inductance (L r ), and resonant capacitance (C r ) combinations can satisfy the necessary voltage conversion range but may VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ result in different operation characteristics and switching frequency range. Generally, the resonance frequency, and switching frequency range is chosen based on the designer's experience and characteristics of the used materials and semiconductor devices. There is no systematic method presented in the literature to choose the resonant elements that are optimized for given DC voltage range and load profile of LEV batteries. The backbone of this systematic approach undoubtedly requires a comprehensive magnetic model that 1) selects the right core material and geometry of the core considering operating flux density, saturation flux and thermal handling of the core from a given database 2) designs windings with optimized wire thickness considering AC losses, 3) modelling of the magnetic components in terms of power loss as the core and copper losses of the magnetic components contribute to a large portion of LLC converter losses [7]. Unlike conventional transformer design, the magnetizing inductance in LLC converter has to be precisely adjusted for the resonant tank to operate as expected. In order to obtain the required magnetizing inductance for ZVS, an air-gap is inserted into the transformer body on the flux-path [8].
To increase the power density of an LLC converter, the switching frequency is pushed higher to shrink the size of magnetics. Even though appropriate ferrite cores exibiting low core loss at high frequencies are preferred, there may still be considerable power losses on the core and the windings due to skin and proximity effects resulting in thermal management issues. In the literature, some researchers proposed topologies with distributed cores instead of using a single core, as illustrated in Fig. 1 [6], [17], [18], [19], [20], [21], [22]. With the distributed core structure, the copper losses and generated heat on the transformer are distributed equally among multiple cores. Likewise, the thermal burden on the secondary-side rectifiers are reduced. The details of the distributed core structure connected in series-parallel are provided in detail in [17] and [18]. In [19] and [20], it was shown that using two small transformers instead of a large one increased the power density of the converter by %24.28 [21], [22]. Substantial research and development efforts have been performed on the optimization of high frequency transformers [9], [10], [11], [12], [13], [14], [15]. The high frequency performance of different core materials and loss models are analyzed in [9]. In [10], the authors recommended a formula for the optimized foil or layer dimensions to estimate winding copper losses under any given arbitrary current waveform. In [11] and [12], the optimum number of strands of Litz wire is obtained by Ferreira and Dowell for slackly and closely packed wires. These studies provide in-depth analyses to estimate power losses and allow developing detailed loss models that are used in the magnetic design models. In [16], it is shown that using both analytical and finite elements aided design procedure is more effective for magnetic design optimization. The flux density and core losses are calculated for a single operating point in analytical methods, while they can be calculated for different operating points in finite elements method (FEM) calculations.
Although detailed studies on various aspects of coil and core design have been provided in the literature, there are very few studies that provides a multi-objective meta-heuristic optimization of high-frequency air-gapped magnetics [10]. The selection of optimal frequency range in LLC considering the transformer volume and weight has been studied in [23], and multi-objective optimization of high frequency transformers using genetic algorithms has been proposed in [24], where the core cross-section, window area product, and the power loss have been chosen as fitness functions while the frequency and maximum flux density have been selected as optimization variables. However, the impact of the stray components have not been considered. In [25], high frequency power transformer has been optimized using Non-dominated Sorting Genetic Algorithm II (NSGA-II) by taking the constraints, such as the core saturation flux density and winding structure, into account. However, only the power loss and leakage inductance have been considered as the fitness function. In [26], the optimization of the planar transformer of LLC converter using a metaheuristic method such as the Genetic Algorithm (GA), Ant Colony Optimization method (ACO), Shuffled Frog Leaping Algorithm (SFLA). According to the analytical design tool proposed in [13], more efficient planar transformers with lower power loss and core volume can be designed. This study, on the other hand, does not consider multi-core structure, different core materials and winding types, cost and thermal handling of the core.
This paper provides a systematic design method for air-gapped magnetics in an LLC converter based on a given core database by optimizing cost, volume and power losses. The presented method can be used as a magnetic optimization tool in a higher level system optimization framework for given L r and L m combinations that satisfy the required voltage conversion gain across whole load range. Different than the previous studies, the algorithm provides; 1) an optimization according to the assigned weights of cost, weight, and volume. The cost function includes core, winding and labour costs. The power loss function includes core loss, temperature dependent copper loss, and AC winding loss due to skin effect; 2) unlike conventional design method where optimal µ is assigned, it is found by sweeping the operating flux density for the cores in the given database that includes various cores with different material each one resulting in different loss coefficients for frequency and flux density; 3) not all operating flux density values are swept but a Big-Bang Big-Crunch algorithm has been used for optimization to cut down the computation time 4) both foil and Litz wires with interleaved structure have been considered; 5) distributed multiple cores at both primary and secondary sides have been considered; 6) analytical methods have been verified with FEM models. The proposed framework with distributed cores is essentially for high output current low output voltage applications. A 3700 W 400 V/48 V LLC converter has been considered as a case study and the results found by analytical and FEM methods have been compared. As a final verification of the inductance, thermal handling, and power loss estimation of the proposed magnetic model, a 3700 W LLC converter having a distributed core transformer with primary-series secondary-parallel structure has been designed.

II. OPTIMIZATION PROCEDURE FOR GAPPED CORE MAGNETICS
The aim of a gapped-core magnetics design optimization procedure is to determine the core size, core window width, magnetic flux density, current density, number of turns, wire size by means of a previously dened fitness function including power losses, volume, cost, etc. This can be regarded as a multi-objective optimization problem.

A. EFFECT OF BATTERY LOAD ON THE MAGNETIC DESIGN
The magnetic design of LLC resonant converter-based lithium-ion battery charger is significantly different from regular passive load applications because of nonlinear load I-V characteristics of battery. While magnetics are designed for a constant load requirement in the passive load case, the lithium-ion chargers are designed by considering the SOC state of the battery. The battery charging process generally consists of two main regions; namely, constant current (CC) and constant voltage (CV) regions. Four critical operation points are identified in the charging profile of Li-ion batteries labeled as ''beginning'', ''nominal'', ''turning'' and ''end''. The beginning point is where the charging power is minimum. The charging power increases in the CC region and reaches to its maximum at the turning point from where on the charging profile enters CV region, and charging power starts decreasing. Therefore, the switching frequency is different at each operating point. The charger starts with high frequency switching and lowers the frequency towards the turning point. Since the resonant current waveform is switching frequency dependent, which is variable throughout the charging profile, it is necessary to express the current as a function of the switching and resonance frequency. As the frequency decreases, the magnetization current increases. Since saturation may occur in the magnetic material, a magnetic design has been made for the point where the frequency is the lowest and the transmitted power is the highest. Afterwards, the power losses were calculated for each aforementioned critical operation point. The selection of the final design can consider the weights assigned to each critical operation point. In this study, the magnetic components have been optimized at the turning point, at which the charging power is maximum.

B. BIG-BANG BIG-CRUNCH ALGORITHM
The proposed magnetic model uses Big-Bang Big-Crunch algorithm to obtain the optimal flux density for a given core geometry and material properties across a wide flux density range across a large-sample design space. Big-Bang Big-Crunch optimization method gets inspired from one of the evolution theories of the universe that is allocated into two phases; named as the big-bang and the big-crunch [28]. Since randomness is thought to be equivalent to the energy distribution in nature, the population created in the Big-Bang phase is randomly generated. In this phase, the candidate solutions are uniformly scattered all over the search space. After the Big-Bang phase, the Big-Crunch phase begins. Since convergence to a local or global optimum point is considered as gravitational attraction, the center of mass is calculated in this phase. The term mass denotes the inverse of the fitness function value and is calculated using the following equation, where x i is a point of a randomly generated N-dimensional search space, f i is a value of the fitness function of x i point, N is the population size in Big-Bang phase [28].
The most suitable individual is calculated using (1). In the next step, new candidates are generated around this center of mass by adding or subtracting a random number, whose value is reduced as the iterations elapse. This process is expressed by (2), where l is the upper limit of the parameter, r is the normal random number, and k is the iteration step [28].
The resulting (X new ) new point is a point bounded from the upper and lower limits. All the iterations described in these steps continue until the stopping criteria are satisfied.

C. DESIGN METHODOLOGY FOR GAPPED CORE MAGNETICS
The design of the resonant elements in LLC is vital to achieve soft-switching. The most widely used method to design the resonant tank is the First Harmonic Approximation (FHA). L m , L r , C r , voltage gain (M ) and switching frequency range values are expressed by (3)-(6) using FHA method, where Q e is the quality factor, L n is the inductance ratio, f 0 is the resonance frequency, F is frequency rate (f s /f 0 ) and R e is the AC equivalent load resistance. During the design phase, the maximum frequency is limited not to exceed 1.8 times of the resonance frequency to avoid inefficient results.
As it can be seen from (7), frequency ratio and gain curve is a 6-root polynomial equation. In other words, there are 6 different frequency values that allow us to obtain a certain gain value. However, LLC converters must operate in the inductive region to provide ZVS, and the largest of these 6 different frequency values is the frequency value that we are interested in.
In order for proper magnetic design, the resonance rms and peak currents, magnetization peak current, and the secondary-side rms currents are required. The operating currents are changing with respect to the operation points of LLC namely, ''below'', ''at'' and ''above'' resonance operating conditions. Therefore, the current quantities need to be expressed in relation with the switching frequency. The necessary current expressions of LLC converter that are needed for the magnetic model are extracted in (8)- (12), where V o and I o are the output voltage and current, f s is the switching frequency and n is the turns ratio.
The flowchart of the proposed optimization algorithm for magnetic components of LLC is given in Fig. 2.
Step -1: The gapped magnetic design starts off with getting input parameters such as required inductance, allowed temperature rise, ambient temperature, heat transfer coefficient (h c ), window utilization factor (k u ) and fringing factor (γ ). For a given resonance frequency and load conditions, resonance frequency and load conditions, the switching frequency range that will provide ZVS can be determined by FHA. Using (8)-(12), the required current values for magnetic design are calculated. It must be noted that the proposed framework also considers the multi-core structure, where the magnetizing inductance is distributed equally among the cores in an attempt to use smaller cores to reduce the penalty function.
Step -2: A core database is created that stores the information of the core sizes and material information such as the operation switching frequency range, the saturation flux density, and the inductance per turn (A L ) information based on the air gap, etc. For given cores in the database, the first step is to check if the core is suitable for the given switching frequency range that satisfies ZVS and voltage gain across all operating conditions.
Step -3: Unlike classical methods where a pre-determined B 0 is selected, an algorithm based on the Big-Bang Big-Crunch method, as explained in Section II-B, determines the operating flux density without sweeping all the points of the design space, thus reducing the computation time.
In comparison to sweeping all the points, the Big-Bang Big-Crunch method saves about 36.7% computation time. This step is critical to cover a wide variety of cores as each core has a different loss-frequency characteristics. The upper bound of the population is limited to 65% of the B sat of the evaluated core in the design loop [30]. In this way, the optimal operating flux density for each core in the database can be found based on the given penalty function consisting of power loss, cost and volume with low computation effort.
Step -4: The current density is calculated for the cores that are provided in the population as given in (13), where A p is defined as the product of the core window area and the crosssection area, k a and k w are the dimensionless coefficients. ρ w is the resistivity of copper, and k u is the window utilization factor, and γ is the total losses excluding the dc copper loss. According to [10], the typical values of k a , k w , k u are 40, 10, 0.6, respectively, which vary in value for different core The cross-sectional areas of the primary and secondary (A w ) windings are calculated in (14)- (15), respectively. The AWG value closest to the calculated conductor dimensions is selected as the conductor diameter (r pri , r sec ), and the new current density values for the primary and secondary windings are updated for the cross-sectional areas of the selected conductors with standard AWGs.
A ws = I sec rms J 0 Step -5: The selection of the operating flux density directly affects the air gap and the number of turns. The optimal relative permeability for each flux density is calculated as in (16). The optimal relative permeability is derived from the relationship between core loss and dissipation. Afterwards, the air gap value (g) is calculated using the core winding length (l c ) as in (17). The empirical formula for A L − gap is provided in the datasheet of the manufacturers, and the A L value corresponding to the calculated air-gap value is determined for each core from this empirical formula. The number of primary turns is calculated as in (18) depending on the desired magnetization inductance, while the secondary turns are determined using the voltage conversion ratio.
The accurate estimation of leakage inductance in transformers is critical for LLC resonant converters because a well-matched resonant frequency is required [29]. The leakage inductance referred to the primary, can be found from the energy stored in the magnetic field as in (19), where a is the winding length, b is the winding build, c is the insulation thickness.
The core window length and width are defined as parameters in the core database in order to check whether the windings fit in the window area. The calculated winding diameters and number of turns that do not fit into the window area are wound in multi-layers with an equal number of turns. The windings VOLUME 11, 2023 are checked one more time to see if they are fitting inside the window area with this arrangement. By using calculated number of winding and conductor diameters, DC resistance can be calculated as in (20) where MLT is the mean length turn of the winding.
The skin effect, which causes an increase in the resistivity of the copper as the effective conductive area is reduced at higher frequencies, is determined by (21). It contributes to the AC resistance of the round wire as given in (22), where σ is the conductivity, and r is the radius of the copper.
R ac round = R dc · 1 + (r/δ) 4 48 + 0.8 · (r/δ) 4 (22) In high current and high frequency LLC converter applications, different winding types (such as foil, round etc. . . ) should be considered particularly at the secondary side. Using (23)- (27), the AC resistance of foil windings, optimal foil thicknesses and lengths are calculated. Here, opt is the coefficient, which is dependent on the current waveform and ψ is the AC resistance coefficient for p layers. d opt and l opt are the optimum foil thickness and length, respectively.
Step -6: After finding the AC resistances and current values, the copper and core losses are calculated by (28)- (29), where V e is the volume of core. The Steinmetz equation is used for core loss calculation and the C m , α and β coefficients were derived by using the P(W /mm 3 )−B(mT ) graphs for different frequency values given by the manufacturer data.
Lr rms (28) Step -7: In order to examine the effect of the losses of the magnetic model on the temperature rise, thermal resistance is calculated for each core by the empirical formula given in (30) for natural convection cooling, and temperature rise is checked by (31) to see if the core temperature rise stays within the given limits.
Step -8: A penalty function consisting of volume, cost and power losses are applied to each designed transformer and inductor. In order to include the cost in the penalty function, the winding Cu cost, material cost based on volume, and labor costs are extracted from different suppliers. Total core, winding and labor costs can be calculated as in (32)- (34), where σ core,x , σ wdg,x and σ lab,x are the specific costs per weight considering different magnetic core and the winding types.
fc core,x , fc wdg,x and fc lab indicate the fixed costs for the magnetic cores including the coil formers, connectors, and labor, where N stack represents the stacking factor. All numerical data retrieved from manufacturer is given in Table 1 [30].
Step -9: After obtaining the penalty function, the center of mass of randomly generated N-dimensional magnetic flux (B) space is calculated and the most suitable individual is chosen by (2).
Step -10: The new magnetic flux (B) candidates around the center of mass is calculated by adding or subtracting a random number.
The process is repeated until stopping criteria where the difference between the calculated values and the values that should be is the least has been met for each core and flux density, where the defined cost function is applied at each iteration. The most optimal core and design of the magnetics minimizing the cost function is selected at the end of the proposed framework.

III. OPTIMIZATION RESULT AND TRANSIENT MAGNETIC MODELLING
In this study, it is aimed to optimize the magnetics of a 400 V/48 V 3700 W LLC design in a LEV charger with a resonance frequency of 370 kHz, the magnetization inductance, resonant inductance, and resonant capacitor are provided as 37.52 µH, 9.38 µH, and 19.73 nF, respectively.
These parameters are fed to the proposed magnetics optimization model from a design space evaluated by a higher level converter optimization tool. The magnetic components are evaluated at 299.8 kHz switching frequency, which is swept by the higher level converter optimization loop. The magnetic components are designed based on the given L m and L r pair. It must be noted that the higher level converter optimization algorithm for the LLC converter is out of the scope of this paper. The remainder of the design specifications are given in Table 2.

A. RESULTS OF THE PROPOSED DESIGN ALGORITHM
The gapped transformer design steps are applied to a number of cores in the database.
The presented framework eliminates the cores that do not provide either the necessary magnetization inductance, thermal limit, B sat limit or frequency limits. This essentially eliminates too-small cores. In addition, too-large cores are also eliminated thanks to the fitness function. Afterwards, among the satisfying transformers, the design that provides the minimum value for the penalty function with assigned weights of 60%, 20%, and 20% to volume, power loss, and cost, respectively, is selected. The parameters of the designed transformers are given in Table 3.
As the volume in the cost function has been given a higher weight, the optimization algorithm selects two parallel 3F36-E422120 with a total volume 45400 mm 3 with 1.272 mm airgap, whose primary windings are connected in series, and secondary windings are connected in parallel. The total core loss is calculated as 9.74 W. Depending on the winding type used in the secondary side, the total Cu loss is calculated as 2.84 W and 2.7 W for round wire and Cu foil, respectively.
In favor of comparing the optimized result with the best result acquired in a single core approach, the optimization algorithm has been run by forcing it to find a solution for FIGURE 3. 3D drawings of transformers, a) single-core 3F36-E552825 round wire, b) two-core 3F36-E422120 round wire, c) single-core 3F36-E552825 Cu foil, d) two-core 3F36-E422120 Cu foil.
only single cores. The algorithm has chosen 3F36-E552825 as the best core with 8 turns and an airgap of 1.027 mm to get the required inductance. The core loss is 15.72 W, while the Cu losses are 1.674 W and 1.59 W depending on using round wire or Cu foil. As seen from the results, the algorithm chooses two of primary series connected and secondary in parallel 3F36-E422120 cores, as the total volume and total loss is smaller than the single 3F36-E552825 core when the assigned weight to volume is 60%.

B. VERIFICATION OF THE OPTIMIZATION RESULTS WITH FEM
In order to verify the model used in the optimization framework, the designed transformers given in Table 3 and LLC circuit model with the specifications given in Table 2 are co-simulated using Ansys Electronics Desktop and Simplorer environment. 3F36-E552825 with 1 parallel and 3F36-E422120 with 2 parallel transformers are selected for FEA and Simplorer Co-simulations. The 3D representations are illustrated in Fig. 3.
Two-dimensional (2D) transient magnetic model is used for FEA to compare the outcomes and operational quantities such as magnetic flux density, current density and losses. All simulations are performed Ansys Electronics Desktop, which uses the actual B-H curve of magnetic material that reflects high frequency saturation effects.
Due to fringing flux in the air gap of the transformer, eddy currents can flow into conductors in proximity. This fringing flux depends on the geometry of the air-gap and it can only be evaluated by FEA method. Moreover, the electric   field generated between primary and secondary windings cause common mode noise to flow through the transformer. The studies in the literature suggest that the power loss due to proximity effect and the primary to secondary winding capacitance can be reduced by interleaving the windings [31]. Interleaved windings can also reduce the leakage inductance. Because of this reason, the selected transformers are designed and analyzed by FEM with interleaving winding configuration.
The resultant magnetic flux distributions are presented in Fig. 4 for each transformer with round wires and copper foils. The most significant effects of high frequency operation are the fringing effect near the air-gap, proximity effect between intra-windings and inter-windings. Fig. 4 shows that the flux is more dense on the section of the core close to the winding window area, where flux path length is minimum, and gets crowded on the inner edge of the core as expected. Fig. 5 shows the magnetic flux lines and the current density distributions in the round and foil windings. As seen from the figure, the current gets crowded on the conductors near the air gap, and also on the surfaces, where primary and secondary windings are facing each other. Obviously, the eddy currents crowding these surfaces contribute to copper losses. FEA tools are used to calculate the copper losses due to the fringing effect and proximity effect.
The resultant magnetizing inductance in Ansys Electronic Desktop and Simplorer co-simulation is found as 35.644 µH, which matches closely with the target specification of 37.52 µH. The analytical and FEA calculation of copper and core losses are given in Fig. 6. According to Fig. 6, there is a slight difference between the FEA measurements and the values calculated from the optimization algorithm. The main reasons of this error can be summarized as follows; 1) the algorithm does not consider a dynamic BH curve, 2) the  fringing effects are more accurately modelled with the FEA tool, 3) there are frequency dependent uncertainties in core loss coefficients in the FEA tool. This error can be further reduced by using higher number of meshes and different mesh types; however, this would require an extensive computation time and processing power.

C. PROTOTYPE AND EXPERIMENTAL RESULTS
A gapped transformer based on the design results has been built to verify the presented methodology. The transformer is placed in the 3700 W LLC resonant converter. The prototype of LLC resonant converter is shown in Fig. 7.
According to the values obtained from the design algorithm, the L m , L k and L r values should be 37.52 µH, 1.3 µH, and 7.78 µH, respectively. Here, the loop inductance of the path was calculated approximately as 0.5 µH. However, there is an error margin of 5-10% in the designs of the manufacturers and the L m , L k and L r values have been measured as 350 kHz. around 350 kHz.
It is not easy to separate the magnetic losses from the overall converter losses during operation. To estimate the transformer power losses, the thermal resistances of the transformer windings have been extracted by passing currents through the primary and secondary windings from 0 A to 20 A at 1 A resolution. Then, the power loss and corresponding temperature data are recorded experimentally. In this way, the transformer power losses are estimated through observing the temperature rise. The curve fitted data for the thermal resistances of primary and secondary windings are given in Fig. 8.
In order to verify the power loss results generated by the proposed algorithm, the calculated core and winding losses are multiplied with the thermal resistances of the core, primary and secondary windings, respectively, as given in (35). Here, T c is the transformer temperature, T amb is   the ambient temperature, P pri loss is the primary winding copper loss, P sec loss is the secondary winding copper loss, P core loss is the core loss, R t pri , R t sec and R t core are the thermal resistances of the primary winding, secondary winding and core, respectively.
T c = T amb + P pri loss · Rt pri +P sec loss · Rt sec +P core loss · Rt core (35) The parameters obtained from the algorithm at different frequencies and load conditions are given in Table 4. At 340 kHz and 2.19 kW operation, the temperature rise of the primary winding, secondary windings and core are estimated as 6.23 • C, 9.74 • C and 17.6 • C, respectively, when the ambient temperature is 25 • C. Consequently, the temperature of the transformer is estimated to be 58.5 • C, which is quite close to the experimentally obtained temperature value of 58.7 • C as shown in the Fig. 9a. Similarly, the tempearture of the transformer is estimated to be 51.1 • C for 1.8 kW operation at 280 kHz as shown in Table 4. The experimentally recorded temperature for this case is shown in Fig. 9b. The results suggest that the output of the proposed algorithm are consistent with the experimental readings, which verifies the power loss equations and analysis incorporated in the proposed magnetic design algorithm.
In order to show the advantage of the proposed algorithm, the proposed optimization algorithm has been run with the design specifications of some of the reference designs and LLC prototypes published in journals [3], [32], [33], [34], [35], and magnetics volume results of the optimization algorithm have been compared in Fig. 10. As it can be seen from the comparison results, magnetics can be designed in smaller volume by using the proposed algorithm.

IV. CONCLUSION
This study proposed a comprehensive magnetics design approach for LLC resonant converter transformers and inductors. For given L r and L m pairs that satisfy the required voltage conversion gain over the entire load range, the ratio between the copper losses and core losses of magnetic components is optimized by minimizing the fitness function consisting of cost, power losses and volume under the thermal and magnetic restrictions such as saturation flux density, current density on the wires, etc. The algorithm then chooses the optimal core, air gap, and the winding design. Unlike conventional approaches, the proposed methodology sweeps the magnetic flux density through a metaheuristic optimization method named as Big-bang Big-Crunch, and considers distributed core structure and different winding types such as Litz wire, round wire, and Cu foils.
The proposed algorithm is used to design the transformer of a 3700 W 48 V LLC resonant charger in a LEV for a given magnetizing inductance of 37.52 µH. The penalty function has weights of 60%, 20%, and 20% assigned to volume, cost and power losses, respectively. The algorithm designed a transformer with distributed cores, in which the primary and secondary windings are connected in series and parallel, respectively. The results show that when the weight of the volume in the fitness function is higher, designs with smaller cores are chosen over single-core designs, as the thermal stress is smaller per core in comparison to a single core design. For this specific design, the algorithm chose two 3F36-E422120 cores and wound 8 turns with round wire on the primary side and 2 turns with 0.2 mm Cu foils on the secondary. Interleaved winding structure reduced the copper and core losses by up to 15% and 5%, respectively, as the proximity and fringing effect diminishes with interleaved winding structure. To verify the analytical equations, a 3-D model has been constructed and co-simulated by Ansys Electronic Desktop and Simplorer. The results closely match with the analytical model. Furthermore, a 3700 W 48 V LLC prototype has been built with the magnetics suggested by the developed algorithm, and the estimated power losses have been verified experimentally through monitoring the temperature rise of the transformer.
The algorithm has been forced to design a transformer with a single core. The model has designed 3700 W, 400 V/48 V 370 kHz resonant frequency. The designed transformer with a single core resulted in 39% higher power losses and 14.53% more volume in comparison to the optimized distributed core design. All in all, the proposed gapped core optimization method can be used as a part of a higher system level LLC converter optimization, where each viable L r and L m combination can be evaluated in order to find the most optimum LLC design.