Inter national J our nal of A pplied P o wer Engineering (IJ APE) V ol. 15, No. 1, March 2026, pp. 328 ∼ 351 ISSN: 2252-8792, DOI: 10.11591/ijape.v15.i1.pp328-351 ❒ 328 Constrained multi-objecti v e optimization of high fr equency transf ormer design f or dual acti v e bridge con v erter in solid state transf ormers using genetic algorithms J ayrajsinh B. Solanki 1 , Kalpesh J . Chudasama 2 1 Electrical Engineering, Gujarat T echnological Uni v ersity , Ahmedabad, India 2 Electrical Engineering Department, A D P atel Institute of T echnology , CVM Uni v ersity , Anand, India Article Inf o Article history: Recei v ed May 16, 2025 Re vised Dec 20, 2025 Accepted Jan 9, 2026 K eyw ords: Dual acti v e bridge con v erter GA-based HFT design High-frequenc y transformer Multi-objecti v e optimization Solid-state transformer ABSTRA CT This study presents a no v el multi-constraint and multi-objecti v e optimization based approach that applies genetic algorithms (GAs) for de v eloping high-frequenc y transformer (HFT) designs for dual acti v e bridge con v erters (D ABs) in solid-state transformers (SSTs). SSTs are incr easingly adopted in modern po wer systems due to their higher ef cienc y , compact structure, and impro v ed operational reliability when compared with con v entional transformers. De v eloping HFTs for SSTs in v olv es se v eral challenges, particularly the need to balance competing objecti v es such as impro ving ef cienc y , limiting losses, and reducing the area product while satisfying multipl e design constraints. T o address these challenges, this w ork applies a constrained multi-objecti v e GA implemented in MA TLAB to optimize the design of an HFT for a D AB con v erter . The me thodology allo ws for the simultaneous optimization of multiple design objecti v es while taking into consideration restrictions lik e ef cienc y , leakage inductance, temperature limits, core winding area, and sizes. Our comparison with particle sw arm optimization (PSO) indicates that the GA achie v es more consistent con v er gence and consistently lo wer total losses. The case studies reinforce this observ ation, gi ving compact and high-performance HFT designs tailored for SST applications. The optimization approach pro vides a reliable and scalable method for de v eloping thermally rob ust and space-ef cient HFTs suitable for ne xt-generation SST platforms and rene w able-ener gy applications. This is an open access article under the CC BY -SA license . Corresponding A uthor: Kalpesh J. Chudasama Electrical Engineering Department, A D P atel Institute of T echnology , CVM Uni v ersity Anand, Gujarat, India Email: ee.kalpesh.chudasama@adit.ac.in 1. INTR ODUCTION Modern po wer systems are shifting to w ard higher ef cienc y , reduced size, and more int elligent ener gy management, lar gely dri v en by the e xpansion of rene w able sources, electric v ehicle (EV) char ging infrastructure, and ongoing grid upgrades. Solid-s tate transformers (SSTs) depart sharply from traditional magnetic-core transformer designs and introduce a fundamentally ne w w ay of handling po wer distrib ution. The inception of po wer electronics and semiconductor materials enabl ed the de v elopment of SSTs, which can switch at high frequencies while maintaining ef cient ener gy management. SSTs are lighter and occup y signicantly less space than con v entional transformers. These transformers are suitable for electric v ehicle J ournal homepage: http://ijape .iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Appl Po wer Eng ISSN: 2252-8792 ❒ 329 char ging stations and rene w able ener gy systems. SSTs are increasingly proposed for EV f ast-char ging architectures due to their modular high-frequenc y isolated s tages and bidirectional control. Furthermore, SSTs of fer adv anced capabilities such as precise v oltage and po wer o w control, thereby enhancing po wer system reliability and ef cienc y . At the core of SSTs is the high-frequenc y transformer (HFT), which is responsible for both po wer transfer and electrical isolation [1]–[6]. Recent studies highli gh t the role of SSTs as smart transformers (STs) which is enhancing grid e xibility and microgrid performance. ST -based microgrids optimized through genetic algorithm s ha v e sho wn impro v ed v oltage re gulation, po wer quality , and f ault resi lience. These results indicate a broader shift from con v entional transformer operation to w ard more i n t elligent transformer systems in modern po wer systems [7], [8]. The y also underline the need for well-optimized high-frequenc y transformers as the main elements responsible for electrical isolation and po wer transfer in solid-state transformer architectures. Designing an HFT for ef cient high-frequenc y operation is not straightforw ard. Thermal beha vior and magnetic stability must be addressed simultaneously , which mak es the problem inherently multidisciplinary within po wer electronics [9], [10]. Figure 1 presents the three-stage conguration of a SST . It comprises: (i) an A C-DC rectication st age, (ii) a high-frequenc y isolated DC-DC con v ersion stage using a D AB, and (iii) a nal DC-A C in v erter stage. Con v entional distrib ution transformers sho w inherent limitations in v oltage re gulation, harmonic mitig ation, and the handling of bidirectional po wer transfer . These limitations are addressed in solid-state transformers through adv anced control strate gies. Operating at higher frequencies helps mitig ate these challenges while impro ving po wer quality and supporting stable grid interaction. Figure 1. Three-stage SST conguration [11] The performance of high-frequenc y transfor mers depends strongly on the materials and technol o gi es used in their construction. T ra nsformer ef cienc y and po wer density mainly depend on the core materials and winding design. Therefore, nanocrystalline cores are considered due to their superior magnetic characteristics that distingui sh them from more typical materials such as silicon steel and ferrite cores. High-frequenc y operation also af fects the performance, e xibility , and reliability of SSTs. It requires careful attention to insulation requirements and the leakage inductance of the HFT [12]–[14]. Additionally , accurate modeling of the phase-shifted full-bridge (PS-FB) ZVS DC-DC con v erter is critical in SST systems. Studies on isolated dual acti v e bridge con v erter (D AB)-based con v erter topologies report that leakage inductance and switching stress introduce measurable ef cienc y and cost penalties, particularly under high-frequenc y operation, which moti v ates the inclusion of e xplicit leakage and thermal constraints during the optimization stage [15]. A recent study highlights the adv antages of using system identication techniques o v er tradit ional a v eraging models, enabling more precise modelling by incorporating parasitic elements and impro ving dynamic performance prediction [16]. Design methodologies for HFT are being de v eloped to impro v e these aspects, maintaining that the transformers are capable of handling the requirements of high-po wer and high-frequenc y functioning for a SST . It is presented that controlling the leakage inductance inside the transformers is essential to ha v e zero v oltage switching (ZVS) without the need for e xtra inductors, thereby optimizing the size and weight of the HFT [17]. Soft-transition designs (e.g., zero v oltage transition (ZVT)) impro v e ef cienc y in non-isolated con v erters; in D AB-HFT stages, controlling transformer leakage to achie v e ZVS pro vides analogous benets without auxiliary inductors [18], [19]. There is a trade-of f between the cost of material, po wer density and thermal ef cienc y . High densi ty of po wer is essential because it reduces the transformer’ s size and weight of core and windings, which is especially useful in space-constrained situations. Higher po wer dens ities, on the other hand, generate more heat, which must be ef ciently handled to a v oid depreciation of transformer performance and its Constr ained multi-objective optimization of high fr equency tr ansformer design for ... (J ayr ajsinh B. Solanki) Evaluation Warning : The document was created with Spire.PDF for Python.
330 ❒ ISSN: 2252-8792 life (SSTs) [20]. Se v eral studies ha v e e xplored transformer optimization using genetic algorithms (GAs) and other e v olutionary techniques. Hoang and W ang [21] proposed a GA-based method for optimizing HFTs, b ut did not consider leakage inductance const raints. Similarly , Mogoro vic and Duji c [17] focused on medium-frequenc y transformer optimization b ut did not incorporate temperature rise constraints and thermal limitations. In contrast, our approach considers multi-objecti v e constraints lik e ef cienc y , leakage inductance, temperature rise, and winding area, which mak es it more suitable for HFT applications in SSTs. Hassan and Hameed [22] studied ho w core geometry af fects transformer ef cienc y us ing MA TLAB-based graphical tools for high-frequenc y transformer design. Their w ork sho ws that both c o r e shape and material choice need to be included when de v eloping optimization frame w orks. Ph ysical and technological constraints limit the de v elopment of HFTs for SSTs for the future smart grid. Thermal capability is a major concern. As operating frequenc y increases, losses in the core and windings raise the operating temperature. This issue is commonly observ ed during design iterations. Thermal management can reduce these ef fects, b ut it increases design comple xity and o v erall system cost. HFT design for SSTs in v olv es bal ancing electromagnetic performance, thermal limits, economic considerations, and implementation constraints. Olo wu et al. [23] proposed a multiph ysics optimization frame w ork, although t heir w ork is focused mainly on medium-frequenc y transformers used in dist rib ution systems. Extending such multiph ysics and multi-objecti v e optimization approaches to HFTs allo ws electromagnetic, thermal, and structural constraints to be treated in a unied design process for SST applications. Bahmani [24], [25] proposed geometric design optimization strate gies b ut lack ed mult i-objecti v e constraint handling. HFT optimization in v olv es carefully re gulating leakage inductance, winding topologies to reduce losses, and core choices based on Area product. These design impro v ements of transformer help to pro vide more ef cient ZVS operation, reduce alternating current (A C) losses to maximizes ef cienc y , and impro ving thermal performance [20], [21], [24]. This w ork presents the design of compact, optimized, and high-ef cienc y transformers for modern po wer electronics applications, such as D AB con v erters and high-po wer con v ersion systems, by incorporating adv anced optimization approaches in MA TLAB. The design of high-frequenc y transformers for solid-state transformer applications in v olv es strong coupling between electromagnetic, thermal, and material-dependent ef fects, which must be addressed simultaneously to ensure reliable operation [26]. Recent adv ances also e xplore AI-assisted or thermally coupled optimization for medium-frequenc y transformers [27], [28]. Building upon prior research in this area,the re vie we d studies demonstrate steady progress in the optimization of HFTs for po wer electronic con v erters. While earlier w orks such as W ang et al. [29] and Hern ´ andez et al. [30] established multi-objecti v e frame w orks using NSGA-II and NSGA-III with electromagnetic modeling, the y did not e xplicitly inte grate manuf acturing or thermal constraints. More recent ef forts, including Shi et al. [31] and Su et al. [32], incorporated articial intelligent (AI)-assisted or rob ust optimization strate gies b ut mainly addressed distrib ution-le v el or lo w-frequenc y designs. Hashemzadeh et al. [33] demonstrated a D AB-based SST e xperimentally , yet lack ed a generalized optimization procedure for transformer parameter synthesis. In contrast, the present w ork introduces a constrained multi-objecti v e GA frame w ork that simultaneously considers ef c ienc y , copper and core losses, leakage inductance, temperature rise, and windo w utilization. This unied approach links electromagnetic beha vior , thermal limits, and manuf acturability constraints within a single design o w . Combining these domains pro vides a clear path to w ard compact, high-ef cienc y ,high-frequenc y transformers for dual-acti v e-bridge stages in ne xt-generation solid-state transformers. Designing high-frequenc y transformers for solid-state transformer applications requires a clear understanding of the go v erning parameters and the constraints imposed by practical implementation. From the initial design stages, emphasis is placed on identifying the rele v ant parameters, go v erning equations, and design criteria, as summarized in section 2. This phase in v olv es careful selection of core and winding materials, detailed estimation of total HFT losses, and the use of appropriate optimization procedures. Section 3 then addresses the k e y design const raints that must be satised to obtain feasible solutions, including ef cienc y requirements, limi ts on leakage inductance, allo w able temperature rise, and winding area constraints for both primary and secondary sides, all of which directly inuence manuf acturabili ty and construction feasibility . The study subsequently e xamines the use of genetic algorithms for constrained multi-objecti v e optimization within a MA TLAB en vironment. The results and discussion section reports the simulation outcomes, including P areto-optimal fronts, which capture the inherent trade-of fs among k e y design objecti v es such as area product, ef cienc y , and total losses. Int J Appl Po wer Eng, V ol. 15, No. 1, March 2026: 328–351 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Appl Po wer Eng ISSN: 2252-8792 ❒ 331 The frame w ork is implemented in MA TLAB and v eried using case studies based on a dual acti v e bridge con v erter . Its performance is e v aluat ed through comparison with particle sw arm optimization during the v alidation stage to assess ef cienc y . The same optimization approach is applicable to modular solid-state transformer congurations used for rene w able ener gy inte gration, where po wer density and isolation requirements are critical. The major contrib utions of this paper are summarized as follo ws: – De v elopment of a constrained multi-objecti v e GA frame w ork that inte grates electromagnetic, thermal, and manuf acturing limits within a single optimization process. – Inclusion of leakage-inductance and temperature-rise constraints to ensure ZVS operation and thermal reliability . – Quantitati v e comparison between GA and particle sw arm optimization (PSO) methods under identical design objecti v es. – V alidation of the optimized design through MA TLAB-based D AB con v erter simulation demonstrating high ef cienc y and compactness. 2. DESIGN METHODOLOGY OF HIGH FREQ UENCY TRANSFORMER The method combines standard transformer design equations with a coordinated handling of k e y constraints. These include ef cienc y , leakage inductance, thermal limits, and winding windo w utilization within a genetic algorithm frame w ork. From the design iterations, the main inputs are the required po wer rating and the primary and secondary v oltage le v els. Core material properties, such as saturation ux density and permeability , along with insulation requirements, are also considered. The go v erning equations determine the number of primary turns, leakage inductance, core geometry , and indi vidual loss components. During optimization runs in MA TLAB, v ariables such as ux density , operating frequenc y , and current density are adjusted to meet the tar get performance. This approach balances electromagnetic objecti v es with practical constraints related to manuf acturability and thermal stability . F ollo wing the main criteria consideration while designing HFT [11]: (i) selection of material for the core and winding, (ii) winding arrangement, (iii) temperature rise considerations, (i v) magnetization and leakage inductance requirement, (v) transformer core loss calculation, and (vi) isolation requirement. Optimizing the HFT design criteria is necessary to achie v e high ef cienc y while reducing both core losses and po wer con v erter losses. The v alue of optimum ux densi ty B opt is compared with the v alue of saturation ux density B sat . So the increased v alue of B opt such that it does not af fect the ef cienc y of the transformer , b ut it can increase po wer density . Figure 2 depicts the optimized HFT design w orko w , starting from the denition of electrical and ma gn e tic input parameters and proceeding through constrained optimization, dimensional calculations, and ef cienc y v alidation. The sequence inc o r po r ates checks on loss minimization, thermal stability , and leakage inductance requirements to guide the selection of an HFT conguration that balances ef cienc y , reliability , and manuf acturability . 2.1. Cor e material and dimension Choosing an appropriate core material for high-frequenc y transformers requires a careful balance between achie v able ux density and o v erall ef cienc y under practical operating constraints. Nanocrystalline cores are widely reported in the literature because of their lo w core losses and high saturation ux density , which mak e them suitable for HFT applications [24]. In practical high-po wer designs, ho we v er , their use is often limited by manuf acturing cost and by the restricted range of a v ailable geometries, since these materials are commonly supplied in toroidal tape-w ound forms. Addressing these material and geometric constraints requires careful thermal management and appropriate wi nding congurations to support higher po wer densities while maintaining compact size. The area product ( A p ) is adopted as the primary parameter for core sizing and is e v aluated using established transformer design equations to satisfy both magnetic and thermal constraints. Empirical v alues of ux density and windo w utilization are applied during the core selection process, guided by manuf acturer datasheets. The area product ( A p ) is computed using the standard e xpression reported in [34]. A p =   √ 2 P V A k v f B 0 k t k f √ k u ∆ T ! 8 7 (1) Constr ained multi-objective optimization of high fr equency tr ansformer design for ... (J ayr ajsinh B. Solanki) Evaluation Warning : The document was created with Spire.PDF for Python.
332 ❒ ISSN: 2252-8792 Where A p is the area product (m 4 ), A c the ef fecti v e core cross-sectional area (m 2 ), A w the winding windo w area (m 2 ), P V A the apparent po wer rating of the transformer (V A), k u the windo w utilization (ll) f actor , k f the w a v eform f actor (e.g., 4 . 44 for sinusoidal EMF), B 0 optimum ux density (T), and f switching frequenc y (Hz). The (1) calculates the product of the windo w area and the cross-sectional area of the core, crucial for determining the ph ysical size of the transformer to handle the magnetic ux. Where k t is gi v en by (2). k t = s h c k a ρ w k w (2) This criterion is used to select a core from manuf acturer datasheets whose dimensions meet or e xceed the calculated area product. When a single core cannot satisfy this requirement, stack ed cores of fer a practical means of achie ving the required area product while accommodating the imposed electrical and thermal loading conditions. This approach adds e xibility to the design process and helps address limitations associated with standard core geometries and the cost of adv anced magnetic materials. The core material properties listed in T able 1 illustrate the dif ferences in loss characteristics among the a v ailable options. Figure 2. Flo wchart of optimization-based HFT design Int J Appl Po wer Eng, V ol. 15, No. 1, March 2026: 328–351 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Appl Po wer Eng ISSN: 2252-8792 ❒ 333 Both ferrite and nanocrystalline core options are e xamined during the design stage. Ferrite material s e xhibit lo w core losses at frequencies abo v e 20 kHz. Ho we v er , their relati v ely lo w saturation ux density limits the achie v able po wer density . Nanocrystalline materials pro vide higher permeability and higher saturation ux density . This allo ws a smaller magnetic v olume and more ef fecti v e utilization in the moderate frequenc y range of 10–20 kHz. Based on these trade-of fs, a nanocrystalline core w as selected for operation at 11 kHz, of fering a practical balance between compact size, acceptable thermal beha vior , and rob ust magnetic performance. T able 1 lists the Steinmetz parameters k , α , and β in mW/cm 3 . These parameters are normalized per unit core v olume. In the loss model as (3). P f e = k V c f α B β m , (3) k is interpreted as a v olumetric loss coef cient and V c (m 3 ) is the ph ysical core v olume, ensuring dimensional consistenc y . During optimization, the nanocrystalline parameter set ( k = 8 . 03 , α = 1 . 62 , β = 1 . 98 ) is emplo yed in all nal GA runs, while ferrite and amorphous data were used in preliminary sensiti vity checks to v erify material-dependent beha vior . This clarication aligns the units and conrms that all core-loss calculations are performed on a per -v olume basis. T able 1. Core material properties Material k (mW/cm 3 ) α β Ferrite 42.8 1.53 2.98 Silicon steel 278.4 1.39 1.80 Amorphous 46.7 1.51 1.74 Nanocrystalline 8.03 1.62 1.98 2.2. W inding material and arrangement W inding arrangements are optimized using litz or foil conductors, depending on current le v els. The design balances A C resistance with thermal handling and ll f actor . The winding windo w utilization is constrained belo w 60% to ensure adequate cooling and pre v ent insulation f ailure. T o ha v e proper design of high v oltage (HV) and lo w v oltage (L V) windings of HFT , a trade-of f between se v eral parameters should be considered: ef fecti v e utilization of winding area, lo w losses, proper electrical isolation, and good thermal beha vior . Lar ge currents require a lar ge conductor cross-section, b ut this leads to higher winding losses. Eddy currents at high frequencies produce a lar ge amount of core losses and will guide the design to w ard reduced core thickness. Subsequently , the primary number of turns N p is calculated as (4). N p = V p k v k f A c B max f (4) Where V p is the RMS primary v oltage (V), f the frequenc y (Hz), B max the peak ux density (T). As a result, litz and foil conductors are the ideal choice for winding conductors in HFTs. The construction of a litz conductor consists of indi vidual insulated wire strands twisted or braided together . The foil or litz wire conductors are used in the winding of HFTs to decrease eddy current loss. These insulated strands adjust all points in the cross-section of the litz conductor to spread the ux linkage and ensure uniform current distrib ution. Due to the high currents in the line, foil conductors are emplo yed in the lo w-v oltage winding of medium-frequenc y transformer (MFT)/ high-frequenc y transformer (HFT). Thin foil conductors are thought to reduce losses due to the skin ef fect. T o reduce proximity ef fect losses, the thickness of foil conductors should decrease as the number of layers increases. After selecting the core, the wire for HFT windings is chosen based on the optimal current density , estimated by GA optimization, as in (5). J o = K t s ∆ T 2 k u · 1 8 p A p (5) The v alue of current density J 0 and the primary and secondary currents in HFT are used to calculate the required primary and secondary bare conductor areas. Due to the skin ef fect, the conducti v e winding Constr ained multi-objective optimization of high fr equency tr ansformer design for ... (J ayr ajsinh B. Solanki) Evaluation Warning : The document was created with Spire.PDF for Python.
334 ❒ ISSN: 2252-8792 cross-section re gion must be considered. The diameter of the Standard W ire Gauge (SWG) wire should be less than the skin depth δ to minimize eddy current losses. The thickness of and equi v alent wire or conductors whose resistance is equal to that of a solid conductor under skin ef fect is kno wn as skin depth. The skin depth, also dened as ef fecti v e depth at which current density f alls to 1 /e of its surf ace v alue due to the skin ef fect, is gi v en by (6) [14], [34]. δ = r ρ π f µ 0 µ r (6) Where ρ is the resisti vity of the conductor ( Ω · m), f is the operating frequenc y (Hz), and µ 0 is the permeability of free space ( 4 π × 10 − 7 H/m). As a consequence, the type of wire SWG can be selected. Then the number of strands is calculated by required total area and selected area of strands with consideration of δ skin depth. It is possible to estimate the required cross-sectional area of the primary or secondary windings from the rated current as (7). A winding = I J 0 (7) Where J 0 is the current density . The strand diameter is then selected to limit the skin ef fect at the chosen switching frequenc y , as (8). d strand = δ 2 (8) Where δ is the skin depth. The strand diameter is chosen between 0 . 5 δ and δ to minimize eddy-current losses while maintaining manuf acturability . The cross-sectional area of one strand is as (9). A strand = π d strand 2 2 (9) The total number of strands required in the Litz b undle is as (10). N strand = A winding A strand (10) The total copper area inside the Litz b undle is as (11). A cu = N strand A strand (11) Ho we v er , due to inter -strand v oids, v arnish coating, and outer insulation, the actual b undle occupies a lar ger cross-sectional area A bundle , related to the copper ll f actor (packing ef cienc y) ν Litz as (12) [14], [34]. A bundle = A cu ν Litz (12) T ypical packi n g f actors for Litz wire range from 0 . 35 to 0 . 55 , depending on strand construction and twist pitch. The equi v alent outer diameter of the b undle, which determines the number of turns per layer , is as (13). D bundle = 2 r A bundle π (13) The number of turns per layer is gi v en by (14). N turns / la y er = W w D bundle (14) And the total number of layers required for each winding is as (15). N la y ers = N N turns / la y er (15) Where W w is the windo w width and N is the total number of primary or secondary turns. Int J Appl Po wer Eng, V ol. 15, No. 1, March 2026: 328–351 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Appl Po wer Eng ISSN: 2252-8792 ❒ 335 Finally , the total ef fecti v e winding area considering both primary and secondary coils is as (16). A w t = N p A bundle , p + N s A bundle , s (16) This formulation enforces the practical wi nding ll constraint ( A w t < 0 . 6 A w ), allo wing insulation thickness, v oid spaces, and Litz-wire b undle packing ef fects to be represented in a realistic manner . From a design standpoint, this constraint links the electrical requirements with thermal beha vior and manuf acturability . Generally it allo ws these three aspects to be addressed together within a mul ti-objecti v e optimization approach. F or dry insulation of the MFT in D ABs [35], the necessary minimum insulation distance between conductors is as (17). D i,min = V iso K iso E ins (17) Where V iso represents the required isolation v oltage le v el, E ins denotes the dielectric strength of the insulation material, and K iso corresponds to the safety mar gin specied by the manuf acturer . The core v olume V c and the winding v olume V w are dened as (18) [36]. V c = l m A c (18) Where l m = mean magnetic path length of the core (m). l m = 2( C w + 2 l ) + 0 . 8 × l w × (2 + π ) (19) Where C w denotes the cross-sectional width of the core, l represents the core cross-sectional thickness, and l w is the winding width for the C-core sheets. Based on the geometric characteristics of the high-frequenc y transformer , an empirical e xpression is used to estimate the core v olume as in (20). V c = W h + C w 2 + C w 2 A c + ( W w × 4) A c 2 + (4 × C w 2 ) A c 2 + (2 × W h ) A c 2 × 1 . 2 (20) Where C w = core width in m, W h = windo w height in m, W w = windo w width in m, C d = core depth in m. Figure 3 sho ws the winding structure and geometric layout of t he high-frequenc y transformer . This conguration i s used to control leakage inductance through adjustments in core dimensions, winding placement, and insulation spacing. These structural choices directly af fect ZVS operation, thermal beha vior , and the o v erall ef cienc y of the transformer . Figure 3. Cross-sectional schematic of HFT core with primary windings and secondary windings The mean length of a turn (ML T) is calculated as (21). M LT = ( C w + ( W w × 0 . 4) × 2) × 2 + ( C d + ( W w × 0 . 4) × 2) × 2 (21) A windings v olume V w is gi v en by (22) [36]. V w = M LT × W a (22) Constr ained multi-objective optimization of high fr equency tr ansformer design for ... (J ayr ajsinh B. Solanki) Evaluation Warning : The document was created with Spire.PDF for Python.
336 ❒ ISSN: 2252-8792 Copper loss is gi v en by (23). P cu = I 2 p R pac + I 2 s R sac (23) Where: I p , I s = primary and secondary currents. R ac = F R × R dc (24) Eddy current loss f actor as (25). F R = 1 + ( r 0 /δ ) 4 48 + 0 . 8 × ( r 0 /δ ) 4 (25) Where r 0 is the radius of the Litz wire of the primary winding and secondary windings. The eddy current loss f actor F R [34] is used in HFT design to account for the increase in ef fecti v e winding resistance due to the skin ef fect. Under DC conditions, the current is distrib uted uniformly across the conductor cross-section. At higher operating frequencies, the current concentrates near the conductor surf ace. This redistrib ution increases the ef fecti v e resistance and results in additional res isti v e losses. F R pro vides the connection between the ideal DC resistance R dc and the higher A C resistance R ac encountered during operation. From a winding design standpoint, this relationship is essenti al for limi ting losses, preserving ef cienc y , and maintaining reliable transformer performance under high-frequenc y conditions. DC resistance calculated as (26) [34]. R dc = N × M LT × r 20 [1 + α 20 ( T max − 20)] (26) Where: – N is the number of turns of primary and secondary windings – r 20 is the tab ulated resistance at 20 ◦ C in Ω / m for the selected wire (A WG/IEC) r 20 = ρ 20 / A cu ; the area is embedded in the wire table v alue [34] – A cu is the conductor cross-section (m 2 ) – ρ 20 is the resisti vity of the conductor Ω × m – α 20 is the temperature coef cient at 20 ◦ C – T max = T + 40 is the maximum operating temperature. A C resistance and proximity ef fects in Litz and l windings: The pre vious eddy-current loss f actor F R in (25) is v alid only for solid round conductors [34] and does not accurately capture proximity ef fects in multi-layer or litz windings. At medium-to-high frequencies, the current distrib ution in each layer is af fected not only by the skin ef fect (within each strand) b ut also by the magnetic eld of neighboring layers—the proximity ef fect. T o account for these, the A C resistance per layer R ac , i is e v aluated using Do well’ s method [37], which models the winding as N L stack ed layers carrying uniform current in the windo w height. F or a transformer winding of N T total turns arranged in N L layers, the normalized A C resistance f actor is as (27). R ac R dc = sinh(2 y ) + sin(2 y ) + 2 sinh 2 ( y ) + sin 2 ( y ) ( N 2 L − 1) 4 N L sinh( y ) cosh( y ) + sin( y ) cos( y ) , (27) Where y = t e p π f µ 0 /ρ , t e is the ef fecti v e layer thickness of one conductor plus insulation, f is the operating frequenc y , µ 0 is the permeability of free space, and ρ is the resisti vity of copper . The (27) is applied separately to the prima ry (Litz) and secondary (foil) windings to estimate the ir layer -dependent R ac /R dc ratios. The resulting correction f actors ranged from 1.06–1.12 for the Litz winding and 1.18–1.25 for the foil winding, which align with typical design curv es reported in [14], [34]. F or Litz wire, the A C loss is reduced because each strand’ s diameter d strand is chosen to be smaller than the skin depth δ ; ho we v er , proximity ef fects between b undles still occur . Hence, the b undle’ s ef fecti v e ll f actor ν Litz (ratio of copper to total b undle area) is included as (28). ν Litz = N strand π ( d strand / 2) 2 A bundle (28) Int J Appl Po wer Eng, V ol. 15, No. 1, March 2026: 328–351 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Appl Po wer Eng ISSN: 2252-8792 ❒ 337 Where A bundle accounts for v arnish, twist pitch, and inter -strand v oids. F or commercially a v ailable Litz constructions, ν Litz ranges between 0.35–0.55. The o v erall winding windo w ll f actor is e xpressed as (29). k u = A cu , tot A w = ν Litz N turns A bundle A w ≤ 0 . 6 , (29) Ensuring that insulation thickness, layer mar gins, and cooling clearances are respected. The copper ll fraction inside the winding itself is typically 60–70% of the b undle area after accounting for enamel and twisting; this corresponds to a total k u of about 0.5–0.55. Maximum ux density in the core is estimated by (30). B m = V p × D T N p × A c × f (30) Where: D T is a duty c ycle. Core loss: P f e = k × V c × f α B β m (31) T otal po wer loss: P T = P cu + P f e (32) The increase in temperature de grades the transformer’ s performance output by di sturbing its electric and magnetic parameters. Therefore, it is v ery important to conduct thermal analysis during the transformer’ s design stage and to estimate t he increase in temperature [38]. Another important constraint is the temperature rise ∆ T (°C) as a function of the surf ace area A t (in cm 2 ) and P T , the total po wer loss. 2.3. Thermal modeling and temperatur e rise estimation The transformer temperature rise results from the total po wer loss P T = P cu + P f e , which is dissipated primarily through natural con v ection and radiation. In practice, both copper and core losses contrib ute to localized heating in the windings and magnetic path, making thermal beha vior strongly dependent on geometry and cooling conditions. Although detailed FEM-based thermal models are possible, empirical methods remain reliable for ferrite and nanocrystalline cores operating under similar cooling en vironments. F ollo wing McL yman [14] and Orenchak [39], the temperature rise is estimated as (33). ∆ T = P T A t 0 . 833 , A t = k s p A p (33) Where k s (20)–(30) is a geometry f actor . The e xponent 0 . 833 is deri v ed from e xtensi v e e xperimenta l correlations on ferrite E-cores under natural con v ection [39]. The model has been v alidated to within ± 10% accurac y for ferrite transformers operating in the 20–200 kHz range [14]. F or the 150 W , 11 kHz D AB prototype considered here, it yields an estimated temperature rise of ∆ T ≈ 25 ◦ C , corresponding to a hotspot temperature of T hot ≈ 50 ◦ C . This v alue remains well belo w the Class B insulation limit, supporting the use of this model as a reliable thermal constraint in the d e sign process. Since transformer performance is go v erned by copper and core losses , the ef cienc y is e v aluated directly from these components as (34). η = P in − ( P cu + P f e ) P in % (34) Constr ained multi-objective optimization of high fr equency tr ansformer design for ... (J ayr ajsinh B. Solanki) Evaluation Warning : The document was created with Spire.PDF for Python.