Surrogate-based optimization of coupled inductors

The ongoing trend toward higher switching frequencies, increased functional integration and greater power densities has made it more challenging to achieve the necessary power ratings and electromagnetic compatibility (EMC) compliance in AC–DC power converters. Advancements into the sub-megahertz (MHz) switching frequency range and beyond (Mohammed and Jung, 2021) enable significant reductions in the size of inductive components, e.g. coupled inductors that use ferromagnetic materials (Rashid, 2024), thereby lowering costs and enhancing power density. However, higher operating frequencies lead to increased electromagnetic emissions within the conducted emission band (Paul, 2006), thus requiring additional filtering stages, such as common-mode chokes, to comply with regulatory limits. These additional components counteract the reductions in size and cost, creating an inherent trade-off between compactness and EMC performance. Addressing this trade-off is essential for developing competitive, high-performance power conversion solutions.

The literature presents a wide range of design approaches for inductive components in AC–DC power converters. In (Rashid, 2024) and (Cheng et al., 2022), analytical methods based on magnetic circuit theory are used to design inductive components, including the determination of core reluctance. Another methodology uses behavioral models derived from circuit simulations and fitted to transient measurement data, as shown in Oliveri et al. (2022). Finite element (FE) simulation-based approaches are reported in Abarzadeh et al. (2023) and Mirza et al. (2023), where the Ansys toolchain is applied to the design process. A comprehensive survey of modeling techniques with varying levels of complexity for magnetic components is provided in Górecki and Detka (2023). While these methods offer valuable insights, they primarily address functional parameters such as power ratings, saturation behavior and thermal performance. In contrast, this work considers essential functional aspects, as well as commercially relevant factors such as component volume and cost, to identify a set of feasible designs that balance performance with economic competitiveness.

The device under test (DUT) is a coupled inductor (CI), which is used in a multilevel interleaved power converter (Anderson et al., 2017) rated at 120 kW, with the converter topology shown in Figure 1. The CI uses a nanocrystalline toroidal core (Vacuumschmelze, 2025), and its geometric features are shown in Figure 2. The main inductance of each coil is denoted as $Lbase$⁠, while $Llong$ represents the longitudinal inductance (Boillat and Kolar, 2012). As shown in Zhang et al. (2010), the longitudinal inductance is beneficial for electromagnetic interference (EMI) suppression when used in an EMI filter. This inductance can be realized either by a dedicated inductor or by exploiting the leakage inductance of the two coils. The latter approach offers a significant advantage by reducing the required volume, decreasing the amount of ferromagnetic material needed and lowering costs. Thus in this work, the CI is designed such that the necessary longitudinal inductance is achieved through coil leakage.

A multiobjective optimization (MOO) strategy is used to find a design of the CI that meets the required power ratings (here 120 kW) while minimizing volume for cost-effectiveness. This optimization strategy considers functional parameters, such as the main inductance $Lbase$ and core losses, as well as the coil’s leakage inductances, to achieve the necessary $Llong$ for EMI filtering. To efficiently solve the MOO problem, analytical models of $Lbase$ and core losses (Möreé and Leijon, 2024) are combined with a surrogate model of the longitudinal inductance $Llong$ developed from 2.5D FE simulations.

The DUT is a toroidal-shaped coupled inductor, with its cross sections and input parameters $x=[x0,…,x5]$ shown in Figure 2. Assuming that the nanocrystalline core exhibits isotropic and linear magnetic behavior, and that the magnetic flux is uniformly confined within the core, the inductance $Lbase$⁠, can be approximated using Ampere’s law as follows (Hurley et al., 2025):

where $μ0$ is the permeability of free space and $μr$ is the relative permeability of the nanocrystalline core. The input parameters $x0$⁠, $x1$⁠, $x2$ are the inner core radius, core width and core height (see Figure 2), while $x3$ represents the number of turns of one coil.

The core losses are estimated based on Steinmetz’s equation (Möreé and Leijon, 2024) as follows:

where $P0$ is the nominal core loss provided in the material datasheet (Vacuumschmelze, 2025) for a given frequency $f0$ and magnetic flux density $B0$⁠. The corresponding values of $P0$⁠, $f0$ and $B0$ are summarized in Table 1. The parameter $Vcore$ is the core volume, defined as:

The parameter f is the switching frequency and B is the magnitude of the magnetic flux density within the core defined as follows (Hurley et al., 2025):

where $IT$ is the DUT’s transverse current defined as $IT(x)=I1−I2$ [see Figure 2(c)]. Figure 3(a) shows the transient switched node voltage $Vsw$ connected between the terminals $T1$ and $T2$ of the DUT and the resulting current $IT$⁠. Applying fundamental circuit theory (Hufschmied, 2020), $IT$ can be obtained as follows:

where $4Lbase(x)$ describes the transverse inductance of the DUT (assuming perfect coupling between the two coils) and T is the duration of the fundamental period of $Vsw$⁠.

Finally, the Fast Fourier Transform of $IT$ yields its spectral components, as shown in Figure 3(b). The Steinmetz equation in (2) assumes a sinusoidal excitation (Möreé and Leijon, 2024). However, Figure 3(b) illustrates that the excitation current exhibits a predominantly triangular waveform. Consequently, using the Steinmetz equation introduces an estimated error of approximately 10% (see  Appendix). Because the primary focus of this paper is the optimization workflow rather than high-fidelity core-loss modeling, this level of accuracy is considered acceptable.

To determine the longitudinal inductance $Llong$ of the DUT [see Figure 2(c)], 2.5D magneto-static FE simulations have been carried out in CST Studio Suite (Dassault Systèmes, 2025). A simplified coil model has been used, considering a homogeneous source current density across the coil cross section, such that the magnetic flux generated by the coil is proportional to the number of turns (Dassault Systèmes, 2025). The prescribed current direction, along with the resulting magnetic flux density and magnetic energy density, is illustrated in Figure 4. As shown, the transverse magnetic flux generated by the two coils within the core material effectively cancels each other out [see Figure 4(c)], resulting in a negligible main inductance $Lbase$⁠. Consequently, the longitudinal inductance can be determined as follows:

where $wmag$ is the magnetic energy density obtained from FE simulations [see Figure 4(d)]. The parameter $IL$ describes the excitation current and it is defined as:

For simplicity $IL=1A$ has been chosen.

The 2.5D FE simulation model used in this paper requires approximately 2 s (using 16 CPU cores on a HPC cluster) to compute $Llong$⁠, hence its computational complexity is very low, if only single input parameter combinations are of interest. However, when the FE model for $Llong$ is combined with analytical models for $Lbase$ and $Pcore,loss$ in the context of an MOO problem – where several thousand evaluations are typically required – using FE simulations becomes computationally impractical. To address this, a surrogate modeling approach is introduced in the following section to significantly reduce the computational cost of estimating $Llong$⁠, enabling the MOO problem to be solved within a feasible time frame.

As already shown in Figure 2, the design space explored in this paper consists of $n=6$ input parameters $x=[x0,…,x5]$⁠, where $x0,x1,x2$ describe various geometric features of the DUT, while $x3,x4,x5$ represent the number of turns, the thickness and the coverage of one coil. To generate the training data set for the surrogate model, Latin Hypercube Sampling (LHS) was applied using $m=500$ input parameter combinations, uniformly distributed across the ranges specified in Table 2. The sample size is selected according to the taxonomy of surrogate modeling methods recommended in Forrester et al. (2008). In addition, 50 test samples were generated using LHS to test the model’s accuracy.

A surrogate model is constructed using Gaussian Process Regression with an isotropic rational quadratic kernel defined as (Pedregosa et al., 2011):

where $d(xi,xj)$ is the Euclidean distance between two points, l is a length-scale parameter and $α$ is a scale mixture parameter. The values of $l=6.28$ and $α=0.00248$ are obtained using the L-BFGS-B optimization algorithm (Byrd et al., 1995) implemented in the Python SciPy library (Virtanen et al., 2020).

The obtained surrogate model achieves a mean-squared-error of only 0.04% for $Llong$ across 50 test samples, suggesting good agreement with the FE model, as shown in Figure 5. This level of accuracy is sufficient for the present study, because small estimation errors have a negligible effect on the intended behavior. In particular, because electromagnetic emissions are typically characterized on a logarithmic scale, and thus minor deviations do not significantly affect the overall predictive reliability of the model. Moreover, the surrogate model can estimate $Llong$ within milliseconds, reducing the computational cost by more than three orders of magnitude compared to the original FE simulations.

In the following section, this surrogate model for $Llong$ is combined with the analytical expressions for $Lbase$ from (1) and $Pcore,loss$ from (2) to formulate a MOO problem.

The objective of the MOO problem presented in this paper is to identify coupled inductor designs that meet the required constraints on power and longitudinal inductance while minimizing total volume. Therefore, the MOO is solved for:

subject to:

using NSGA-II, implemented in the Python pymoo library (Blank and Deb, 2020). The constraints on $Pcore,loss$ and $Jcu$ have been defined based on experience gained from previous preliminary prototypes. The constraint on $Llong$ was derived from system-level EMI simulations reported in Hussain et al. (2025). The geometric constraints on $x0$⁠, $x1$ and $x2$ were arbitrarily selected to exclude designs that are difficult to manufacture and to prevent the design space from expanding to impractically large configurations. The constraint $Jcu$ describes the maximal allowed current density within the coil and it is defined as:

where $k=0.5$ accounts for the coil’s fill factor. The parameter $IDC=90A$ is the maximum load current through the coil, determined by the maximum power rating of the converter (here 120 kW 3-phase multilevel interleaved power converter, see Figure 1).

The resulting Pareto front, together with the corresponding input parameters $x$ and the convergence behavior, is shown in Figure 6. The Pareto front was obtained using a population size of 5,000 individuals over 100 generations, with a total optimization time of approximately 7 min. As illustrated in Figure 6(a), a wide range of designs satisfies the optimization problem defined in (9)–(17), clearly demonstrating the trade-offs between core losses, volume and longitudinal inductance. The volumes of the Pareto-optimal solutions span from 22 to 311 cm³, indicating significant potential for space reduction in practical applications. Core losses range from 0.25 to 5.3 W, highlighting opportunities to reduce losses and thereby relax cooling requirements. The longitudinal inductance varies from 10 to 31.2 µH, demonstrating the potential to enhance EMI filtering performance when required.

Figure 6(b) shows the convergence behavior of the MOO, evaluated using the hypervolume performance indicator (Pedregosa et al., 2011). The hypervolume is computed based on objective values normalized to the range [0,1], enabling a scale-independent assessment of convergence. An increasing hypervolume value over successive generations indicates progressive improvement in both convergence toward the Pareto front and diversity among the solutions.

Finally, Figure 6(c) summarizes the scatter matrix of the input parameter combinations corresponding to the Pareto-optimal solutions shown in (a). As illustrated, multiple feasible solutions exist for the parameters $x0$⁠, $x1$⁠, $x2$ and $x5$⁠, whereas only marginal variation is observed for $x3$ and $x4$⁠. The main diagonal of the scatter matrix displays the probability density functions of the individual design parameters $x$⁠, while the off-diagonal elements depict the joint variation between pairs of parameters, thereby revealing correlations and dependencies within the Pareto-optimal set.

Three Pareto-optimal solutions from Figure 6 are summarized in Table 3, along with their respective input parameters $x$⁠. These solutions highlight the balance between functional performance and cost efficiency. For comparison, Table 3 also includes a reference design manufactured by VAC (Vacuumschmelze, 2025) with a power rating of 120 kW, enabling a comparative assessment of core losses, volume, longitudinal inductance and cost.

Finally, Table 4 compares the longitudinal inductance $Llong$ predicted by the surrogate model with the values obtained from the FE model. The relative error remains below 3.4%, confirming the surrogate model’s accuracy. To illustrate the differences in size and volume among the three Pareto-optimal solutions in Table 3, their corresponding FE models are shown in Figure 7, scaled by a factor of 4.

The design for minimal core losses (⁠$Pcore,loss=0.25W$⁠) achieves the lowest loss, but at the expense of a large core volume (⁠$Vcore=245cm3$⁠), approximately double that of the reference. The increased magnetic material reduces the core losses, yet drives up cost by 60% compared to the reference design. In contrast, the minimal-volume design (⁠$Vcore=22cm3$⁠) reduces the used material by an order of magnitude but results in substantially higher core losses (4.6 W). The maximal-inductance design yields the highest longitudinal inductance (⁠$Llong=31.2$ µH), more than twice compared to the reference design. Achieving this requires an even higher core volume (311 cm³), which again increases material cost by 70%.

The cost analysis highlights a clear trade-off: designs optimized for low losses or high longitudinal inductance demand greater material and manufacturing effort, while the compact configuration minimizes cost only by compromising magnetic efficiency and inductive capability. It is important to note that the core material was not included as a design variable in the optimization process, the relative permeability was therefore held constant. Future investigations may consider alternative core materials to further explore this design space.

In practice, the optimal solution is determined not only by application-specific constraints such as allowable temperature rise, geometric footprint and inductance requirements, but also by the manufacturability of the Pareto-optimal designs. Realizing the proposed configurations generally necessitates multiturn and multilayer winding arrangements, which are known to increase leakage inductance and reduce core utilization, thereby complicating the winding process and introducing additional parasitic effects. To ensure design reliability, a comprehensive winding model that accurately captures turn distribution and magnetic coupling effects is required. Moreover, winding capacitance has not been considered in this work. In high-frequency operation, this parasitic element can significantly influence resonant behavior and EMI characteristics. Similarly, the thermal behavior of the designs is not yet included; because higher core losses or compact configurations can result in elevated temperatures, future work should include detailed thermal–electromagnetic coupling to evaluate practical limits.

The choice of conductor type has also a significant influence on both manufacturability and electromagnetic performance. In this study, a defined filling factor is assumed to represent the effective packing density of the windings. In practice, however, different designs may use alternative winding technologies – such as round-wire windings (potentially connected in parallel), foil windings or shell-type configurations – each characterized by distinct filling factors. These variations directly affect achievable packing density, thermal dissipation and AC loss behavior. The volume footprint of the designs also remains a critical consideration for integration into constrained assemblies. While the minimal-volume solution achieves the smallest footprint, its higher losses may necessitate additional cooling or spacing, offsetting the initial size advantage. Consequently, the results presented provide a quantitative and practical basis for selecting a balanced design, while emphasizing the need to extend future optimization studies to include winding layout, parasitic capacitance and coupled thermal–electromagnetic behavior for a more comprehensive assessment of performance and reliability.

This paper has presented a multiobjective optimization framework for the design of a coupled inductor used in multilevel interleaved power converter, targeting minimization of core losses (power density constraints), volume (cost considerations) and maximization of longitudinal inductance (EMI performance). The approach combines analytical considerations with a surrogate model trained on 2.5D FE simulations, to significantly reducing the computational burden of the optimization process. The resulting Pareto-optimal designs were validated against FE simulations and demonstrated a relative error of less than 3.4%, thereby confirming the accuracy of the surrogate model. Overall, the proposed framework supports application-specific inductor optimization and can be readily extended to include manufacturability and cost models, thereby bridging performance-driven and economically constrained design objectives.

The authors would like to express their sincere gratitude to Vacuumschmelze for providing the material data for their nanocrystalline cores and for sharing their valuable expertise. This work has been jointly supported by Vacuumschmelze GmbH & Co. KG, Hanau, Germany, AVL List GmbH, Graz, Austira, CBMM Technology Suisse SA, Geneve Switzerland and by Silicon Austria Labs (SAL), owned by the Republic of Austria, the Styrian Business Promotion Agency (SFG), the federal state of Carinthia, the Upper Austrian Research (UAR), Industry and the Austrian Association for the Electric and Electronics (FEEI).

Error of the Classical Steinmetz Equation for Triangular Excitation.

The classical Steinmetz equation (SE) expresses the specific core loss $Pv$ under sinusoidal flux excitation as Arruti et al. (2023): 

where f is the excitation frequency, $B^$ is the peak flux density and $kSE$⁠, $α$ and $β$ are empirical Steinmetz parameters obtained from sinusoidal loss measurements.

The improved generalized Steinmetz equation (iGSE) extends this model to arbitrary periodic waveforms $B(t)$ with period TArruti et al. (2023): 

where $ΔB=Bmax−Bmin$ is the peak-to-peak flux swing and $ki$ is an effective loss coefficient that is related to $kSE$⁠, $α$ and $β$ by enforcing that (19) and (20) coincide for a pure sinusoidal waveform. Let us consider a sinusoidal flux:

with peak-to-peak swing $ΔB=2B^$⁠. Its time derivative is:

Substituting this into (20) yields:

With the change of variables $θ=ωt$⁠, $T=2π/ω$⁠, we obtain:

where:

Thus:

Requiring consistency with the classical SEequation (19):

gives the relationship:

Next, we consider a symmetric triangular flux waveform with the same period T and peak value $B^$ as the sinusoid. Over one period, $Btri(t)$ ramps linearly from $−B^$ to $+B^$ in $T/2$ and back to $−B^$ in the remaining $T/2$⁠, so the magnitude of the slope is constant:

and the peak-to-peak swing is again $ΔB=2B^$⁠. Inserting this into (20) yields:

Using $4α=22α$⁠, this simplifies to:

Substituting $ki$ from (29) gives:

Therefore, the ratio between the iGSE prediction for triangular excitation and the SE prediction (which corresponds to sinusoidal calibration) is:

where $Pv,SE=kSEfαB^β$ as in (19). Note that the exponent $β$ cancels out in the ratio; the deviation depends only on $α$⁠.

For the parameters used in this work, $α=1.51$ and $β=2.4$⁠, the constant $Cα$ defined in (26) evaluates numerically to:

Inserting this into (37) yields:

Hence, for a symmetric triangular flux waveform with the same peak value $B^$ and frequency f as the sinusoidal excitation used to identify the Steinmetz parameters, the classical SE overestimates the core loss by approximately:

This shows that, for $α=1.51$ and $β=2.4$⁠, the modeling error introduced by using the classical SE for symmetric triangular excitation is on the order of $10%$⁠, with the SE providing a conservative (slightly pessimistic) estimate of the core loss.