Accurate Calculation of High-Frequency Transformer Leakage Inductance Based on Dowell’s Model and Analysis of Influencing Factors

Abstract

The leakage inductance of high-frequency transformers (HFTs) is a critical parameter affecting the performance of power electronic equipment, such as DC-DC converters. During the transformer design phase, by precisely calculating and retaining an appropriate amount of leakage inductance, the independent inductors in the original topology can be replaced, thereby reducing the size of the converter. This paper derives the analytical expression for the magnetic field distribution in the core window, based on Dowell’s one-dimensional model system. A leakage inductance calculation model is established using the square integral of the magnetic field strength. The study investigates the effects of winding spatial distribution and operating frequency on leakage inductance. Finally, the model’s accuracy is verified through experimental measurements, with the results aligning closely with those obtained from the analytical method, and the error falling within a reasonable range.

1. Introduction

HFTs play a significant role in grid regulation, power quality improvement, and renewable energy integration. With the growing integration of large-scale photovoltaic (PV) plants, battery energy storage systems (BESS), and wind energy conversion systems (WECS) into high-voltage direct current (HVDC) networks, the demand for efficient and reliable grid interfaces has become increasingly critical. These systems impose stringent requirements, including galvanic isolation, fault ride-through capability, and wide-range voltage conversion [1,2,3,4]. In this context, high-performance isolated DC-DC converters, such as modular multilevel DC-DC converters (DC-DC MMC) based on dual active bridge derivations, have emerged as key solutions. The high-frequency transformer serves as the heart of these converters, where its leakage inductance directly influences the system’s power transfer capacity, efficiency, and dynamic performance, underscoring the vital importance of the accurate design and control of this parameter.

It is important to note that the nonlinear characteristics of magnetic components are a key factor limiting the performance of high-frequency power converters. This is evident not only in the high-frequency transformers of the isolated converters studied in this paper, but also significantly affects the power inductors in non-isolated DC-DC converters [5,6]. Therefore, accurate modeling and measurement of magnetic component parameters are prerequisites for optimizing the performance of various converter topologies, from non-isolated to isolated configurations. The precise calculation method for high-frequency leakage inductance proposed in this study provides an important experimental basis for such accurate modeling.

Compared with traditional low-frequency transformers (LFTs), HFTs offer the possibility of miniaturization and a high efficiency of electrical equipment with their lighter weight, more compact size, and higher power density, and they are the core components for the highly reliable operation of modern power systems [7,8,9]. However, HFT leakage inductance leads to higher voltage stresses and additional switching losses in switching devices, which is a key parameter affecting the efficiency and reliability of converter operation.

The research on the design and precise measurement methods of HFT leakage inductance presented in this paper is strongly oriented towards practical engineering applications. Its significance is particularly evident in advanced resonant or phase-shifted converter topologies, such as the dual-active-bridge (DAB) converter. As illustrated in Figure 1, the leakage inductance (L_σ) of the transformer serves as a crucial element for the power transfer and soft-switching realization in the DAB converter. H1 and H2 represent two H-bridge circuits. Conventional design approaches often treat leakage inductance as a parasitic parameter to be minimized, or rely on a separate, bulky resonant inductor. In contrast, modern high-frequency, high-power-density design philosophies advocate for magnetic integration—that is, the proactive and precise design of the transformer’s leakage inductance to a specific value that meets the converter’s requirements, thereby directly replacing the discrete resonant inductor [10,11,12,13]. This approach can significantly reduce the number, volume, and cost of magnetic components, highlighting the practical value of accurate leakage inductance modeling. Building on this, the leakage inductance of an HFT exerts multi-faceted influences on the converter design, not only directly affecting the system efficiency and stability but also playing a key role in circuit topology selection and dynamic response characteristics. For example, in resonant converters such as the LLC topology, leakage inductance is often intentionally utilized as an essential series inductor, eliminating the need for a separate inductive component. This configuration helps simplify the circuit structure, while the interaction between leakage inductance and resonant capacitance contributes to a higher power density and reduced manufacturing costs [14,15].

In flyback converter applications, the HFT leakage inductance will lead to current waveform distortion and increase the harmonic content, so it is necessary to reduce the leakage inductance, thereby reducing the voltage spikes caused by the switching device at the time of switching off, reducing the switching losses, and improving the operating efficiency of the converter [16]. Therefore, accurate calculation of the leakage inductance of HFTs is crucial for the design and performance optimization of various converters, which not only affects the efficiency and stability of the converter, but also relates to the electromagnetic compatibility of the circuit [17].

The calculation methods of HFT leakage inductance include numerical and analytical methods [18]. Among them, the numerical method can accurately calculate the leakage inductance value, but it takes a long time to solve the analytical model constructed by the numerical method [19,20]. In order to reduce the consumption of computational resources, the actual physical model can be simplified, such as by assuming a linear material or an equivalent uniform medium, but it affects the accuracy of the results. The analytical method is a calculation method based on electromagnetic field theory, which solves the leakage inductance by analyzing the geometrical structure and electromagnetic characteristics of the transformer windings. However, for the high-frequency and high magnetic density characteristics of the transformer, the traditional analytical method neglects the decrease in high-frequency leakage inductance due to the magnetic field inhomogeneity caused by the eddy current effect and proximity effect, which affects the accuracy of the calculation [21]. In [17], the high-frequency effect is considered comprehensively, and the leakage inductance is calculated by introducing the Helmholtz differential equation and analyzing the magnetic field strength and current distribution at a high frequency in detail. At present, the analytical method is used to calculate the leakage inductance of high-frequency transformers that are mostly based on the energy method, which means that the leakage inductance is obtained by solving the magnetic field energy in the transformer window. In [22], based on Dowell’s one-dimensional magnetic field assumption, a leakage inductance calculation method for foil windings is proposed to improve the calculation accuracy by considering the nonlinear distribution of magnetic field strength and current density at high frequencies. In [23], for the evaluation of leakage inductance parameters of round conductor transformers, the area equivalence method is adopted, and the wire structure of round conductor transformers is regarded as a copper foil, which realizes the extension of the analytical model of leakage inductance of traditional transformers with copper foil windings, and its applicability is significantly enhanced.

Most of the current studies on leakage inductance focus on high-frequency transformers with foil windings or round conductor windings [9,23], while few references address high-frequency transformers with Litz wire windings [24,25]. Ref. [24] converted Litz wire windings to foil windings, based on the principle of area equivalence, in order to apply Dowell’s frequency-dependent leakage inductance model [22], and then indirectly calculated the leakage inductance. However, this area-equivalent method overestimates the high-frequency effect and ignores the influence of the Litz wire filling factor, resulting in a small value after the equivalence. Reference [25] proposed a method to accurately calculate the leakage inductance of a HFT over a wide frequency range, which takes into account the filling factor and high-frequency effect of the Litz wire; however, the method ignores the magnetic field energy in the irregular region between the turns of the Litz wire and the internal proximity effect of the Litz wire. As proposed in the ref. [18], a non-area-equivalent analytical method, which takes into account the magnetic field energy in the part of the inter-turn irregular region of a circular Litz wire, analyzes the two-dimensional magnetic field inside the window, and indirectly calculates the leakage inductance of the HFT.

This paper focuses on the study of HFT leakage inductance as a key parameter, based on Dowell’s one-dimensional model, to derive the analytical expression of magnetic field distribution in the core window, to establish the leakage inductance calculation model, to explore the influence of winding space distribution and operating frequency on the leakage inductance, to enhance the accuracy of the calculation through the introduction of the Rogowski coefficient, and to validate the model’s accuracy with experimental measurements, so that it can provide a theoretical basis for an accurate calculation of the leakage inductance of HFTs. It provides a theoretical basis and method for the accurate calculation of the leakage inductance of HFTs and helps to optimize the design of HFTs and improve their performance and reliability.

2. Leakage Inductance Calculation Model

The leakage inductance of the HFT exists mainly in the core window, and its corresponding leakage energy is also distributed in this region. Specifically, the leakage energy exists not only inside the winding but is also distributed in the winding interlayer insulation and the primary and secondary side winding between the main insulation. This leakage energy is closely related to the magnetic field strength. In HFTs, the windings are affected by the skin effect and the proximity effect, which generate eddy currents. The distribution of these eddy currents is affected by the frequency and winding structure, which in turn change the magnetic field distribution in space.

As shown in Figure 2, d₁ and d₂ denote the thicknesses of the primary and secondary windings, respectively. d_ins1 is the thickness of the interlayer insulation of the primary winding, and d_iso is the thickness of the insulation between the primary and secondary windings. The values of each parameter are shown in

Table 1. Parameters of model.

Parameter	Value
d1	4 mm
d2	4 mm
dins1	2 mm
diso	4 mm

. Assuming that the permeability of the core material is infinite (μ_r → ∞, σ → 0), from Figure 2, it can be observed that the magnetic field strength in the winding varies with frequency, and the higher the frequency, the smaller the magnetic field strength. However, the magnetic field strength between the insulation layers basically does not vary with frequency. Therefore, the core window is divided into three parts to calculate the leakage inductance, including the following: the winding part, the interlayer insulation part within the primary and secondary windings, and the insulation part between the primary and secondary windings.

The distribution of the copper foil windings is shown in Figure 3, and this paper calculates the magnetic field strength in the insulation and copper foil windings based on Dowell’s model. To simplify the calculation, it is assumed that the height of the winding and the core window are equal and the end effect is ignored. At the same time, the magnetic field strength, H, has no horizontal component, and the direction of the electric field, E, is perpendicular to the xy-plane inward—that is, the z-axis direction—and there is no magnetic voltage drop within the xy plane. The two fields are divergence-free inside the conductors, and so it follows that the E and Hare functions are of x only [17]. Therefore, the leakage inductance calculation is performed under the assumption of neglecting displacement currents. This simplification introduces an error, which remains within an acceptable range of less than 5%.

This is obtained from Maxwell’s system of equations:

$$\nabla \times E=-{\mu }_0{\displaystyle \frac{\partial H}{\partial t}}$$

(1)

$$\nabla \times H={\sigma }_{w}E,$$

(2)

where σ_w is the electrical conductivity of copper and μ₀ is the magnetic permeability in vacuum.

In practical engineering, each layer of copper foil winding usually consists of multiple narrow copper foils, invoking Ampere’s law for the closed loops 1 and 2 in Figure 3 in a high permeability core (μ_r →∞, σ → 0). Assuming that the primary winding in Figure 3 consists of N₁ narrow copper foil windings per layer, and the secondary winding consists of N₂ narrow copper foil windings per layer, the n₁-th layer of the primary winding boundary magnetic field strength is as follows:

$$H(x_{\mathrm{in}}^{n_1})={\displaystyle \frac{m_2{I}_2{N}_2-(n_1-1)I_1{N}_1}{h_{\mathrm{wl}}}},$$

(3)

$$H(x_{\mathrm{out}}^{n_1})={\displaystyle \frac{m_2{I}_2{N}_2-n_1{I}_1{N}_1}{h_{\mathrm{wl}}}},$$

(4)

At this time, the magnetic field strength inside the primary side winding of the n1-th layer and the magnetic field strength inside the secondary side winding of the n₂-th layer are shown in Equations (5) and (6) [17].

$$H^{n_1}\left(x\right)={\displaystyle \frac{H(x_{\mathrm{out}}^{n_1})\sinh \left[\gamma (x-x_{\mathrm{in}}^{n_1})\right]+H(x_{\mathrm{in}}^{n_1})\sinh \left[\gamma (x_{\mathrm{out}}^{n_1}-x)\right]}{\sinh (\gamma d_p)}},$$

(5)

$$H^{n_2}(x)={\displaystyle \frac{H(x_{\mathrm{out}}^{n_2})\sinh \left[\gamma (x-x_{\mathrm{in}}^{n_2})\right]+H(x_{\mathrm{in}}^{n_2})\sinh \left[\gamma (x_{\mathrm{out}}^{n_2}-x)\right]}{\sinh (\gamma d_{\mathrm{s}})}},$$

(6)

$$\gamma =\sqrt{j\omega {\mu }_0\sigma }={\displaystyle \frac{1+j}{{\delta }_w}},$$

(7)

where δ_w is the skin depth and ${\delta }_w={\displaystyle \frac{1}{\sqrt{\pi f{\mu }_0\sigma }}}$.

The variation in the magnetic field strength inside the copper foil winding with the distance inside the window is shown in Figure 4. In the figure, the brown section represents the primary winding, and the green section represents the secondary winding. In the paper, the model was meshed into 67,994 elements, with a total of 543,952 nodes. Using second-order Lagrange interpolation polynomials, the accuracy of the numerical method is within 5%. The magnetic field strength first decreases and then increases with the increase in the inside distance of the window. The magnetic field strength inside the copper foil winding, calculated by equations, is consistent with the trend of the finite element analysis results. At 10 kHz, the difference between the calculated and simulated magnetic field strength in the middle of the winding is 0.7 A/m, while the difference between the calculated and simulated magnetic field strength in the middle of the winding at 50 kHz is only 0.2 A/m. This is due to the increase in frequency that is caused by the increase in the skin effect of the winding, and the internal magnetic field becomes smaller, so that the error between the calculated value and the simulated value decreases.

The leakage inductance size was calculated with the HFT leakage inductance energy through the primary and secondary winding leakage energy, the primary and secondary interlayer insulation part of the leakage energy, and the primary and secondary insulation leakage energy within the main insulation layer of the superposition of the three parts of the solution, and then based on the leakage energy leakage inductance of the relationship between the leakage inductance. The leakage energy model will be analyzed according to the structure shown in Figure 5, where red section represents the secondary winding and green section represents the primary winding. The key parameters are defined as follows: d_ins1 and d_ins2 are the primary and secondary winding interlayer insulation thicknesses, d_iso is the primary and secondary primary insulation distance, h_w1 and h_w2 are the primary and secondary winding heights, N₁ and N₂ are the number of wires in each layer of the primary and secondary windings (N₁ = N₂ = 1 in the case of the copper-foil winding), and d_s and d_p are the number of conductors per layer of the primary and secondary windings. The thickness of the primary and secondary windings, m₁ and m₂, are the number of layers of primary and secondary windings.

The leakage energy within the primary side winding is shown in Equation (8):

$$\begin{array}{l}{W}_P={\displaystyle \frac{1}{2}}\mu l_p{h}_{w1}{\displaystyle \sum _{n_1=1}^{m_1}{\displaystyle {\int }_{x_{in}^{n_1}}^{x_{out}^{n_1}}{H}^{n_1}}}{(x)}^2\mathrm{d}x\\ ={\displaystyle \frac{\mu N_1^2{l}_p\delta }{12h_{w1}}}{m}_1\left[\left(4m_1^2-1\right)F_{p1}-2\left(m_1^2-1\right)F_{p2}\right]I_1^2\end{array},$$

(8)

$$F_{p1}={\displaystyle \frac{\sinh (2{\Delta }_p)-\sin (2{\Delta }_p)}{\cosh (2{\Delta }_p)-\cos (2{\Delta }_p)}},$$

(9)

$$F_{p2}={\displaystyle \frac{\sinh ({\Delta }_p)-\sin ({\Delta }_p)}{\cosh ({\Delta }_p)-\cos ({\Delta }_p)}},$$

(10)

where l_p is the average turn length of the primary winding, I₁ is the rms value of the current flowing through the primary winding, and μ is the magnetic permeability of the desired region, which is the copper magnetic permeability.

The leakage energy within the secondary winding is shown in Equation (11):

$$\begin{array}{l}{W}_s={\displaystyle \frac{1}{2}}\mu l_s{h}_{w2}{\displaystyle \sum _{n_2=1}^{m_2}{\displaystyle {\int }_{x_{in}^{n_2}}^{x_{out}^{n_2}}{H}^{n_2}}}{(x)}^2\mathrm{d}x\\ ={\displaystyle \frac{\mu N_2^2{l}_s\delta }{12h_{w2}}}{m}_2\left[\left(4m_2^2-1\right)F_{s1}-2\left(m_2^2-1\right)F_{s2}\right]I_2^2\end{array},$$

(11)

$$F_{s1}={\displaystyle \frac{\sinh (2{\Delta }_s)-\sin (2{\Delta }_s)}{\cosh (2{\Delta }_s)-\cos (2{\Delta }_s)}},$$

(12)

$$F_{s2}={\displaystyle \frac{\sinh ({\Delta }_s)-\sin ({\Delta }_s)}{\cosh ({\Delta }_s)-\cos ({\Delta }_s)}},$$

(13)

where l_s is the average turn length of the primary winding, I₂ is the rms value of the current flowing through the primary winding, and μ is the magnetic permeability of the desired region, which in this equation is the magnetic permeability of copper.

If both primary and secondary windings are copper foil windings, the normalized thickness Δ is expressed as follows:

$$\Delta ={\displaystyle \frac{d}{\delta }},$$

(14)

where d is the thickness of the primary or secondary winding and δ is the skin depth.

If both the primary and secondary windings are round wire windings, Δ is represented as follows:

$$\Delta ={({\displaystyle \frac{\pi }{4}})}^{{\displaystyle \frac{3}{4}}}{\displaystyle \frac{d_r}{\delta }}\sqrt{{\displaystyle \frac{N_1{d}_r}{h_w}}},$$

(15)

where d_r is the diameter of the circular wire.

If both the primary and secondary windings are Litz windings, Δ is represented as follows:

$$\left\{\begin{array}{l}\Delta ={\displaystyle \frac{d_s}{2\delta }}\sqrt{\pi \eta }\hfill \\ \eta ={\displaystyle \frac{\sqrt{\pi N_s}{d}_s}{2(d_{Litz}+d_L)}}{\displaystyle \frac{h_w}{h_c}}\hfill \end{array}\right.,$$

(16)

where d_s is the diameter of a single strand, d_litz is the diameter of the Litz wire, h_c is the window height, and N_s is the number of strands contained in each Litz wire.

At the same time, the primary and secondary winding layers, m₁ and m₂, of the Litz winding can be equivalent to m₁′ and m₂′.

$$m\prime =\sqrt{N_s}\times m,$$

(17)

The primary-side winding interlayer leakage energy can be expressed as follows:

$$\begin{array}{l}{W}_{p\_ins}={\displaystyle \sum _{n_1=1}^{m_1-1}\frac{1}{2}}{\mu }_0{h}_w{l}_{ins1}{\displaystyle {\int }_{x_{out}^{n_1}}^{x_{out}^{n_1}+d_{ins1}}H{(x_{out}^{n_1})}^2}dx\\ ={\displaystyle \frac{{\mu }_0{l}_{ins1}{d}_{ins1}}{12h_{w_1}}}{m}_1(m_1-1)(2m_1-1)I_1^2\end{array},$$

(18)

where l_ins1 is the average length of the insulating region of the primary side winding and μ₀ is the magnetic permeability of the insulation layer.

The interlayer leakage energy of the secondary winding is expressed as follows:

$$\begin{array}{l}{W}_{s\_ins}={\displaystyle \sum _{n_2=1}^{m_2-1}\frac{1}{2}}{\mu }_0{h}_w{l}_{ins2}{\displaystyle {\int }_{x_{out}^{n_2}}^{x_{out}^{n_2}+d_{ins2}}H{(x_{out}^{n_2})}^2}dx\\ ={\displaystyle \frac{{\mu }_0{l}_{ins2}{d}_{ins2}}{12h_{w_2}}}{m}_2(m_2-1)(2m_2-1)I_2^2\end{array},$$

(19)

where l_ins2 is the average length of the insulation region of the secondary winding and μ₀ is the magnetic permeability of the insulation layer.

The primary and secondary primary insulation leakage energy can be expressed as follows:

$$W_{\mathrm{iso}}={\displaystyle \frac{1}{2}}{\mu }_0{h}_{w2}{l}_{iso}{\displaystyle {\int }_{x_{out}^{m_2}}^{x_{out}^{m_2}+d_{iso}}{\left(\frac{m_2{I}_2{N}_2}{h_{w2}}\right)}^2}dx={\displaystyle \frac{{\mu }_0{N}_2^2{l}_{iso}{d}_{iso}}{2h_{w2}}}{m}_2^2{I}_2^2,$$

(20)

where l_iso is the length of the main insulation layer and μ₀ is the magnetic permeability of the insulation layer.

The leakage inductance value normalized to the primary side is calculated, as shown in (21):

$$L_{\delta }={\displaystyle \frac{2}{I_1^2}}(W_{iso}+W_p+W_s+W_{p\_ins}+W_{s\_ins}),$$

(21)

When the primary and secondary windings are used with the round wire winding HFT structure shown in Figure 6a, the primary side current rms value is 8A. The HFT leakage inductance with frequency change patterns is shown in Figure 6b. When the transformer works at a high frequency, the high-frequency effect of the Litz wire influences the magnetic field distribution in the Litz bundle region, causing the attenuation of magnetic field energy in the Litz bundle region, and there are only the impacts of the strand-level internal proximity effect and the strand-level external proximity effect to be considered [18]. Leakage inductance with frequency increases and decreases, and as frequency increases, the leakage inductance’s rate of change decreases. This is because as the frequency increases, the copper foil winding leakage energy declines, but the primary and secondary winding interlayer insulation area leakage energy and the original secondary main insulation layer leakage energy do not have the frequency effect, and thus, the insulation layer leakage energy ratio increases with the frequency increase: the frequency of more than 500 kHz leakage inductance is approximately constant. In addition, the results of the leakage inductance separation model are consistent with the trend of the finite element results. The calculated value of the winding leakage inductance of the formula at 500 kHz and above is stable at about 41 μH, which is 0.95 times that of the finite element simulation, and the maximum error between the FEM calculated value and the formula method is less than 7.9%, which verifies the validity of the leakage inductance separation model.

3. Leakage Sensing Influence Factor Analysis

Since the amplitude of the magnetic field strength in the insulation area does not change with frequency, but only with the size of the current passing through the winding and the number of turns, the thickness of the insulation area affects the magnitude of the magnetic leakage energy in the case of transformer turns and current determination. From Equations (18)–(20), the primary winding interlayer leakage energy and the secondary winding interlayer leakage energy are proportional to the primary interlayer insulation distance d_ins1 and the secondary interlayer insulation distance d_ins2, respectively. At the same time, the primary side and secondary side main insulation layer leakage energy is proportional to the primary side and secondary side main insulation distance, d_iso2.

Taking the transformer structure in Figure 6a as an example, if the rest of the parameters are fixed and only d_ins2 is changed, the transformer’s leakage inductance is shown in Figure 7. The transformer leakage inductance decreases with increasing frequency, and a frequency greater than the 500 kHz leakage inductance is almost unchanged. When the frequency is unchanged, with the increase in the insulation distance between the layers, the leakage inductance gradually increases. As can be seen from the figure, the model calculation results are consistent with the finite element simulation results, verifying the effectiveness of the leakage inductance calculation model. The primary side winding leakage magnetic variation law and secondary side winding consistently will not repeat the description.

Taking the transformer in Figure 8a as an example, when using a Litz wire winding, its leakage inductance varies with the thickness of the insulation layer, as shown in Figure 8b. As can be seen from the figure, the trend of the leakage inductance of the Litz wire winding is consistent with that of the round wire winding, which decreases with the increase in frequency. Litz wire is composed of multiple individually insulated strands twisted together, with each strand completing a full cross-over transposition within a single twist pitch. This configuration promotes a more uniform current distribution among the strands, thereby reducing the impact of skin and proximity effects on the leakage inductance. At 500 kHz, the leakage inductance of the main insulation thicknesses of 2 mm, 4 mm, and 6 mm are 0.28 μH, 0.43 μH, and 0.61 μH, respectively, and the FEM values are 0.27 μH, 0.42 μH, and 0.59 μH, which verifies the accuracy of the modeling of leakage inductance under the different winding structures.

The law of influence of winding width ds on transformer leakage inductance is shown in Figure 9. The leakage inductance gradually increases as the winding width increases. Meanwhile, the leakage inductance gradually decreases with the increase in frequency. In addition, the leakage inductance model calculation results are basically consistent with the FEM simulation results, and the maximum error is 4.3%, which in turn verifies the accuracy of the leakage inductance model. In addition, when the transformer is operated at 10 kHz and 1 MHz, the change rule of the leakage inductance with the winding width is shown in Figure 10. When the operating frequency is low, the transformer leakage inductance increases with the increase in width. However, when the frequency is higher, the transformer leakage inductance is almost unchanged under different winding widths. In Equations (5)–(7), this can be seen: the winding internal magnetic field strength with the frequency increases gradually tends to 0, and thus, the winding internal leakage energy is 0. Therefore, in high-frequency conditions, the winding width does not affect the size of the leakage inductance.

In contrast, the winding height, h_w, is also a key factor affecting the leakage inductance. From the leakage inductance model, it can be seen that the leakage energy is inversely proportional to the winding height, and the smaller the winding height is, the higher the leakage energy is, and the larger the leakage inductance is. However, in the leakage inductance model, it is assumed that the winding height is always equal to the window height, which means that the filling rate of the winding is 100%. If the winding height is smaller than the window height, the model calculation error increases, and it is necessary to introduce the winding height correction factor, K_R (Rogowscki coefficient), to solve the equivalent winding height, h_weq, which makes the equivalent winding height equal to the window height.

The equivalent winding height, h_weq, can be expressed as follows:

$$h_{weq}={\displaystyle \frac{h_{\mathrm{c}}}{K_{\mathrm{R}}}},$$

(22)

$$K_{\mathrm{R}}=1-{\displaystyle \frac{1-e^{-\pi h_{\mathrm{c}}/\left(d_p^{all}+d_{i\mathrm{so}}+d_{\mathrm{s}}^{\mathrm{all}}\right)}}{\pi h_{\mathrm{c}}/\left(d_p^{all}+d_{i\mathrm{so}}+d_{\mathrm{s}}^{\mathrm{all}}\right)}},$$

(23)

where h_c is the window height, $d_p^{all}$ is the total thickness of the primary winding and insulation, and $d_s^{all}$ is the total thickness of the secondary winding and insulation.

Figure 11 shows the changed rule of transformer leakage inductance at different frequencies under the winding filling rate of 70%. If we do not consider the winding height equivalent, the leakage inductance model calculation results and the finite element results of the error are 17.7%, and when we consider the winding height equivalent calculation, the error is only 6.7%, effectively improving the leakage inductance model calculation accuracy.

In transformer optimization design, different structural parameters exert varying degrees of influence on the leakage inductance. Therefore, conducting parameter sensitivity analysis is crucial for identifying key variables and enhancing optimization efficiency. By distinguishing between high-sensitivity and low-sensitivity parameters, the optimization process can focus on primary influencing factors, effectively reducing the search space dimension. This enhances convergence efficiency and conserves computational resources. To this end, this paper introduces the Pearson correlation coefficient as a sensitivity evaluation metric to systematically assess the impact of each geometric parameter on the leakage inductance, as shown in the following equation.

$${\rho }_{X_i,Y_i}={\displaystyle \frac{N\sum X_i{Y}_i-\sum X_i\sum Y_i}{\sqrt{N\sum X_i^2-{\left(\sum X_i\right)}^2}\sqrt{N\sum Y_i^2-{\left(\sum Y_i\right)}^2}}},$$

(24)

where N denotes the sample size, X_i represents the value of the design parameter, and Y_i denotes the value of the i-th objective function.

During the optimization process, we minimized the leakage inductance by adjusting the geometric structures of two high-frequency transformer prototypes to improve the magnetic flux distribution and reduce the leakage flux. Taking the structure shown in Figure 12 as an example, the following variables were selected as optimization parameters: winding diameter, d_l; gap distance, d_cw, between the winding and core; spacing distance, d_iso, between primary and secondary windings; number of turns, N_w, in a single winding layer; primary winding turn pitch, d_ins1; and secondary winding turn pitch, d_ins2. The optimization ranges for each variable of the two HFTs (round conductor and Litz wire) are shown in

Table 2. Range of optimization variable values.

Parameter	Round Conductor	Litz Wire
dl	[0.4–1.1] mm	[0.99–4.2] mm
dcw	[1–5] mm	[1–4.5] mm
diso	[1–7] mm	[1–7] mm
Nw	[10–48]	[8–18]
dins1	[0.1–0.32] mm	[0.1–0.4] mm
dins2	[0.1–0.32] mm	[0.1–0.4] mm

.

Figure 13 presents the absolute value analysis results of the sensitivity coefficients for leakage inductance across each optimization variable in the round conductor and Litz wire transformers. It can be observed that for both winding types, the primary–secondary winding spacing (d_iso), single-layer winding turns (N_w), and winding diameter (d_l) are all highly sensitive parameters with the most significant impact on leakage inductance. This result indicates that despite differing winding configurations, these geometric parameters play a dominant role in leakage inductance formation.

Further investigation into the error sources suggests that in the round conductor configuration, the proposed model may underestimate the additional magnetic field dissipation caused by skin and proximity effects, leading to calculated values that are lower than the measured ones. In the Litz wire structure, however, the model likely overestimates the effective magnetic path length or fails to fully capture the complex distribution effects of internal insulation gaps—despite the inherent ability of Litz wire to suppress high-frequency eddy current losses. As a result, below 1.4 MHz, the calculated values are generally higher than the measured ones. When the frequency exceeds 1.4 MHz, high-frequency eddy current effects become more pronounced, and the electromagnetic coupling behavior between strands and among turns within the Litz wire grows increasingly complex. The model’s simplified description proves inadequate for capturing the actual three-dimensional eddy current paths, local magnetic field distortions, and distributed capacitance effects introduced by the insulating medium at very high frequencies, which causes the calculated leakage inductance values to fall below the measured data in this frequency range. The problems have been modified in the revised paper.

4. Experimental Measurement

This paper constructed two HFT prototypes, as shown in Figure 14, which were used to validate the accuracy of the transformer winding loss model. Both cores employed E-type ferrite material, with one wound using round wire and the other using Litz wire. The specific prototype parameters are shown in

Table 3. Structural parameters of prototypes.

Wire Type	Round Conductor	Litz Wire
Primary Turns	45	16
Secondary Turns	45	16
Primary Layers	1	1
Secondary Layers	1	1
Turn-to-Turn Distance	0.23 mm	0.34 mm
Outer Diameter	1 mm	3.96 mm
Specifications		0.1 mm × 800
Resonant Frequencies	5.3 MHz	6.2 MHz
Apparent Power	1.8 kVA	7.7 kVA

.

The actual test results indicate that both prototypes exhibit resonance frequencies exceeding 5 MHz. Since the test frequency is significantly lower than the resonance frequency, measurement errors that are attributable to resonance can be considered to be negligible. In the DAB converter design corresponding to this study, the switching frequency is set to 10 kHz. As shown in the table, the resonant frequency is significantly higher than the switching frequency. This ensures that the DAB converter consistently operates in the inductive region, creating the necessary conditions for achieving zero-voltage switching (ZVS).

To verify the direct measurement method used in this paper, the equipment used is the “Keysight E4990A” impedance analyzer (Keysight Technologies, Santa Rosa, CA, USA), with the fixture model “16047E” (Keysight Technologies, Santa Rosa, CA, USA). Prior to testing, the equipment undergoes calibration: first, open-circuit calibration, followed by short-circuit calibration, and finally, load calibration using a standard 50 Ω resistor. After calibration, the transformer under the test is connected to the fixture for testing. The validity of the error model formed after calibration directly depends on the quality of the calibrated components and the performance of the instrument. When properly calibrated, the 16047E fixture provides accurate measurements of up to 30 MHz. The leakage inductance measurement frequency in this paper is well below this limit, ensuring the validity of the calibration and the reliability of the data.

Using a fixed step size of 9.95 kHz, a total of 201 data points were acquired across the 10 kHz to 2 MHz range. This resolution is sufficient to clearly capture the impedance curve of the transformer, particularly the smooth variation in the leakage inductance with frequency. The selected step size also allows for accurate identification of potential resonance points, as the width of the resonance peaks is typically much greater than 9.95 kHz, ensuring that no key features are missed due to under-sampling.

The leakage inductance measurement of prototype 1 and prototype 2, and the comparison of the measurement results with the results of the formula analysis method are shown in Figure 15; Figure 15a shows the comparison between the measured value and the formula analysis value of prototype 1. It can be seen that the difference between the measured value and the analyzed value is about 0.2 mH when the frequency is between 10 kHz and 100 kHz, and the measured value and the analyzed value tend to be stable at about 16% before 1 MHz, and the error is between 3% and 10% between 10 kHz and 100 kHz. After 1 MHz, the measured value and the analyzed value tend to level off with a difference of 0.7 mH, and the error is stabilized at about 16%; before 1 MHz, the error is less than 16%; and between 10 kHz and 100 kHz, the error is between 3% and 10%. Figure 15b compares the measured values of Prototype 2 with the analytical formula values. The figure shows that between 10 kHz and 2 MHz, the decrease in the leakage inductance is smaller than that of the circular conductor, amounting to only about 0.25 mH. At 10 kHz, the analytical formula value exceeds the measured value by 0.05 mH., while at 2 MHz, the analytical formula value is 0.1 mH lower than the measured value. The average error is approximately 2%.

To evaluate measurement uncertainty, we conducted five replicate measurements on both round wire and Litz wire transformer prototypes, focusing on leakage inductance parameters at three critical frequency points: 100 kHz, 500 kHz, and 1 MHz.

The statistical analysis results of the leakage inductance data from replicate measurements are presented as probability density distribution plots in Figure 16. The figure indicates that the standard deviation of the leakage inductance measurements for the Litz wire winding is lower than that of the round wire winding across all frequencies, demonstrating superior measurement repeatability and greater stability for the Litz wire prototype. This distribution diagram visually reflects the consistency and reliability differences in leakage inductance performance between the two winding structures, further validating the credibility of the experimental data presented herein.

To further evaluate their parameters, AC resistance measurements were conducted on the two manufactured high-frequency transformer prototypes within a frequency range of 10 kHz to 300 kHz. The results for the round wire transformer are illustrated in Figure 17a. It can be observed that the discrepancy between the measured values and Dowell’s equivalent model increases with frequency. At 10 kHz, the errors of Dowell’s model and the FEM model compared to the measured value were 0.23% and 0.21%, respectively. At 100 kHz, the errors were 8.1% and 10.2%, and at 300 kHz, they were 8.4% and 6.5%, respectively.

The winding losses of the Litz wire HFT are shown in Figure 17b. The calculated values from Dowell’s equivalent model are relatively close to the measured values, though the discrepancy between different models gradually increases with frequency. At 10 kHz, the errors between Dowell’s model, the FEM model, and the measured value were 0.42% and 0.41%, respectively. At 100 kHz, the errors were 2.5% and 6.4%, and at 300 kHz, they were 4.4% and 4.7%, respectively.

5. Conclusions

Based on Dowell’s equivalent model, this paper analyzes the magnetic field inside the winding and the insulation layer, derives the leakage energy formula for different winding types, calculates the leakage inductance size of the HFT, and accurately analyzes the trend of the influence of each key parameter on the leakage inductance. For the winding filling rate, which is a key factor affecting the leakage inductance calculation error, the Rogowski coefficient is introduced for correction, which effectively improves the calculation accuracy. The accuracy of the model is further verified through experimental measurements, which provide a reliable theoretical basis and method for the accurate calculation of the leakage inductance of HFTs, and helps to optimize the design of HFTs and improve their performance and reliability.