A modelling technique to determine the high frequency transformer leakage inductance using the winding structure | Scientific Reports

Scientific Reports volume 15, Article number: 2373 (2025) Cite this article

661 Accesses

Metrics details

The operation and efficiency of isolated DC-DC converters, critical components in solid-state transformers, are significantly impacted by leakage inductance in high-frequency transformers (HFTs). Different converters have varying demands regarding leakage inductance, ranging from precise control to minimal inductance. For instance, the performance of bidirectional isolated converters (BIDCs) and resonant converters is dependent on leakage inductance for the delivery of power. This study proposed a method for the control of leakage inductance by altering the winding configuration. The technique involved positioning the primary and secondary windings at predetermined heights to increase the separation between some winding turns, and thus, enhance leakage inductance. By not varying the average length of turns in the winding configuration, the proposed method was able to maintain a consistent copper loss. A modified mathematical model has been put forward to determine the leakage inductance effectively and precisely. The viability and accuracy of this method were validated through simulations and experiments using a three-phase HFT (3P-HFT) in a 3P-BIDC. The results showed that the leakage inductance that was calculated using the theoretical model was closely correlated to that of the simulation model, with only a variance of 3.9% and an error of 4.53% from the experimental results.

The use of power electronics technology can enhance the efficiency of power converters and lead to significant energy savings. This technology is employed in a variety of devices, including high-frequency inverters and DC/DC converters. In particular, the DC/DC converters used in aircraft and automobiles must have high power densities to fit into limited spaces. Consequently, several researchers have been exploring techniques to reduce the size of passive components by increasing the switching frequency to MHz values, as reported in1.

Dual active bridge (DAB) or bidirectional isolated DC-DC (BIDC) converters are high-power converters that are commonly utilised in various applications, such as smart grids, traction systems, and offshore wind farms2. Their utilisation as solid-state transformers for all-electric aircraft and ships, as well as energy storage devices for electric cars and renewable energy installations, has been widely established in multiple studies2,3,4,5,6,7. BIDCs are categorised as either single-phase (1P-BIDC) or three-phase (3P-BIDC) converters. Both converters provide bidirectional power flow, isolation, and soft switching. However, three-phase (3P) converters perform better than single-phase (1P) converters because they have higher power densities, lower input and output filter sizes, and lower transformer root mean square (RMS) currents. As a result, 3P-BIDCs are significantly more popular than their 1P counterparts8,9,10 because they offer higher efficiency and power density while decreasing power backflows and switching stress on the switching components.

The performance of a BIDC, which is essential for power delivery, is significantly affected by leakage inductance11. However, a large leakage needs a large value of phase shift angle which can limit the transfer of the power, and the zero-voltage switching (ZVS) cannot be achieved to reduce the switching losses, particularly at high frequencies. To maximize the power density of a converter, it is critical to incorporate the necessary inductance into the transformer without the use of external components12,13. Therefore, an accurate leakage inductance design for an HFT is crucial for optimal and efficient converter performance.

The leakage flux of a transformer is defined as the magnetic field that leaks from the core and returns through the atmosphere instead of through the connection between the two windings. Usually, numerical, and analytical methods are used to estimate the leakage inductance. The numerical method, which utilises finite element simulations to solve Maxwell’s equations in countless subregions14, boasts high computational accuracy but suffers from low efficiency and high complexity. Consequently, the current research employed an analytical modelling method based on the ideal geometry of the HFT, winding height, and winding thickness15,16.

Leakage inductance, which is directly related to winding arrangements, is a critical consideration in the design and operation of HFTs. Adjusting the winding-to-winding distance or primary number of turns, as proposed in17, is a classic approach for regulating leakage inductance. However, such adjustments can impact the magnetic core window utilisation factor and may not provide constant regulation of the leakage inductance. Furthermore, changing the number of turns may affect the operating magnetic intensity and transformer loss distribution, leading to increased losses18. Once the core structure has been identified, modifications to the winding-to-winding distance and number of primary turns may be limited. In19, it was observed that increasing the number of turns and winding distance to produce leakage inductance in a 3P-HFT resulted in an increase of copper loss by at least 15% and raised the active mass of the device by 20%.

Interleaved windings are commonly used to minimise copper loss and control the leakage inductance as in16, the authors presented a partial interleaved winding method that enabled the degree of interleaving to be adjusted by selectively interleaving only a few turns. However, because this approach reduces the magnetic field and may not be sufficient to achieve the desired leakage inductance, the addition of a magnetic leakage layer has been proposed. Although a leakage layer increases the leakage inductance, it also leads to significant power loss. To address this issue, the authors of20proposed the use of a simple rectangular layer spanning the entire winding width. This was found to increase the winding loss by only 6% compared to the case without a leakage layer. On the other hand, a leakage layer with minimum losses was observed in21, which also led to an increase in the leakage inductance of the transformer. However, this increase in leakage inductance with the presence of a leakage layer cannot be controlled, while a converter requires a certain leakage inductance value.

In22, an investigation was conducted into the use of two opposing winding arrangements to control the amount of magnetic flux leakage that occurred in an HFT. The first arrangement involved completely and symmetrically separating the windings on both sides to enable the formation of a high magnetic field, resulting in a high overall leakage flux. Conversely, the second arrangement involved completely covering one winding with the other and tapping the magnetic flux between the windings to direct and share the magnetic flux leakage between the windings, thereby altering the overall amount of leakage flux.

In23, the researchers examined the impact of concentric and separate winding arrangements. It was found that the concentric windings yielded negligible leakage inductance, whereas the separate windings yielded a leakage inductance that exceeded the amount required by the BIDC. A degree of freedom in the winding arrangement, such as the turn-to-turn distance of the secondary winding, as proposed in24, can affect the overall leakage inductance within a wide range. However, this approach only decreases the leakage inductance. In25, a proposal was made to minimize the external field of the transformer and inductor in a series resonant converter. The idea was to employ three sections of windings with varying coupling coefficients to reduce the external field. However, this winding structure results in an increase in the magnetic volume and a decrease in the leakage flux.

A variable leakage and magnetizing inductance were proposed in a variable inductance transformer (VIT)26, where the leakage inductance was varied by moving one of the windings vertically, while the magnetizing inductance was varied by the length of the air gap. A few iterations of the leakage inductance mathematical model were proposed for the circular core leg and circular windings only, and a relative error of 4.5% was achieved compared to the FEM. However, this technique is based on bobbins with a certain thickness, which tends to increase the mean length turn (MLT) of the windings and copper loss, as well as the iteration calculations, leading to a high percentage error as the overlapping distance increases. Similarly, in27, a method for evaluating the VIT based on the magnetic image was proposed with a percentage error of 4.5%. Later, in VIT28, a frequency-dependent hybrid model of variable leakage inductance was proposed with a maximum error of 7.1% at 50 kHz with an overlapping distance of 11 mm between both windings.

In29, the authors presented a flux diverter cap containing minimal ferrite to increase the leakage flux by lowering the path inductance. However, this approach may also increase the core volume, thereby leading to higher core loss. In30, researchers utilized the phase-shift angle of the BIDC as a free parameter to optimize the HFT design based on the leakage inductance produced by the transformer. However, increasing the phase-shift angle resulted in an elevated reactive power, which subsequently led to higher converter losses. In31, researchers employed a wire guide in a toroidal transformer to regulate leakage inductance by altering the height and thickness of the wire guide. The advantage of this technique lies in its ability to reduce the stray capacitance between the windings, particularly with an increase in the distance between the windings. However, this approach also leads to an increase in the active mass of the transformer owing to the addition of a wire guide.

Several studies have investigated the impact of the tapping position on the leakage inductance of 3P-transformers. For instance32, utilised finite element analysis (FEA) to examine five different tapping positions and found that positioning the tapping 15% above and 15% below the winding resulted in the highest leakage inductance. Additionally33, optimised the tapping positions through experimental verification and found that a tapping position of 12.5% yielded the highest leakage inductance. It was also found that side tapping had the least impact on the leakage inductance and was determined to be the best position for a transformer with two windings. However, it should be noted that changing the tapping position within a transformer can significantly affect copper loss, leakage reactance, and core loss.

It is imperative to accurately model the leakage inductance of a transformer during the design stage, as insufficient leakage inductance can negatively impact soft switching and the efficiency of a BIDC, whereas excessive leakage inductance can result in the unnecessary circulation of reactive power and decrease the efficiency and output power of the converter. According to the literature mentioned above, most of the previous approaches led to additional losses or changes to other geometric and electromagnetic parameters of the HFT. Furthermore, there is no mathematical formula for obtaining the desired leakage inductance in a single step with good accuracy. Therefore, this paper proposes an improved mathematical model for tuning the leakage inductance based on moving primary winding upward and secondary winding downward. In this way, the proposed method will help to assign the desired leakage inductance in a single step with high accuracy. Moreover, the winding structure will keep the MLT of the windings constant, resulting in constant copper loss.

This paper is structured as follows: Sect. 2 discusses the leakage inductance general analytical model; Sect. 3 elaborates on the method employed to model the leakage inductance utilizing the proposed winding arrangements. Section 4 presents the theoretical, simulation, and experimental results of a 3P-HFT, based on the modelled leakage inductance, and also discusses the impact of the designed HFT on the operation of a 3-kW, 20-kHz, 3P-BIDC; and finally, Sect. 5 presents the conclusions of the study.

The leakage magnetic energy in the transformer winding and its insulation, as stated in Eq. (1), is typically used to calculate the transformer leakage inductance.

where\(\:{\:E}_{\text{k}}\) is the total leakage magnetic field energy, \(\:{\mu\:}_{\text{o}}\) is the air permeability, \(\:\left|\overrightarrow{H}\right|\) is the magnetic field intensity surrounding the insulation area and transformer windings, and \(\:dV\)is the winding volume34.

The transformer windings are composed of solid or Litz wire, and the authors of15 pointed out that they are comparable to foil windings. If the magnetic potential drops to zero and the magnetic permeability of the core is high, the MMF distribution in the window can be found using the ampere-loop theorem. To make the MMF in the transformer winding area easier to understand, Fig. 1 illustrates a 1-D cross section variable with a trapezoidal wave distribution along the width of the windings and insulation.

The distribution of the magnetic field depending on the width of the windings and insulation.

According to Eq. (1), the leakage magnetic field energy of the insulation area per unit length can be found as shown in Eq. (2).

where \(\:{N}_{1}{I}_{1}\) is the primary ampere turns, \(\:{d}_{\text{i}\text{n}\text{s}}\) is the insulation thickness, and \(\:{h}_{\text{w}}\)is the winding height34. Similarly, the leakage magnetic field energy of primary and secondary windings per unit length can be obtained using Eqs. (3) and (4), respectively.

where \(\:{d}_{\text{w}\text{p}}\) is the primary winding thickness, \(\:{h}_{\text{w}\text{p}}\) is the height of the primary winding, \(\:{d}_{\text{w}\text{s}}\) is the secondary winding thickness, \(\:{h}_{\text{w}\text{s}}\) is the height of the secondary winding, as shown in Fig. 2 (a), and, \(\:{F}_{\text{w}\text{p}}\) and \(\:{F}_{\text{w}\text{s}}\) are the frequency-dependent coefficients. The Dowell model was used to incorporate the frequency-dependent coefficient, \(\:{F}_{\text{w}}\) to improve the magnetic field energy of the winding region because the skin and proximity effect have an impact on how the MMF of the windings is distributed under high-frequency conditions. However, with selection of the Litz wire based on the operating frequency, the impact of the skin, and proximity effect can be disregarded. Thus, the frequency-dependent coefficient of this winding was \(\:{F}_{\text{w}}\approx\:1\)34. Based on Eq. (2) to (4), Eq. (5) was used to analytically solve the total magnetic leakage energy, while Eq. (6) was used to calculate the overall leakage inductance.

As shown in Fig. 2 (b), the \(\:MLT,\) which is the mean length turn of the winding, was calculated using Eq. (7):

The geometry of transformer.

where, \(\:x\) is the core leg width, \(\:y\) is the core leg depth, and \(\:{d}_{\text{w}\text{p}}\), \(\:{d}_{\text{i}\text{n}\text{s}}\), \(\:{d}_{\text{w}\text{s}}\), are the thickness of primary winding, insulation, and secondary winding, respectively. The estimation of the leakage inductance in Eq. (6) is accurate when concentric windings were placed at the same height as the primary and secondary windings. To attain the intended leakage inductance with the HFT window area, the winding height in Eq. (6) does not always need to be fixed; instead, it can be changed depending on the particular winding arrangement being utilized. Therefore, the proposed modelling technique in the subsequent section based on a specific winding height that was appropriate for the modelling of the leakage inductance.

As depicted in Fig. 2 (a), the transformer windings were typically wound concentrically and positioned at the centre of the window height \(\:\left({H}_{\text{w}}\right)\). This configuration limited the leakage inductance, which was influenced by factors such as the number of turns, insulation thickness, winding thickness, and winding height. The leakage magnetic field of the longitudinal was determined by specifying the primary number of turns, core geometry, and insulation thickness. By adjusting the energy of the transverse leakage magnetic field, which was associated with the trapezoidal distribution of the MMF, the leakage inductance value could be changed, therefore, the winding arrangement could be modified to alter the leakage inductance value inside HFT34.

The modelling of leakage inductance typically involves the separation distance between the primary and secondary windings. This distance also affects the coupling coefficient and, consequently, the leakage inductance. This is reflected in Eq. (8) below, where a decrease in the coupling coefficient resulted in an increase in leakage inductance35.

where \(\:{L}_{\text{k}}\) is the leakage inductance, \(\:{k}_{12}\) is the coupling coefficient between the primary and secondary windings, and \(\:{L}_{\text{p}}\) is the primary self-inductance. The conventional approach to raising the leakage inductance primarily depends on thickening the insulation between the primary and secondary windings. The proposed method is based on previously developed analytical techniques and the findings of a finite element analysis which involved positioning the winding along the window height by shifting the overall primary winding upward and secondary winding downward relative to the core’s window height. This is increasing the separation distance between some of the primary and secondary turns and increases the leakage inductance.

Figure 3 presents a flowchart of the proposed method, which included the definition of HFT parameters such as core window height \(\:\left({H}_{\text{w}}\right)\), number of turns \(\:\left({N}_{\text{p}}\right)\), winding thickness and MLT, as well as specify the required leakage inductance value for 3P-BIDC converter operation.

In order to shift the primary winding upward and secondary winding downward according to the proposed method, the upper and lower empty distance of the core window height as shown in Fig. 2 (a), should be calculated first as in Eq. (9) to specify the maximum shifting distance could be implemented inside the 3P-HFT core.

The flowchart of the proposed transformer leakage inductance tuning.

According to the energy calculation methods presented in Eqs. (2)-(4), the height of concentric winding structure was found to be equal to the actual winding height \(\:({h}_{\text{w}\text{p}}={N}_{\text{p}}\times\:{d}_{\text{w}\text{p}})\), and by shifting the primary winding upward and secondary winding downward along the core window height by a specific shifting distance, the concentric winding height \(\:\left({H}_{\text{c}\text{o}\text{n}}\right)\) will be decreased as shown in Fig. 4, and thus creating a separation distance between some of primary and secondary turns leading to increase the leakage inductance. Therefore, the new value of concentric winding height equal to the winding height minus the shifting distance upward for primary and downward for secondary, and by considering both windings having the same winding height then the \(\:{H}_{\text{c}\text{o}\text{n}}\)

will be:

The proposed arrangement of the windings.

where \(\:{H}_{\text{c}\text{o}\text{n}}\) is the new concentric winding height, \(\:{h}_{\text{w}\text{p}}\:\)is the primary winding height, and \(\:{d}_{\text{s}\text{d}}\) is the shifting distance. In this manner the actual winding height \(\:\left({h}_{\text{w}\text{p}}\right)\) will not be valid to estimate the leakage inductance as per Eq. (6). Therefore, an equivalent height \(\:\left({H}_{\text{e}\text{q}}\right)\) is proposed here based on the differences between the concentric height, shifted distance, and the percentage of the actual winding height to the concentric height as shown in Eq. (11), to determine the leakage inductance based on a specific value of shifting distance of primary winding upward and secondary winding downward.

Lastly, by substituting the equivalent height \(\:\left({H}_{\text{e}\text{q}}\right)\) in Eq. (11) into Eq. (6) instead of the actual winding height \(\:{(h}_{\text{w}\text{p}})\), the total leakage inductance obtained through the proposed method was determined as per Eq. (12):

This method employed a shifting distance to determine the spacing between the primary and secondary turns and to position the winding, either up or down, within the range of the height of the core. As shown above, the proposed method deals with vertical shifting of the winding and keeps the insulation thickness (\(\:{d}_{\text{i}\text{n}\text{s}})\) constant, resulting in a constant inter-winding distance and constant MLT. Therefore, the proposed approach tunes the transformer leakage inductance without compromising the copper loss.

Figure 5 shows the 3P-BIDC circuit, which operated in a symmetric 6-step switching mode, with each phase leg having a 50% duty cycle. At the switching frequency, the 3-phase legs were voltage-shifted from one another by 120°. The 3P-BIDC utilised two voltage-sourced bridges that were galvanically isolated by the 3P-HFT.

Three-phase BIDC converter topology.

Table 1 outlines the specifications of 3P-BIDC converter, where the 3P-HFT was designed and constructed to satisfy the converter requirements. Table 2 displays the design parameters of the 3P-HFT. The traditional area-product design method was employed for the core and winding design, as outlined in36. Ferrite PC40 was selected as the core material37 due to its low core loss under high frequency, and insulated Litz wire was utilised for both windings for minimal copper loss.

Figure 6 depicts the design points of the 3P-HFT. The dotted red curve represents the variation in the number of turns with flux density when the core cross-sectional area, operating frequency, and applied voltage are fixed. Besides, the dotted blue curve represents the increase in transformer loss with the number of turns. Most of the increased losses were related to the copper loss, which increased with the number of turns. A two-layer winding structure was implemented to accommodate all the 30 turns within the core window area, which also increased the leakage inductance of the HFT and eliminated the need for an external inductor to control the flow of power in the 3P-BIDC.

The design point of 3P-HFT.

As stated in the introduction section, the leakage inductance is responsible for power transfer and ZVS of 3P-BIDC converter, and this can be done based on the selected phase shift angle of 3P-BIDC converter. As shown in Fig. 7, the ZVS of both converter bridges can be achieved over a wide range of phase shift angle. However, at high value of phase shift angle the converter reactive power will increase and the power factor is decreased as shown in Fig. 8, which in turn increases the overall converter loss and reduce its efficiency.

Therefore, in this paper, the design point of 3P-BIDC converter has been selected at phase shift angle δ = 9° at which the power transfer and ZVS are ensured, as well as to prevent high reactive power and converter losses. According to the design point at δ = 9°, the required leakage inductance is about 24.5 µH and based on the core and winding details of 3P-HFT in Table 2 , the maximum vertical shifting distance can be achieved is 6 mm as calculated using Eq. (9). Therefore, a range of leakage inductance is calculated based on the range of shifting distance from 0 mm to 6 mm as presented in Table 3.

Variation of transfer power and ZVS along with phase shift angle.

Variation of reactive power and power factor along with phase shift angle.

As shown in Table 3, the leakage inductance of 3P-HFT increase with the shifting distance as calculated using Eq. (12) without manipulating the inter-winding distance, which in turn increase the copper loss with leakage inductance increasing. Moreover, the increase of leakage inductance is demonstrated by using ANSYS Maxwell software used to perform the finite element analysis simulation as shown in Fig. 9. The magnetic field energy increases as the shifting distance increases resulting in increasing the stored magnetic energy.

The magnetic energies simulated using various shifting distances.

In terms of experimental results, Fig. 10 shows the proposed 3P-HFT prototype according to the design details in Table 2. In order to achieve the required leakage inductance inside the transformer windings which is about 24.5 µH, and according to the results of the proposed method as in Table (3), the required value of leakage inductance can be achieved at 5 mm shifting distance. Therefore, the primary winding shifted upward and secondary windings downward by 5 mm.

(a) The details of the windings structure (b) The prototype transformer.

A GW Instek®-LCR 8101G meter with frequency measurement range of 20 Hz to 1 MHz, was used to measure the AC resistance and leakage inductance of the 3P-HFT prototype windings. To measure the equivalent transformer leakage inductance referred to the primary side, the secondary winding was short-circuited to the neutral point. Table 4 presents the measurements for each 3P-HFT phase using the LCR meter.

In order to evaluate the proper operation of the 3P-BIDC converter with the designed 3P-HFT, the converter was constructed and tested. The 3P-BIDC prototype uses six Infineon IGBT dual modules with the voltage rating of 1.2 kV and current rating of 50 A, 12 snubber capacitors, and six film capacitors. Figure 11shows the complete prototype of the converter with ratings of 20 kHz, 3 kW, and 300 V. To operate the 3P-BIDC converter under the full-load condition, parallel connection of the secondary bridge DC output terminals to the primary-side DC input terminals was performed as in38,39, as displayed in Fig. 12. This configuration is feasible only when the transformer turns ratio is 1:1. Consequently, a DC power supply is required to supply the 3P-BIDC losses.

3P-BIDC converter full prototype.

Full load experimental setup.

An oscilloscope was employed to measure the input and output powers as well as the AC voltages and currents of the 3P-BIDC. As depicted in Fig. 13, the experimental waveform of the AC terminal phase voltages across the primary and secondary bridges was demonstrated, where an output power of 3 kW was achieved at a phase shift angle of =10.3°. The waveform revealed that the primary voltage led the secondary voltage by =10.3°, indicating that power flowed from the primary bridge to the secondary bridge. Furthermore, the peak phase voltage of the primary and secondary bridges was 200 V, which is equal to \(\:\frac{2{V}_{in}}{3}\).

Phase shift angle between primary and secondary voltages.

In order to perform the power transfer of three-phase BIDC converter based on the achieved leakage inductance inside transformer windings, Fig. 14, shows the three-phase current of the primary side, where the peak value is achieved at the rated power. However, there are some variations among the three-phase current because the achieved leakage inductance value in each phase is not equal as seen in Table 4.

Three-phase current of the primary side.

According to the experimental setup in Fig. 12, the input voltage and output voltage along with input and output current are measured as shown in Fig. 15. The input and output voltage are the same as 300-V, while the achieved output current is about 9.78 A and the input current is about 0.477 A, which is used to supply the converter losses including the 3P-HFT and power semiconductor switches losses. Therefore, the total input power from the DC power supply is about 143.1 W which is considered to be the total converter losses, and according to the measured core and copper losses of 3P-HFT, the total 3P-HFT losses is about 39.14 W, which indicates that a large portion of power losses are related to switching and conduction loss in the semiconductor switches, which is about 103.95 W. The total converter efficiency of 95.35% was achieved and this is demonstraed by using power analyzer for input and output DC power measurement as presented in Table 5.

Measured DC voltage with input and output current.

The ZVS of both converter bridges was taken into consideration with regard to the gate voltage and collector-emitter voltage of the IGBT. The measured waveforms of S1 in the primary bridge and S7 in the secondary bridge are shown in Figs. 16 and 17, respectively. Both switches were operating under ZVS, based on the operating phase-shift angle, and the achieved leakage inductance inside each transformer windings of 3P-HFT.

The collector-emitter voltage across switch S1 is zero when the gate-source voltage is above the threshold turn-on voltage of S1.

The Switching Waveforms of S7 in the Secondary Bridge.

In this section, the merits of the proposed transformer leakage inductance model have been compared according to calculation accuracy with respect to the experimental prototype. Moreover, the proposed transformer vertical-winding-shift method has minimal effect on increasing the AC winding resistance. The proposed winding-shift method and the mathematical model are compared with the traditional transformer leakage inductance adjustment methods, which are based on the shifting the secondary winding horizontally17and vertical winding shifting26,27,28.

As shown in Table 6, the proposed leakage inductance control method keeps the AC winding resistance constant due to the vertical winding shifting. Moreover, the mathematical model of the transformer winding results in a small calculation error of 3.9%. However, using the conventional method which is based on increasing the inter-winding distance horizontally, the winding resistance is increased by 3.76% which leads to significant increase in the copper losses of 3P-HFT. Moreover, the transformer model results in high percentage error in leakage inductance calculation.

Furthermore, the method with vertical shifting of one winding and keeping the other winding constant as proposed in26 achieved a slight increase in the AC winding resistance due to using a bobbin with a certain thickness, which in turn increases the inter-windings distance. This paper also uses iteration method to calculate the leakage inductance at a certain shifting distance resulting in an error of 4.5%. The magnetic image method was employed to improve the accuracy of leakage inductance estimation. However, it produces the same error value. Moreover, by using the frequency dependent method, the error value increased by 7.1% especially when the shifting distance is increased.

According to the three-phase transformer structure the vertical windings shifting is more convenient for leakage inductance modelling due to the shared window area between two windings. Furthermore, the proposed mathematical model outperforms the previous research works in terms of having minimal effect on the increase in AC winding resistance, at a certain shifting distance as well as minimum percentage error was achieved with the proposed mathematical model.

Furthermore, in order to perform the effectiveness of the proposed method as a leakage inductance calculation method with a concentrated winding arrangement (at zero shifting distance), Table 7 presents a comparison between the proposed method and other calculations methods, considering the calculation accuracy, and complexity.

The proposed method and the conventional method are based on the energy method and the calculation parameters are the transformer winding geometry due to this the low complexity and computational efforts are achieved in both methods. However, the proposed method achieved a lowest percentage error as compared to the conventional method based on the three-phase HFT with concentrated winding arrangement in40, the leakage inductance achieved is 11.23 µH while the value calculated using the proposed method is 11.38 µH in which achieve a minimum percentage error about 1.3%, while the calculated value using the conventional method is about 10.86 µH with a percentage error of 3.4%.

The Dowell method41which is considered as the popular method to estimate the leakage inductance achieved the worst percentage error as compared to others, due to this, the hybrid model15 was proposed based on the integration the Dowell model with Rogoski’s coefficient to estimate the equivalent winding height for leakage inductance calculation with an error of 4%, leading to increase the complexity of leakage inductance calculation model.

Furthermore, non-ideal geometric factor for unequal winding height has been considered in leakage inductance estimation based on the integration of superposition theorem with Rogoski coefficient34. However, the calculation error is high at 6.78% with high complexity since it is based on the integration of magnetic energy inside and outside of the transformer core.

The frequency dependent method in42 uses a closed form expression for leakage inductance considering the skin and proximity effects. A moderate percentage error with medium level computational complexity is reported in the paper. The value of leakage inductance will fall depending on the relative size of Litz wire diameter and skin depth at a given frequency. However, the reduction of leakage inductance value will be negligible when the diameter of the Litz wire is lower than the value of skin depth at the operated frequency, and this is usually attained to reduce the winding losses at high frequency.

According to Tables 6 and 7 and, it can be concluded that the proposed mathematical model of the leakage inductance overcomes the previous research in terms of improved model accuracy and reduced computational complexity and maintaining a constant copper loss.

Figure 18 presents the validation of the theoretical and simulation results of the proposed vertical winding shifting along with the percentage error. The maximum error of about 3.9% is seen at shifting distance of 2 mm. Whereas the error at the maximum shifting distance of 6 mm is about 3.5%, showing a constant calculation error even when the shifting distances is increased.

Theoretical and Simulation validation with Parentage Error.

Finally, Table 8 presents the experimental validation of the proposed HFT, where the primary and secondary windings are shifted upward and downward by 5 mm. The lowest percentage error between the calculation value and experimental value is achieved at the phase B. On the other hand, the maximum error occurs at phase C which is due to the required shifting distance is not executed accurately.

Moreover, by using the proposed mathematical model, any value of leakage inductance can be attained inside the transformer windings with no additional cost, as no additional material is required to be added in order to achieve a specific value of leakage inductance, considering the shifting distance should be less than or equal to the core window height limit which is equal to \(\:\frac{{H}_{w}-{h}_{wp}}{2}\).

This paper proposed a method for adjusting the leakage inductance value of 3P-HFTs by manipulating the winding arrangement. Specifically, the separation between certain turns of the primary and secondary windings was increased by strategically placing them at a specific height within a core window. Mathematical proof was provided to support this approach, and a practical and versatile method for continuously adjusting the leakage inductance from a minimum value was developed. Simulation and experiment verifications were conducted to validate the proposed method and confirm its accuracy. Unlike traditional methods that require adjustments to the insulation thickness and number of primary turns, the proposed method utilised a degree of freedom in the arrangement of the windings, which had previously been overlooked. A significant advantage of this method is its ability to continuously adjust the leakage inductance within a specified range without sacrificing copper loss. In addition, it can be implemented at no additional cost, making it a highly accessible and practical method for the design and optimisation of HFTs.

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Matsuura, K., Yanagi, H., Tomioka, S. & Ninomiya, T. Power-density development of a 5 MHz-switching DC-DC converter, in Twenty-Seventh Annual IEEE Applied Power Electronics Conference and Exposition (APEC), 2012, pp. 2326–2332. (2012). https://doi.org/10.1109/APEC.2012.6166147

Sugimoto, K., Watakabe, M., Okuno, M. & Homma, T. Stabilization of Impedance Matching System using RF Transformer. IEEJ Trans. Fundamentals Mater. 136 (7), 404–408. https://doi.org/10.1541/ieejfms.136.404 (2016).

Article ADS Google Scholar

De Doncker, R. W., Divan, D. M. & Kheraluwala, M. H. A Three-phase soft-switched high power density dc/dc converter for high power applications, in Conference Record - IAS Annual Meeting (IEEE Industry Applications Society), Publ by IEEE, pp. 796–805. (1988). https://doi.org/10.1109/ias.1988.25153

Nadia, M. L., Tan, S., Inoue, A., Kobayashi & Akagi, H. An Energy Storage System Combining a 320-V, 12-F Electric Double Layer Capacitor Bank with a Bidirectional Isolated DC-DC Converter, in Annual IEEE Conference on Applied Power Electronics Conference and Exposition (APEC), IEEE, (2008).

Nguyen, D. D., Bui, N. T. & Yukita, K. Design and optimization of three-phase dual-active-bridge converters for electric vehicle charging stations, Energies (Basel), vol. 13, no. 1, Dec. (2019). https://doi.org/10.3390/en13010150

Lei, T., Wu, C. & Liu, X. Multi-objective optimization control for the aerospace dual-active bridge power converter. Energies (Basel). 11 (5). https://doi.org/10.3390/en11051168 (2018).

Tan, N. M. L., Abe, T. & Akagi, H. Design and performance of a bidirectional isolated DC-DC converter for a battery energy storage system. IEEE Trans. Power Electron. 27 (3), 1237–1248. https://doi.org/10.1109/TPEL.2011.2108317 (2012).

Article ADS MATH Google Scholar

De Din, E., Siddique, H. A. B., Cupelli, M., Monti, A. & De Doncker, R. W. Voltage Control of Parallel-Connected Dual-Active Bridge Converters for Shipboard Applications, IEEE J Emerg Sel Top Power Electron, vol. 6, no. 2, pp. 664–673, Jun. (2018). https://doi.org/10.1109/JESTPE.2017.2786350

Segaran, D., Holmes, D. G. & Mcgrath, B. P. Comparative Analysis of Single and Three-phase Dual Active Bridge Bidirectional DC-DC Converters, in Australasian Universities Power Engineering Conference, AUPEC, (2008).

Baars, N. H., Everts, J., Wijnands, C. G. E. & Lomonova, E. A. Performance Evaluation of a Three-Phase Dual Active Bridge DC-DC Converter with Different Transformer Winding Configurations, IEEE Trans Power Electron, vol. 31, no. 10, pp. 6814–6823, Oct. (2016). https://doi.org/10.1109/TPEL.2015.2506703

Zhao, B., Song, Q., Liu, W. & Sun, Y. Overview of dual-active-bridge isolated bidirectional DC–DC converter for high-frequency-Link Power-Conversion System. IEEE Trans. Power Electron. 29 (8), 4091–4106. https://doi.org/10.1109/TPEL.2013.2289913 (2014).

Article ADS MATH Google Scholar

Yang, B., Lee, F. C., Zhang, A. J. & Huang, G. LLC resonant converter for front end DC/DC conversion, in APEC. Seventeenth Annual IEEE Applied Power Electronics Conference and Exposition (Cat. No.02CH37335), pp. 1108–1112 vol.2. (2002). https://doi.org/10.1109/APEC.2002.989382

Mogorovic, M. & Dujic, D. 100 kW, 10 kHz medium-frequency Transformer Design optimization and experimental Verification. IEEE Trans. Power Electron. 34 (2), 1696–1708. https://doi.org/10.1109/TPEL.2018.2835564 (2019).

Article ADS Google Scholar

Zhang, K. et al. Optimization Design of High-Power High-Frequency Transformer Based on Multi-Objective Genetic Algorithm, in IEEE International Power Electronics and Application Conference and Exposition (PEAC), 2018, pp. 1–5. (2018). https://doi.org/10.1109/PEAC.2018.8590371

Mogorovic, M. & Dujic, D. Medium frequency transformer leakage inductance modeling and experimental verification, in IEEE Energy Conversion Congress and Exposition (ECCE), 2017, pp. 419–424. (2017). https://doi.org/10.1109/ECCE.2017.8095813

Pavlovsky, M., de Haan, S. W. H. & Ferreira, J. A. Partial Interleaving: A Method to Reduce High Frequency Losses and to Tune the Leakage Inductance in High Current, High Frequency Transformer Foil Windings, in IEEE 36th Power Electronics Specialists Conference, 2005, pp. 1540–1547. (2005). https://doi.org/10.1109/PESC.2005.1581835

McLyman, C. W. T. Transformer and Inductor Design Handbook (CRC, 2011).

Google Scholar

Margueron, X., Keradec, J. P. & Magot, D. Analytical calculation of Static Leakage inductances of HF transformers using PEEC formulas. IEEE Trans. Ind. Appl. 43 (4), 884–892. https://doi.org/10.1109/TIA.2007.900449 (2007).

Article Google Scholar

Kauder, T., Hameyer, K. & Belgrand, T. Design strategy and simulation of medium-frequency transformers for a three-phase dual active bridge, in IECON –44th Annual Conference of the IEEE Industrial Electronics Society, 2018, pp. 1158–1163. (2018). https://doi.org/10.1109/IECON.2018.8592689

Pavlovsky, M., de Haan, S. W. H. & Ferreira, J. A. Winding losses in high-current, high-frequency transformer foil windings with leakage layer, in 37th IEEE Power Electronics Specialists Conference, 2006, pp. 1–7. (2006). https://doi.org/10.1109/pesc.2006.1711861

Cougo, B. & Kolar, J. W. Integration of Leakage Inductance in Tape Wound Core Transformers for Dual Active Bridge Converters, in 7th International Conference on Integrated Power Electronics Systems (CIPS), 2012, pp. 1–6. (2012).

De, D. et al. Achieving the desired transformer leakage inductance necessary in DC-DC converters for energy storage applications, in 6th IET International Conference on Power Electronics, Machines and Drives (PEMD 2012), pp. 1–6. (2012). https://doi.org/10.1049/cp.2012.0212

Baek, S., Du, Y., Wang, G. & Bhattacharya, S. Design considerations of high voltage and high frequency transformer for solid state transformer application, in IECON 2010–36th Annual Conference on IEEE Industrial Electronics Society, pp. 421–426. (2010). https://doi.org/10.1109/IECON.2010.5674991

Hoang, K. D. & Wang, J. Design optimization of high frequency transformer for dual active bridge DC-DC converter, in Proceedings – 2012 20th International Conference on Electrical Machines, ICEM 2012, pp. 2311–2317. (2012). https://doi.org/10.1109/ICElMach.2012.6350205

Yang, J., Wu, X., Muhammad, F. & Deng, Z. External magnetic field minimization for the Integrated Magnetics in Series Resonant Converter. IEEE Trans. Power Electron. 37 (1), 498–508. https://doi.org/10.1109/TPEL.2021.3095491 (2022).

Article ADS MATH Google Scholar

Sharma, A. & Kimball, J. W. Novel Transformer with Variable Leakage and Magnetizing Inductances, in 2021 IEEE Energy Conversion Congress and Exposition (ECCE), pp. 2155–2161. (2021). https://doi.org/10.1109/ECCE47101.2021.9595797

Sharma, A. & Kimball, J. W. Evaluation of Transformer Leakage Inductance using magnetic image Method. IEEE Trans. Magn. 57 (11), 1–12. https://doi.org/10.1109/TMAG.2021.3111479 (2021).

Article MATH Google Scholar

Sharma, A. & Kimball, J. W. New Hybrid Model for Evaluating the Frequency-Dependent Leakage Inductance of a Variable Inductance Transformer (VIT), in 2022 IEEE Energy Conversion Congress and Exposition (ECCE), pp. 1–7. (2022). https://doi.org/10.1109/ECCE50734.2022.9947451

Jung, T. U., Kim, M. H. & Yoo, J. H. Design optimization of high frequency transformer with controlled leakage inductance for current fed dual active bridge converter. AIP Adv. 8 (5), 56646. https://doi.org/10.1063/1.5007773 (2018).

Article MATH Google Scholar

Li, X., Huang, W., Cui, B. & Jiang, X. Inductance Characteristics of the High-Frequency Transformer in Dual Active Bridge Converters, in 22nd International Conference on Electrical Machines and Systems (ICEMS), 2019, pp. 1–5. (2019). https://doi.org/10.1109/ICEMS.2019.8921741

Shintai, R., Orikawa, K., Ogasawara, S. & Takemoto, M. High Frequency Transformers with Adjustable Leakage Inductance using Wire Guides, in 23rd International Conference on Electrical Machines and Systems (ICEMS), 2020, pp. 1547–1552. (2020). https://doi.org/10.23919/ICEMS50442.2020.9291023

Dawood, K., Işik, F. & Kömürgöz, G. Analysis and optimization of leakage impedance in a transformer with additional winding: a numerical and experimental study. Alexandria Eng. J. 61 (12), 11291–11300. https://doi.org/10.1016/j.aej.2022.05.014 (2022).

Article MATH Google Scholar

Dawood, K., Sönmez, O. & Kömürgöz, G. Optimization and analysis of tapping position on leakage reactance of a two winding transformer. Alexandria Eng. J. 69, 363–369. https://doi.org/10.1016/j.aej.2023.02.006 (2023).

Article MATH Google Scholar

Guo, X., Li, C., Zheng, Z. & Li, Y. General Analytical Model and Optimization for Leakage Inductances of medium-frequency transformers. IEEE J. Emerg. Sel. Top. Power Electron. 10 (4), 3511–3524. https://doi.org/10.1109/JESTPE.2021.3062019 (2022).

Article MATH Google Scholar

Ayachit, A. & Kazimierczuk, M. K. Transfer functions of a transformer at different values of coupling coefficient. IET Circuits Devices Syst. 10 (4), 337–348 (2016).

Article MATH Google Scholar

Hurley, W. G. & Wolfle, W. H. TRANSFORMERS AND INDUCTORS FOR POWER ELECTRONICS (John Wiley & Sons Ltd., 2013).

Book MATH Google Scholar

TDK Material characteristics, May Accessed: Jan. 16, 2024. [Online]. (2022). Available: https://product.tdk.com/system/files/dam/doc/product/ferrite/ferrite/ferrite-core/catalog/ferrite_mn-zn_material_characteristics_en.pdf

Inoue, S., Akagi, H., Bidirectional Isolated, A. & DC–DC Converter as a Core Circuit of the Next-Generation Medium-Voltage Power Conversion System. IEEE Trans. Power Electron. 22 (2), 535–542. https://doi.org/10.1109/TPEL.2006.889939 (2007).

Article ADS MATH Google Scholar

Sharifuddin, N. S. M., Tan, N. M. L., Toh, C. L. & Ramli, A. Q. Implementation of Three-Phase Bidirectional Isolated DC-DC Converter with Improved Light-Load Efficiency, in IEEE Industrial Electronics and Applications Conference (IEACon), 2021, pp. 224–229. (2021). https://doi.org/10.1109/IEACon51066.2021.9654573

Dira, Y. S., Ramli, A. Q., Ungku Amirulddin, U. A., Tan, N. M. L. & Buticchi, G. Design and analysis of a three-phase high-frequency transformer for three-phase bidirectional isolated DC-DC Converter using Superposition Theorem. Sustainability 16 (21). https://doi.org/10.3390/su16219227 (2024).

Villar, I. Multiphysical Characterization of Medium-Frequency Power Electronic Transformers ( EPFL Lausanne, 2010).

MATH Google Scholar

Zhang, K. et al. Accurate calculation and sensitivity analysis of Leakage Inductance of high-frequency transformer with Litz Wire Winding. IEEE Trans. Power Electron. 35 (4), 3951–3962. https://doi.org/10.1109/TPEL.2019.2936523 (2020).

Article ADS MATH Google Scholar

Download references

This work was supported by Universiti Tenaga Nasional (UNITEN) through the Energy Transition Grant Fund under Project 202205002ETG.

Institute of Power Engineering, Universiti Tenaga Nasional, Kajang, 43000, Malaysia

Yasir S. Dira, Ahmad Q. Ramli & Ungku Anisa Ungku Amirulddin

Key Laboratory of More Electric Aircraft Technology of Zhejiang Province, University of Nottingham Ningbo China, Ningbo, 315100, China

Nadia M. L. Tan

You can also search for this author in PubMed Google Scholar

You can also search for this author in PubMed Google Scholar

You can also search for this author in PubMed Google Scholar

You can also search for this author in PubMed Google Scholar

Conceptualization, Y.D. and A.Q.; resources, Y.D. and A.Q.; writing original draft preparation, Y.D. and A.Q.; writing review and editing, Y.D., A.Q., U.A. and N.T.; supervision, A.Q., U.A. and N.T.; project administration, A.Q.; funding acquisition, A.Q. All authors have read and agreed to the published version of the manuscript.

Correspondence to Yasir S. Dira or Ahmad Q. Ramli.

The authors declare no conflict of interest.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.

Reprints and permissions

Dira, Y.S., Ramli, A.Q., Amirulddin, U.A.U. et al. A modelling technique to determine the high frequency transformer leakage inductance using the winding structure. Sci Rep 15, 2373 (2025). https://doi.org/10.1038/s41598-025-86816-z

Download citation

Received: 17 May 2024

Accepted: 14 January 2025

Published: 18 January 2025

DOI: https://doi.org/10.1038/s41598-025-86816-z

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative