Calculation Of Leakage Inductance Research Articles

Analytical Double-2D Frequency-Dependent Leakage Inductance Model for Litz-Wired High-Frequency Transformer

Accurate and fast calculation of frequency-dependent leakage inductance is critical for the optimization of the high-frequency transformer with Litz wire winding. The analytical method is widely used due to its high computation speed. However, its computation accuracy is limited due to the following factors. i) One-dimensional (1D) or two-dimensional magnetic field model (2D) has unsatisfying accuracy for the transformer with a rotationally asymmetric structure. ii) The skin and proximity effect factors based on the multilayer foil winding are inaccurate for the Litz wire winding in the calculation of frequency-dependent magnetic energy. This paper proposes a double-2D frequency-dependent leakage inductance model for the transformer with Litz wire winding. It uses the image method to calculate the 2D leakage magnetic field inside and outside the core window, respectively, which considers the rotationally asymmetric structure in leakage inductance calculation. The round conductor model is proposed for the frequency-dependent magnetic energy calculation. The single strand in a Litz wire is taken as the basic modeling object of the round conductor model, which is more detailed than the multilayer foil winding model for the Litz wire winding. The comparison between the experiment, FEM, state-of-the-art models and the proposed model is conducted for eight transformers with E-core, U-core, round or rectangular Litz wire winding. The proposed model shows satisfying computation accuracy in the whole frequency band. And the computational speed of the proposed model is 100 times that of the FEM.

Improved methods for stator end winding leakage inductance calculation

PurposeCalculating the stator end-winding leakage inductance, taking into account the rotor, is difficult due to the irregular shape of the end-winding. The end-winding leakage may distribute at the end of the active part and the fringing flux of the air gap. The fringing flux belongs to the main flux but goes into the end-winding region. Then, not all the magnetic flux occurring in the end region is the end-winding leakage flux. The purpose of this paper was to find a method to accurately separate the leakage from the total flux, taking into account the rotor.Design/methodology/approachIn this paper, two methods based on energy calculation are presented. Both methods require the assumption that the machine is symmetrical. The first method depends on the total leakage inductance and the machine’s active region length. The second method is based on the energy stored in the end region of the machine. In this case, removing the energy produced by the fringing flux of the air gap is necessary. The model should have a volume-closing fringing flux to remove the part of energy belonging to the end of the air gap.FindingsThe method presented in the paper does not require rotor removal. The values of the end-winding leakage inductance computed based on the proposed method were compared with values computed using the method with the removed rotor. The computations show that the proposed method is closest to the results from the method presented in the literature. Results obtained in the first method present that rotor influence on the value of end-winding leakage inductance exists. The model of the stator end-winding described in the paper is general. Therefore, the proposed methods are suitable for calculating the end-winding leakage inductance of other electric machines.Originality/valueThe method presented in the paper considers the rotor in end-winding leakage inductance calculation. It is not necessary to remove the rotor as in the similar method presented in the literature. The authors elaborated a parametric model with a volume-closing fringing flux to remove the part of energy belonging to the end of the air gap. The authors also elaborated their 3D model of the machine winding for calculations in Opera 3D.

The Structure and Its Leakage Inductance Model of Integrated LLC Transformer With Wide Range Value Variation

The integrated magnetics technology is famous in academic and industrial applications to achieve low profile and high power density of switch-mode power supply (SMPS). The integrated LLC transformer is widely used in LLC converter, where the transformer and resonant inductor are integrated into a magnetic component. However, the value Variation range of leakage inductance of conventional integrated transformer structures is limited due to the low profile and small volume in some planar applications. In this paper, a new integrated LLC transformer structure and its leakage inductance adjustment method are proposed. Based on the two-dimensional equivalent model of the structure, the distribution of the leakage magnetic field and the leakage inductance calculation model of the proposed transformer structure are established. The proposed structure can improve the value variation range of leakage inductance in some special applications and is easily achieved in the planner structure. Finally, the design guideline of the proposed integrated LLC transformer is obtained. Simulation results and prototype experiment results verify the correctness and flexibility of the theoretical analysis.

Transformer Leakage Inductance Calculation Method with Experimental Validation for CLLLC Converter Topology

Leakage inductance is one of the key parameters of a transformer, and it is often intentionally integrated into transformers. Rogowski’s equation is generally used for leakage inductance calculation; however, it is only applicable to concentric winding transformers where windings have the same height. Consequently, it has limited applications. This paper proposes a transformer leakage inductance calculation method using a double Fourier series. The limitation of Rogowski’s leakage inductance equation was analyzed in practical applications, and a new model for calculating the leakage inductance of a double-group-overlapping winding transformer was derived. Experimental prototypes of transformers were developed, and their simulation models were built in Ansys. The correctness of the proposed calculation method for transformer leakage inductance using a double Fourier series was verified by comparing the calculation results with the simulation and measured ones. A sensitivity analysis was also conducted by studying the variations in different parameters that might affect the leakage inductance value. The proposed calculation model gives an intuitive and simple method with less calculation and design effort while maintaining reasonable accuracy for leakage inductance calculation. In addition, the featured double Fourier series approach has a wider range of applications than Rogowski’s equation.

Analysis and Verification of Leakage Inductance Calculation in DAB Converters Based on High-Frequency Toroidal Transformers under Different Design Scenarios

High-frequency transformers are becoming an essential component in the integration of power resources that rely on power electronic converters; their efficiency and performance are influenced by parasitic characteristics in the interface. In this article, the design of a high-frequency toroidal transformer has been explained in detail using the ANSYS Maxwell platform. Various parameters, such as leakage inductance, magnetic flux density, magnetic field strength and uniform magnetic flux line are analyzed using Finite element analysis. High-frequency transformers using a toroidal core with different winding configurations are examined and all parameters obtained through simulation are validated by an analytical approach. Analysis of each design is based on its leakage inductances, which will aid in the appropriate selection of transformers as a function of their operating frequency. This analysis is expected to guide designers to optimize the high-frequency transformer parameters based on practical applications. The optimized parameters are then applied for a dual active bridge (DAB) converter within MATLAB/Simulink to verify the design process. A prototype has been built to validate the simulation and design procedure. The results obtained from both simulation and experiments are compared and show great correlation.

Module Integral Method for the Calculation of Frequency-Dependent Leakage Inductance of High-Frequency Transformers

The accurate calculation of frequency-dependent leakage inductance of high-frequency transformers is critical for the optimized design of power electronic transformers. The leakage inductance can be obtained through the magnetic energy, usually calculated by the phasor integral of the magnetic energy density in the literature. This article analyzes the difference between the module and the phasor integral based on frequency-domain energy and proposes to take the module integral of magnetic energy density as the periodic averaged magnetic energy. Furthermore, the closed-form formula of the leakage inductance is given accordingly. The modules under different frequencies correspond to the intrinsic constant magnetic densities. Thus, the proposed method has a better physical interpretation compared with the previous phasor integral method. The finite element method simulation and experimental results of five transformers are taken as benchmarks to compare the proposed method with the phasor integral method. The comparison shows that the proposed module integral method has satisfying effectiveness and accuracy in the whole frequency band for all the cases involving both noninterleaved and interleaved windings.

Induction Machine Sensorless Control Based on Saliency Extraction That Uses One Single SVPWM Active State

Some induction motors present saliencies that periodically modify the air-gap flux following a sinusoidal shape. Consequently, these modulations are reflected in the transient leakage inductance. Therefore, transient leakage inductance can be expressed as a constant term or offset, which comes from the symmetrical machine, plus all motor saliency sinusoids, modulated by the mentioned saliencies. Since saliencies are linked to the rotor position or flux, calculation of transient leakage inductance enables encoderless vector control. A way to calculate transient leakage inductance is to excite the machine with voltage steps and estimate the resulting phase current slopes since they are inversely proportional to phase transient inductance. For creating a vector, only the three current slopes resulting from a single active state are sufficient. The vector sum of the three current slopes leads to the so-called saliency-offset vector, which is a vector composed of an offset, which is created due to the contribution of the transient inductance offset, plus several saliency phasors, superposed to the offset. To access saliencies, it is critical to accurately identify the offset of the saliency-offset vector, especially in induction motors since offset is much bigger than saliencies. This article presents an advanced signal processing scheme that identifies the offset in the saliency-offset vector, assuming offset amplitude differsdepending on the voltage step applied. Using a feedforward scheme, the offset is eliminated, leading to a saliency vector where rotor slotting can be extracted from. A novel excitation is proposed to reduce additional current distortions coming from the one-active excitation.

Comparison of Analytical Method and Different Finite Element Models for the Calculation of Leakage Inductance in Zigzag Transformers

A zigzag transformer is a key segment of the electric power system. The optimal design of the zigzag transformer is important for transformer designers to provide a required return path for earth faults and to ensure proper operation of a power system. The two most important parameters of the zigzag transformers are no-load losses and leakage impedance. The accurate calculation of both factors helps to minimize the overall cost of the transformer. Therefore, the prediction of leakage reactance in the zigzag transformer using analytical or numerical methods is an essential part of the early designing stages of the transformer. This paper provides several two- and three-dimensional finite element models. The main purpose of these models is to evaluate the accuracy of the different models for the calculation of the leakage reactance. An analytical formula and a complete procedure for the calculation of the leakage reactance in the zigzag transformer are also provided, which will help the researchers and transformer designer to optimize this type of transformer. The prototype is also manufactured and tested to verify the accuracy of the analytical method and finite element models for the calculation of the leakage reactance. The simulation and experimental results show that the finite element calculation cannot only obtain accurate leakage reactance values (magnetostatic analysis), but also provides better accuracy in the no-load losses (transient analysis).

Inductance Calculation of HTS Transformers with Multi-segment Windings Considering Insulation Constraints

High-temperature superconducting (HTS) transformers with multi-segment windings have been proposed in earlier literatures for the hysteresis loss reduction and short circuit electromagnetic force mitigation. The optimum distributive ratios for these multi-segment windings have been determined in earlier literatures. Asymmetrical seven segment winding with its optimum distributive ratio (k = 1/12) results in minimization of short circuit electromagnetic forces under normal tap and any adjusted tap. However, insulation design and leakage inductance calculation for the multi-segment winding of an HTS transformer have not been addressed in past literatures. Insulation design is one of the most crucial aspects in the transformer design, especially for high voltage power transformers. The analysis and optimization of the electric field distribution over the winding’s segments insulation in these HTS transformers provide the required dimensions for precise determination of the leakage inductances. In this paper, for the first time, winding insulation design and leakage inductance calculation of a 200 MVA, 230/20 kV HTS transformer with asymmetrical seven segment winding (and k of 1/12), are carried out. The analytical method besides a two-dimensional (2D) finite element method (FEM), through COMSOL Multiphysics software, is utilized for insulation design and calculation of leakage inductances of this transformer.

An Accurate Analytical Method for Leakage Inductance Calculation of Shell-Type Transformers With Rectangular Windings

This paper presents an accurate analytical method for calculating the leakage inductance of shell-type E-core transformers with rectangular windings. For this purpose, first, an expression for calculating the leakage inductance per unit length inside the core window considering the core walls as the flux-normal boundary condition is derived. Then, a new accurate method for determining the Mean Length of Turns (MLT) based on the total stored energy is presented. The MLT is needed for the leakage inductance calculation using 2-D methods. By dividing the MLT into three partial lengths and calculating the corresponding leakage inductances using three different core window arrangements, the effect of core structure on the total leakage inductance is considered. The method is verified by 3-D FEM simulations as well as the leakage inductance measurements on two different fabricated transformer prototypes. The superiority of the method is also confirmed by comparisons with the previous analytical approaches. The proposed method enables the leakage inductance calculation with an error less than 1%, compared to the 3-D FEM results. Using the presented method, the leakage inductance calculations can be performed rapidly and accurately in the design stage without the need for time-consuming 3-D FEM simulations.

An Accurate Method for Leakage Inductance Calculation of Shell-Type Multi Core-Segment Transformers With Circular Windings

The leakage field in shell-type transformers is strongly affected by the boundary conditions introduced by the core walls and thus the effect of the core should be considered properly in the leakage inductance calculation. In this paper, a new method for accurate calculation of the leakage inductance of shell-type multi core-segment transformers with circular windings is presented. For this purpose, first, the expressions for self and mutual inductances are derived in cylindrical coordinates considering the core walls as the flux-normal boundary condition. Then, a new approach is proposed for calculating the leakage inductance considering the number and dimensions of the used core segments. The method is developed at first for single and double core-segment transformers (known also as E-core and U-core transformers) and then adopted for shell-type segmented-core transformers. The method is verified by 3-D FEM simulations. The comparisons with the previous analytical methods demonstrate the superiority of the proposed method. A transformer prototype has been built and verification tests have been conducted. The comparisons show that the leakage inductance can be estimated with an error less than 1%, demonstrating a very high accuracy with the proposed method.

Calculation of Leakage Inductance in Toroidal Core Transformer With Non-Interleaved Windings

Recently, transformers with toroidal core have received much attention. The compactness and weight of these kinds of transformers make them suitable for power electronics applications. In some areas, it is necessary to have a toroidal transformer with particular parasitic elements such as leakage inductance. So, there is a need for a precise method to calculate the exact amount of leakage inductance for designing the desired system. This article presents a pure analytical method based on Maxwell’s equations to accurately determine the leakage inductance of a toroidal transformer which its windings are wounded separately. The proposed expression for the leakage inductance calculation considers the influence of the core specifications, such as dimensions and material. In order to validate the presented method, 3-D finite-element simulation and experimental results of the prototype transformer have been done.

Analytical Calculation of Resistance, Leakage Inductance and Parasitic Capacitance for High Frequency Transformer

The design of a high-frequency transformer, which is the main component of a DC/DC converter, is very important. In the case of a DC transmission system, such as HVDC, the transformer should use high power and high frequency, but there are few studies in this category. Therefore, this study examined the design of a high frequency transformer. When designing the transformer covered in this paper, it is necessary to consider the frequency components differently from a traditional transformer design. Therefore, winding resistance, leakage inductance, and parasitic capacitance were calculated according to the frequency change, and finite element method (FEM) analysis and comparative analysis were conducted for validation.

Improved calculation of the slot leakage inductance of different slot shapes

The slot leakage inductance calculation is an important element of the electric machine’s design. The empirical definitions are described in the literature and widely used until today. The crucial parameter is the slot permeance factor which defines the influence of slot dimensions on the leakage inductance. The traditional approach to the slot leakage inductance calculation is used in this paper, but improved flux line length and active slot area calculation are presented. The main objective of the paper is the definition of relatively simple relation to the calculation of the slot permeance factor and comparison with the finite element calculation and literature-based equations. Therefore, an overview of different slot types is presented in this paper, and obtained results are in general more reliable than literature-related relations based on the finite element validation.

Accurate Calculation and Sensitivity Analysis of Leakage Inductance of High-Frequency Transformer With Litz Wire Winding

Leakage inductance is an important parameter in a high-frequency transformer for its influence on power transmission performance and the soft switching of converters. Nowadays, Litz wire is widely used to control winding loss and reach high efficiency. However, it is not an easy task to accurately calculate its leakage inductance. This article presents a detailed model of leakage energy in Litz wire, and proposes an accurate closed-form expression for calculating the leakage inductance in a high-frequency transformer. The skin effect and the proximity effect of Litz wire are taken into account in two-dimensional polar coordinates, and the filling factor of Litz wire is also considered. A global sensitivity analysis is performed to assess the influence of each parameter on the overall leakage inductance. Finally, two transformer prototypes are made to verify the accuracy of the proposed analytical expression.

Finite-Element Analysis Modeling of High-Frequency Single-Phase Transformers Enabled by Metal Amorphous Nanocomposites and Calculation of Leakage Inductance for Different Winding Topologies

The solid-state transformer (SST) is an emerging technology which is gaining increasing importance for future power distribution systems. Here, we present 2-D finite-element analysis (FEA) of single-phase SSTs for operating frequencies of 50 kHz. We consider designs to benchmark materials aimed at controlling high-frequency losses to achieve higher power densities. The FEA model is solved in the time domain for frequencies of 50 kHz, at a fixed 800 V input voltage and 10 kW output power for the first four cycles. This paper analyzes the influence of leakage inductance on the waveform with an increasing number of turns and discusses how to calculate the leakage inductance of the different winding topologies by FEA. In addition, transformer topologies are coupled to a power analysis using a Steinmetz parameterization of magnetic losses capturing induction and field scaling for transformer losses for FeNi-based metal amorphous nanocomposites (MANCs) for SST applications and their benchmarking with amorphous and other MANCs and ferrites. Recently, discovered FeNi-based MANCs allow smaller transformers at equivalent power as compared to Si steels, Metglas, and Co-based MANCs. These Fe-rich and non-Co containing MANCs also offer economies based on lower raw materials costs as compared with Co-based MANCs and significantly improved mechanical properties with respect to commercially available Fe-based MANCs.

Leakage Inductance Calculation of the Transformers With Disordered Windings

In switching power supplies, the transformer leakage inductance (TLI) is of considerable practical significance. Ordered winding is the certain assumption in previously reported works for the calculation of the TLI value. However, disordered windings are usually unavoidable in practice and their effect on the value of the TLI is considerable. The problem becomes more complex when the number of winding turns increases. In this paper, the effect of winding disorder on the TLI value is investigated. In disordered wrapping, the winding turns arrangement (WTA) of each produced transformer is different from the others. Thus, each produced transformer has an exclusive TLI value in a disordered wrapping. Moreover, the number of possible WTAs is high, especially in transformers with a large number of winding turns. The proposed method utilizes an analytical tool for estimating the distribution of the TLI values for all of the possible WTAs. This estimation is carried out based on the limited number of the WTAs. The required number of WTAs is calculated and the validity of the proposed method is checked with Kolmogorov–Smirnov theorem. The TLI value is calculated by means of the stored magnetic energy in different parts of the transformer using the magnetomotive force variation method. Finite element simulation and experimental results are presented to show the effects of disorder on the TLI value and the validity of the proposed analytical method.

High-Frequency <italic>LLC</italic> Resonant Converter With Magnetic Shunt Integrated Planar Transformer

Achieving high efficiency and high power density is emerging as a goal in many power electronics applications. LLC resonant converter has been proved as an excellent candidate to achieve this goal. To achieve smaller size of passive components, the resonant inductor in the LLC converter is usually integrated into the transformer by utilizing its leakage inductance. However, the leakage inductance of the transformer is usually insufficient and thus the LLC converter has to be operated in a limited frequency range (this limits the input voltage range accordingly), otherwise the power efficiency will drop dramatically. Therefore, a larger resonant inductance in the LLC converter is expected to operate in a wider input voltage range. This paper proposes a new method to create a larger resonant inductance by using a magnetic shunt integrated into planar windings. The accurate leakage inductance modeling, calculation, and optimal design guideline for LLC planar transformer, including optimal magnetic shunt selection and winding layout, are presented. A 280–380 V input and output 48 V–100 W half-bridge LLC resonant converter with 1 MHz resonant frequency is built to verify the design methodology. A comparison is made between two converters with the same parameters, one using magnetic shunt integrated transformer and the others using traditional planar transformer and external inductor. Experimental result shows the proposed converter with magnetic shunt is capable to achieve comparable high efficiency and regulation capability with the other under a wide input voltage, which verifies the optimal design methodology. Above all, this magnetics integration methodology reduces the whole converter's volume and thus increases the power density.

Semi-Analytical Methods for Calculation of Leakage Inductance and Frequency-Dependent Resistance of Windings in Transformers

Leakage inductance and frequency-dependent resistance of windings are the key parameters for the design of magnetic devices. This paper focuses on analytical or semi-analytical calculation techniques to be integrated into a medium frequency transformer optimization process. To this extend, a review of the currently most used method to calculate the inductance and resistance of windings transformer is established. Based on this review, new calculation methods are developed. Magnetic field cartography is evaluated on the basis of 2-D structures thanks to a combination of Ampere’s law and the image method. From this magnetic field map, leakage inductance can be calculated. The obtained accuracy with this method is equivalent to 3-D finite-element modeling (FEM). For frequency-dependent resistance evaluation, new methods based on 2-D or 1-D approach are compared to FEM and reviewed models. An analysis of the behavior of each model is done, showing that newly developed models are more suitable in the case of a medium frequency transformer design in terms of accuracy, robustness, and calculation time. This paper could be useful to the designers of magnetic devices because newly developed models could easily be adapted to various geometries, such as toroidal structures or even inductors.

Mathematical model of high-voltage instrument autotransformer intended for use in smart grid networks

The object of the research is a mathematical model of the active part of the high-voltage instrument autotransformer with several output windings. The most challenging task in this model is leakage inductance calculation of single winding turns or groups of autotransformer winding turns. Also, a significant problem when calculating the parameters of the autotransformer are those operating conditions, that are close to no-load conditions. To create a mathematical model of the active part of a high-voltage autotransformer, taking into account the leakage inductance of each single winding turn (or groups of winding turns), and also the magnitude and type of the load, it is proposed to modify the known system of transformer equations detailed to the level of single turns (or groups of turns) and use the proposed method for determining the partial self- and mutual- leakage inductances by numerical methods. With a help of the developed mathematical model of the active part of a high-voltage autotransformer, a prototype of a 10 kV autotransformer with high metrological characteristics was designed and manufactured. Positive results of metrological certification of the created high-voltage autotransformer confirmed the possibility of calculating the parameters of such primary high-voltage transformers that can be used in Smart Grid networks in conjunction with analog-to-digital converters, thus unifying the electromagnetic primary high-voltage transformers. It is shown that the operating conditions of high-voltage autotransformers, close to no-load conditions, allow easily achieve high accuracy of high voltage transformation.