Generalized Steinmetz Equation Research Articles

The Composite Improved Generalized Steinmetz Equation (ciGSE): An Accurate Model Combining the Composite Waveform Hypothesis With Classical Approaches

The Composite Improved Generalized Steinmetz Equation (ciGSE): An Accurate Model Combining the Composite Waveform Hypothesis With Classical Approaches

A novel estimation model for hysteresis loss based on semi-minor loops

This paper proposes an estimation model of hysteresis loss based on the mapping of the hysteresis loop conducted through tiny material samples with the aid of the Superconducting Quantum Interference Device (SQUID). This approach allows precise measurement of hysteresis loops. The aim of this paper is to develop a precise estimation model for hysteresis loss, one component of core loss, so as to improve the overall estimation accuracy of the entire core loss under complex excitations. First, with the help of the SQUID system and measurements conducted on tiny samples, the proposed model manages to minimize the impact of eddy current throughout the measurement and the modeling of hysteresis loss. This contrasts with the use of the conventional Steinmetz model and its modifications. Moreover, compared with methods including the modified Steinmetz equation (MSE) and the generalized Steinmetz equation (GSE) which merely focuses on the overall iron loss under simple non-sinusoidal excitations, the model presented here enables the estimation of hysteresis loss under complex excitations where multiple peaks and DC-bias take place. This approach will be very useful in situations such as those requiring pulse-width modulation (PWM) excited converters which are quite common on occasions including wind power systems. The estimation error of this model on the actual coil is within 15% under given excitations.

Integrated Inductor Design for a Highly Compact Embedded Battery Charger

An integrated inductor for a highly compact embedded battery charger is designed in this article. A zero-voltage switching (ZVS) buck converter is the selected topology to achieve high power density. ZVS is achieved by sizing the inductor based on energy balance and introducing the <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> factor. A novel four-leg magnetic core structure is proposed as the integrated inductor. By introducing a modified parameter <inline-formula><tex-math notation="LaTeX">$\alpha _{\mathrm{eff}}$</tex-math></inline-formula> to the improved generalized Steinmetz equation method, the optimal switching frequency for core losses can be found. From the constructed inductor model and semiconductor loss calculation, the design tradeoff and optimization is analyzed. The optimal <inline-formula><tex-math notation="LaTeX">$f_{\mathrm{sw}}$</tex-math></inline-formula> is found to be around 2 MHz, which is when the performance factor for the core material peaks. As a proof of concept, a highly compact prototype, featuring stacked printed circuit board solution, with 16.2 &#x00D7; 19.2 &#x00D7; 6 mm<inline-formula><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> dimension is built. The input voltage ranges from 6 to 12 V, and the converter is capable of delivering up to 10 A at the 4-Voutput. Converter peak efficiency of 96&#x0025; is achieved, and the inductor design has a power density of 113.6 W/cm<inline-formula><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula>. This shows improvement over several state-of-the-art designs. This article is accompanied by a video demonstrating the flux distribution in the magnetic core over the switching period.

Improvement on Loss Separation Method for Core Loss Calculation Under High-Frequency Sinusoidal and Nonsinusoidal Excitation

The original loss separation formula introduced by Bertotti has long been utilized mainly for low-frequency sinusoidal applications. In this article, the traditional method is improved to fit into the core loss calculation under higher frequency sinusoidal and nonsinusoidal excitations, verified by the measured data of nanocrystalline material that is often used in medium-frequency transformers (MFTs). For high-frequency sinusoidal waveform, it is proposed that the exponent of the excess loss needs to be reidentified, and it is necessary to consider the skin effect for hysteresis and eddy current loss calculation under broader frequency scope. When the core is supplied with nonsinusoidal excitation (square and rectangular waveform), a new empirical method is derived based on the concept of effective frequency and the loss separation results, with which the estimation accuracy is greatly guaranteed compared to empirical methods originated from the Steinmetz equation such as improved generalized Steinmetz equation (iGSE). Due to clear separation of loss components, the new method has the potential to precisely predict losses for higher frequency applications.

Influence of Non-Linearity in Losses Estimation of Magnetic Components for DC-DC Converters

In this paper, the problem of estimating the core losses for inductive components is addressed. A novel methodology is applied to estimate the core losses of an inductor in a DC-DC converter in the time-domain. The methodology addresses both the non-linearity and dynamic behavior of the core magnetic material and the non-uniformity of the field distribution for the device geometry. The methodology is natively implemented using the LTSpice simulation environment and can be used to include an accurate behavioral model of the magnetic devices in a more complex lumped circuit. The methodology is compared against classic estimation techniques such as Steinmetz Equation and the improved Generalized Steinmetz Equation. The validation is performed on a practical DC-DC Buck converter, which was utilized to experimentally verify the results derived by a model suitable to estimate the inductor losses. Both simulation and experimental test confirm the accuracy of the proposed methodology. Thus, the proposed technique can be flexibly used both for direct core loss estimation and the realization of a subsystem able to simulate the realistic behavior of an inductor within a more complex lumped circuit.

Comparison of low-frequency non-sinusoidal measurement for transformer no-load characteristics based on different interpolation methods

As the calculation of transformer no-load characteristics measured by low-frequency test method is not perfect, different interpolation methods are explored to fit and convert the transformer's volt-ampere characteristics under power frequency (50 Hz) excitation to find the best interpolation fitting conversion method. The no-load test of transformer is carried out by using 10 Hz, 15 Hz, 20 Hz low-frequency square wave and triangular wave excitation. Then the measured excitation voltage, excitation current and no-load loss are converted to those of 50 Hz sine wave excitation. The converted volt-ampere characteristic curve is calculated by the proposed algorithm, and compared with the measured results under the 50 Hz sine wave excitation. Experiments show that under the condition that the magnetic flux amplitudes are equal at each frequency of non-sinusoidal and sine waves, whether it is square wave or triangular wave excitation, the volt-ampere characteristic curve converted to the 50 Hz sine wave excitation by the proposed method can better approximate the curve of the measured 50 Hz sine wave excitation. The average values of the relative errors of the converted excitation current of the three frequencies are within 4%, and those of the converted excitation voltage of the square wave are within 1.1%, and those of the converted excitation voltage of the triangular wave are within 0.47%. Compared with the generalized Steinmetz equation method and the multi-frequency method, the proposed method has a better conversion effect and can almost replace the no-load test results of the 50 Hz sine wave, which has practical value in engineering.

Novel iGSE-C Loss Modeling of X7R Ceramic Capacitors

Due to the large relative permittivity of Class II dielectrics, ceramic capacitors (CCs) from these materials promise significant volume and weight reductions in inverter and rectifier sine-wave filters, and are especially attractive in mobile applications that demand ultrahigh power density. While previous literature found large low-frequency losses in these components, no extensible loss model was proposed to accurately characterize these ferroelectric losses. In this article, we take advantage of prior art on ferromagnetic components in power electronics to propose a Steinmetz parameter-based loss modeling approach for X7R CCs, named the Improved Generalized Steinmetz Equation for CCs, or iGSE-C. This model is verified using the Sawyer-Tower circuit to measure losses in a commercially available X7R capacitor across excitation magnitude, dc bias, temperature, excitation frequency, and harmonic injection. Losses are shown to scale according to a power law with charge, with the resulting Steinmetz coefficients valid across dc bias and slightly varying as the temperature is increased. The iGSE-C accurately predicts losses for typical nonsinusoidal phase voltage waveforms with an error under 8%. Finally, the loss modeling technique is demonstrated for the sine-wave output filter of a bridge-leg arrangement with both low- and high-frequency excitations, with total capacitor losses predicted within 12% accuracy.

On the Optimization of a Class-E Power Amplifier With GaN HEMTs at Megahertz Operation

Class-E power amplifiers have regained academic interest over the past decades due to the introduction of new high-performance wide-bandgap semiconductor devices and the increasing demand for high-efficiency power amplifiers. While these power devices, notably GaN HEMTs, have exceptional performance at megahertz operation, they also display an additional loss component due to $C_{{\rm oss}}$ at this frequency. Unfortunately, the dependence of this loss term on the peak voltage and the device size is the opposite of that of the conduction loss. In this article, we mathematically analyze the operation of a Class-E amplifier to find the optimal input voltage and device sizing where the sum of these two losses is minimized. From this analysis, we have found that the common design approach of maximizing device voltage rating and area to get the best efficiency no longer holds for some operating conditions. Furthermore, we examine the constraints that dictate when this optimization equations can and cannot be used, as well as propose a distributed loss model for the $C_{{\rm oss}}$ loss based on a generalized Steinmetz equation (GSE) to allow this new loss term to be easily simulated. To verify our mathematical analysis and to demonstrate the applications of the proposed GSE-based $C_{{\rm oss}}$ loss model, two design examples are provided. They consist of a choke-input Class-E amplifier at 10 MHz and a variable-resistance Class-E amplifier at 40.68 MHz. The experimental results on these two design examples show good agreements with our analysis.

Comparative Core Loss Calculation Methods for Magnetic Materials under Harmonics Effect

The general data of industrial electromagnetics materials are given only for sinusoidal waveform, and mostly load is non-linear which may cause the harmonics effect. Steinmetz parameters, the well-known empirical equation, also differ under different frequencies and only for sinusoidal. Generalized Steinmetz equation and improved generalized Steinmetz equation are used to predicting the core loss under non-sinusoidal excitation. In this paper, a brief introduction of core loss calculation methods under non-sinusoidal excitations is discussed, and comparative analysis of under different harmonics orders and contents are presented. Ferrite N87 and Nano-crystalline 1K107 ring samples are magnetized as for experimental results. The experimental results show that the core loss depends not only frequency but also on the harmonics effect.

Comparative Analysis of Core Loss Calculation Methods for Magnetic Materials Under Nonsinusoidal Excitations

Magnetic components in power electronic equipment usually suffer from nonsinusoidal excitations. In this paper, a brief description about the existing core loss calculation methods for nonsinusoidal excitations is given. Then calculation formulas of aforementioned methods for two typical excitation waveforms, that is, square and rectangular with variable duty cycle D, are derived. Core loss of nanocrystalline FT-3KS, under above two nonsinusoidal excitations, is measured on an established test setup. The accuracy and applicability of each method are discussed theoretically and verified experimentally, especially in terms of different duty cycles and frequencies. It is found that the waveform coefficient Steinmetz equation (WcSE) is more applicable for the case of relatively small harmonics content of magnetic field intensity H (normal duty cycle region), while the improved generalized Steinmetz equation (IGSE) has better precision where the harmonics count (extreme duty cycle region). In addition, the IGSE can better follow the changing trend of core loss with D and have better frequency applicability than WcSE, which is proven to be not valid in low frequency (where the static hysteresis loss dominates). The results can provide reference for core loss prediction under nonsinusoidal excitations.

Closed-Form Solution for Core Loss Calculation in Single-Phase Bridgeless PFC Rectifiers Based on the iGSE Method

This letter proposes a closed-form solution for core loss computation exemplarily for the boost inductor of a single-phase power factor correction boost bridgeless converter. The core loss calculation is based on the improved generalized Steinmetz equation (iGSE) method, which consists in the separation of the contributions of major and minor loops in the B – H curve. However, the computational procedure based on loop separation routine of the iGSE is replaced here with an integral-based computation. This leads to the advantage that the resulting expressions are in a closed form that is suitable for optimization routines. The proposed method can be extended to other converters. Three different design results are compared with two standard numerical iGSE routines, one made by the authors and the original routine made by the iGSE authors. Experimental results for a fourth case are presented. The importance of modeling the major loop losses is also discussed.

RGSE-Based SPICE Model of Ferrite Core Losses

A SPICE model for estimating and simulating core losses of a ferrite magnetic device is presented. The approach is based on a proposed revised generalized Steinmetz equation (RGSE) and requires only the use of the Steinmetz coefficients and core geometry data. The letter presents a RGSE-based SPICE simulation strategy that can both estimate and simulate the effect of the core losses on the circuit. Primary advantages of RGSE approach are that it does not require switching frequency data and is applicable to arbitrary excitation and not just sinusoidal excitation as the Steinmetz equation approximation does. RGSE model is also insensitive to dc bias, this is an improvement over the earlier model. Comparison of previously reported measured and calculated core loss values with the simulation results of the presented SPICE RGSE model shows excellent agreement in the sinusoidal case and good agreement in the rectangular pulse case.

Core loss calculation for magnetic materials employed in SMPS under rectangular voltage excitations

Magnetic materials are widely used in switching-mode power supplies (SMPS) and magnetic components in SMPS usually work under two typical rectangular excitations (with or without the period of zero voltage). Extensive experimental results have shown that there is quite a difference of core loss between sinusoidal excitations and rectangular excitations, which means the traditional core loss calculation methods are no longer applicable. In this paper, two formulas for core loss calculation under the above rectangular excitations are derived based on the Improved Generalized Steinmetz Equation (IGSE). Core loss of different magnetic materials, under both sinusoidal excitations and rectangular excitations with different frequencies and duty cycles, are measured. Experimental results show that the formulas are accurate enough and very useful to predict the core loss.

An Improved Empirical Formulation for Magnetic Core Losses Estimation Under Nonsinusoidal Induction

Calculation of core loss is essential in the design of magnetic components especially in high frequency applications. Existing empirical approaches still present some limitations such as the inaccuracy and the difficulty to apply under nonsinusoidal waveforms. In particular, these methods fail to predict core loss with low duty cycle and when there is a significant change in the frequency. In addition to that, the use of different solutions of Steinmetz parameters for different frequency range can present some discontinuity problems at the boundary of each frequency interval. The main contribution of this study is to develop a new empirical method to estimate magnetic core losses under nonsinusoidal induction. The developed method is enough accurate and user-friendly to apply by designers. The effects of the frequency and the duty cycle are considered. The developed model is verified and compared with the improved generalized Steinmetz equation and measurement data from literature with 3F3 and N67 ferrite materials.

鉄損の直流磁界バイアス特性を考慮した降圧チョッパ回路用インダクタの低損失設計

In order to increase the power density of power converters, reduction of both the loss and the volume of the inductor is essential. However, it is difficult to calculate the iron loss of the inductor with the Steimetz equation or the improved generalized Steinmetz equation (iGSE) because those calculation methods do not consider the loss dependency of pre-magnetization. Hence, we have to repeat the re-design process in order to reduce the inductor loss or the inductor volume. In this paper, the authors propose an accurate loss calculation method for the inductor used in the buck-chopper circuit that is based on the iron loss characteristic considering pre-magnetization. In addition, the overall loss of the inductor is compared among three typical magnetic materials, such as iron powder, sendust, and ferrite, which have different iron loss characteristics for pre-magnetization. As a result, the sendust-based inductor achieves both the same loss reduction and a smaller volume compared with the ferrite-based inductor. From these studies, it is concluded that the proposed loss calculation method, which takes into account the loss dependency of pre-magnetization, is useful for optimum design of the inductors.

Iron Loss Calculation of AC Filter Inductor for Three‐Phase PWM Inverters

SUMMARYIn recent years, remarkable advancement of new power semiconductor devices, such as SiC and GaN, enables the increase of switching frequency of power converters, and hence the volume of passive components, such as ac filters and transformers, can be reduced. However, temperature rise caused by the inductor loss is increasing, and hence iron loss evaluation of the inductor is one of the most important issues to realize high power density converters. Conventionally, an improved generalized Steinmetz equation (iGSE) has proposed in order to calculate the iron loss under a pulse voltage magnetizing condition. However, accurate iron loss calculation of the ac filter inductor used in a PWM inverter cannot be realized. The authors have proposed two methods of iron loss evaluation of ac filter inductors. The first one is a loss map method which can calculate the iron loss without using a real PWM inverter. Another one is an ILA (Inductor Loss Analyzer) which can measure the iron loss in every switching period in a real PWM inverter. In this paper, comparisons of the iron loss between the ILA and the loss map method on both the single‐phase and three‐phase inverters are studied. It is found that iron loss of the ac filter inductor in the three‐phase PWM inverter which is calculated by the loss map method cause a large error on a specific condition. In order to prevent the calculation error, the authors proposed a revised loss map method and proved the effectiveness of the method.

Iron Losses in a Medium-Frequency Transformer Operated in a High-Power DC–DC Converter

The three-phase dual-active bridge is a dc-dc converter, which is highly suitable for high-power applications. Among others, this is due to the medium-frequency transformer in the ac link, which provides galvanic isolation. The transformer is operated with a square-shaped voltage waveform. The flux density in the transformer core is piecewise linear. However, for the sake of simplicity, the magnetic flux is often assumed sinusoidal. Thereby, the actual iron losses generated in the core material are misinterpreted. This paper discusses the difference between the exact piecewise linear and the sinusoidal course in terms of iron losses. Silicon steel with a thickness of 0.18 mm is measured at a frequency of 1000 Hz, comparing the sinusoidal excitation with the actual one. The measurements are validated using the improved generalized Steinmetz equation. Finally, the transformer core losses are evaluated when the dc-dc converter is operated under load. The results are confirmed through measurement.

Saturable Thermally-Representative Steinmetz-Based Loss Models

Core losses in power magnetic components are important factors in their design. These losses are influenced by factors including frequency, temperature, and flux density waveform. A common empirical model to predict these losses is the Steinmetz equation and its derivatives such as the modified Steinmetz equation and the generalized Steinmetz equation. In this paper, these three models are modified to address magnetic saturation and temperature dependence. The work focuses on manganese zinc ferrite samples for low to medium frequency operation. Experimental hardware verification is provided and results obtained demonstrate that the proposed model modifications significantly reduce model error.

Iron Loss Calculation of AC Filter Inductor for Three Phase PWM Inverters

In recent years, remarkable advancement of new power semiconductor devices, such as SiC and GaN, enables the increase of switching frequency of power converters, and hence the volume of passive components, such as AC filters and transformers, can be reduced. However, temperature rise caused by the inductor loss is increasing, and hence iron loss evaluation of the inductor is one of the most important issues to realize high power density converters. Conventionally, an improved generalized Steinmetz equation (iGSE) has proposed in order to calculate the iron loss under a pulse voltage magnetizing condition. However, accurate iron loss calculation of the AC filter inductor used in a PWM inverters cannot be realized. The authors have proposed two methods of iron loss evaluation of AC filter inductors. The first one is a loss map method which can calculate the iron loss without using a real PWM inverter. Another one is an ILA (Inductor Loss Analyzer) which can measure the iron loss in every switching period in a real PWM inverter. In this paper, comparisons of the iron loss between the ILA and the loss map method on both the single-phase and three-phase inverters are studied. It is found that iron loss of the AC filter inductor in the three-phase PWM inverter which is calculated by the loss map method cause a large error on a specific condition. In order to prevent the calculation error, the authors proposed a revised loss map method and proved the effectiveness of the method.