Analytical Loss Model for Magnetic Cores Based on Vector Magnetic Circuit Theory

Currently, some empirical formula methods, such as the Steinmetz equation, have been widely used in calculating high-frequency magnetic core loss. Although these methods may provide satisfactory accuracy under some certain conditions, they not only cannot present clear physical interpretation and fail to account for the origin and variation of losses, but also offer much lower accuracy for wide operation ranges. In this article, a new analytical model is proposed for the first time to predict core loss by the new vector magnetic circuit theory. In addition to reluctance, two new components, i.e., magductance and hysteretance are proposed in the vector magnetic circuit, which are used to describe the eddy-current loss and the hysteresis loss, respectively. Thus, the clear physical significance is available with the proposed loss model. Given the geometrical and physical parameters of cores, the loss can be predicted. Furthermore, the uneven distribution of magnetic flux inside the magnetic core is characterized by parallel vector magnetic circuits. By comparing with the available empirical formulas, it has been proven that the proposed analytical model not only offers clear physical concepts, but also has much higher accuracy.

SECTION I. Introduction

Magnetic components, such as transformers and inductors, are essential components in circuit systems, especially in power electronics [1]. In recent years, the rapid development of silicon carbide(SiC) and gallium nitride(GaN) switching devices have improved power electronics, making them smaller and faster [2]. However, in a modern power converter, magnetics are approximately half of the volume and weight, and generate most of the power loss [3], [4]. In addition, fast switching devices make accurate calculation of magnetic core losses under high-frequency conditions even more important. Due to the nonlinear changes in magnetic core losses and limitations in calculation methods, engineers have to leave more than necessary margin when designing magnetic cores, which makes them unable to be fully utilized [5], [6], [7]. Therefore, accurately predicting magnetic core losses is crucial to achieving optimal design of the passive components [8].

Currently, there are three main methods used to calculate these losses: Steinmetz equation-based methods, loss separation model, and hysteresis model [9]. The loss separation model with the terms eddy-current loss, hysteresis loss, and excess loss is primarily based on the G. Bertotti model [10]. On the other hand, the hysteresis model is based on the Preisach and Jiles-Atherton models [11], [12].

The Steinmetz equation (SE) is widely used for calculating magnetic core losses under sinusoidal excitation. The equation is expressed as follows [13]: $$\begin{equation*} P = k{{f}^\alpha }B_{\mathrm{m}}^\beta \tag{1} \end{equation*}$$ where, B_m is the amplitude of magnetic flux density of a sinusoidal excitation with frequency f, and k, α, and β are coefficients, which can be obtained from sinusoidal measurements without bias using a least-squares curve fitting method.

The Steinmetz equation-based methods, due to its few parameters and simplicity, are convenient for industrial applications [13], [14], [15], [16], [17]. Therefore, many manuals for magnetic core materials come with Steinmeiz parameters. However, the operating frequency of magnetic cores is generally wide, and (1) is difficult to maintain a certain degree of accuracy over a wide frequency range [18]. In order to improve this situation, in recent years, the emerging methods like neural networks are now becoming popular for their higher accuracy [19], [20], [21], but this also makes the calculation of magnetic core losses a black box. Moreover, more importantly, the SE is an empirical coefficient equation that does not reflect the mechanism and distribution of losses inside the magnetic core, which is not conducive to further study by researchers on magnetic core losses.

The loss separation model is widely used in low-frequency applications, such as silicon steel sheets. Traditionally, the total losses are divided into classical eddy-current losses and hysteresis losses as: $$\begin{equation*} P = {{k}_e}{{f}^2}{{B}^2} + {{k}_h}f{{B}^n} \tag{2} \end{equation*}$$ where, k_e, k_h, and n are coefficients which are fitted based on material properties. There is a huge divergence between the result from (2) and experimental data, so excess eddy-current losses is proposed by G. Bertotti to match this difference. Some studies [10] based on the statistical loss theory derived the excess losses from the flux density change due to the motion of random distributed domain walls, with the total losses expressed as: $$\begin{equation*} P = {{k}_e}{{f}^2}{{B}^2} + {{k}_h}f{{B}^n} + {{k}_{ex}}{{f}^{1.5}}{{B}^{1.5}} \tag{3} \end{equation*}$$ where, the coefficient k_ex is a microstructure parameter influenced by material thickness, the cross-sectional area, and the conductivity. Similar to the Steinmetz equation, this model is also difficult to accurately predict core losses over a wide magnetic flux density range. Furthermore, more importantly, excess loss is sometimes considered as the difference between experimental and calculated values, and sometimes as excess eddy current loss. However, in practical applications, this part of the loss is usually obtained through fitting, which also makes it difficult to explain the source of excess loss and loses its physical significance [22], [23], [24].

Previously, some researchers have tried to define more magnetic parameters in addition to reluctance, in order to model the characteristics of magnetic circuits such as losses more clearly [25], [26], [27]. The study of magnetic circuit theory originated in 1840, when J. Joules first discovered the phenomenon of magnetic-resistance [28], that is, the hindrance of magnetoresistance to constant magnetic flux. In order to better distinguish magnetic-resistance from electrical resistance, O. Heaviside clarified the concept of magnetic-resistance in 1880 and used "reluctance" instead of the original "magnetic-resistance" [29]. Subsequently, J. Hopkinson verified Ohm's law of the magnetic circuit on this basis, that is, the magnetic circuit only contains a single reluctance component to describe the amplitude relationship between the magnetomotive force (MMF) and the magnetic flux. However, a single reluctance component cannot reflect the active power loss generated by the magnetic circuit. Therefore, in 1946, K.A. Macfadyen and others proposed the complex magnetic permeability [27]. Although this method can solve the problem of loss calculation in mathematics, it cannot explain the source of the loss and lacks actual physical meaning.

In 1967, E.R. Laithwaite analyzed the characteristics of the secondary winding connected with resistors, inductors, or capacitors of a single-phase transformer based on the duality between magnetic circuit and electrical circuit, introduced "quadrature-axis" reluctance, and name it as “transference” [30]. However, this parameter was introduced to characterize the energy conversion of the secondary winding of a transformer. It only equates the electric field energy of the secondary winding to the magnetic circuit, and cannot be used to explain the eddy-current effects and hysteresis effects that appear in the magnetic circuit.

The Buntenbach magnetic circuit model can be regarded as a deformation of the Laithwaite magnetic circuit model [31], [32]. In order to pursue the duality between magnetic circuit and electrical circuit, the Buntenbach magnetic circuit model uses the derivative of magnetic flux as the “magnetic current” in magnetic circuit, defines the “transference” to the magnetic “resistance” and the reciprocal of reluctance to the “magnetic capacitance”. However, the limitation of this model is that the defined magnetic circuit parameters lose their direct correspondence with the physical phenomena in the magnetic circuit, thereby losing the most essential connotation of the magnetic circuit parameters and conflicting with the traditional scalar magnetic circuit theory.

In order to consider the nonlinear behavior of the magnetic core under high-frequency conditions and accurately calculate the magnetic core loss, Zhu et al. [33], [34], [35] proposed a dynamic circuit model of the magnetic core. The basic modeling concept is to treat the magnetic material as a series of zones. Each zone is modeled as a resistance-inductance (RL) element, in a network in the magnetic domain, as shown in Fig. 1 where arrows indicate the direction of eddy currents. This ladder L-R network, commonly used in modeling nonlinear diffusion in magnetic material, is adopted to represent the magnetic core. This method is still based on traditional scalar magnetic circuits and is similar to the models of Laithwaite, Carpenter et al., equating magnetic components to electric components, characterizing magnetic field characteristics through equivalent electric circuits. The effects of eddy currents and hysteresis on phase of magnetic flux is not considered in this method, the spatial differences in magnetic permeability caused by the uneven distribution of magnetic flux is also not taken into account, and more importantly, the calculation of magnetic core losses still requires the use of the Preisach model and loss separation model [36], [37]. Therefore, the accuracy of loss calculation may depend more on the loss calculation model rather than the equivalent circuit model.

FIGURE 1.

Zones and equivalent core loss circuit model [35].

Recently, Prof. Cheng et al. from Southeast University proposed a new vector magnetic circuit theory [38], [39], which merges the magnetic circuit and electromagnetic field theories, introducing a vector magnetic circuit model. Different from the previous scalar magnetic circuit with only reluctance, this new model comprises three components, i.e., reluctance, magductance, and hysteretance. The model enables a deeper understanding of the characteristics and physical significance of magductance and hysteretance components, as well as the magnetoelectric power law. The theory attempts to explain the origins of eddy-current loss, hysteresis loss, and the phase shift between MMF and magnetic flux within the magnetic circuit.

The main purpose of this article is to propose an analytical calculation model for magnetic core losses under sinusoidal excitation based on the vector magnetic circuit theory. This analytical calculation method does not require the magnetic circuit to be equivalent to the electrical circuit to calculate the active power loss of the magnetic core, nor does it require the use of empirical formulas to fit the magnetic circuit loss. Instead, it directly starts from the magnetic circuit and considers the nonlinear behavior inside the magnetic core, providing a new method for calculating the eddy-current loss and hysteresis loss of the magnetic core. This article first recalls the concept of vector magnetic circuit theory, then presents the analytical loss model for magnetic circuit, demonstrates the application process in the loss calculation of magnetic core. Finally, the experimental data of magnetic cores 3E6 and N87 are given to verify the effectiveness of the proposed analytical loss model. Theoretical analysis and data validation are both compared with existing methods, pointing out the shortcomings of existing methods and the advantages of the proposed method in this article.

SECTION II. Vector Magnetic Circuit Theory

According to vector magnetic circuit theory [39], a vector magnetic circuit can be represented by Fig. 2. Assuming that all the magductance ${\mathcal{L}}_e$ of the entire magnetic circuit is concentrated on the magductance component in the bc segment, all the hysteretance ${\mathcal{C}}$_h of the entire magnetic circuit is concentrated on the hysteretance component in the cd segment, all the reluctance of the entire magnetic circuit is concentrated on the reluctance component in the de segment. H is the magnetic field strength. J is the current density on the magductance component. ${\boldsymbol{\mathcal F}}$ is the MMF. Φ is the magnetic flux. N is the number of turns of the magductance component.

FIGURE 2.

Vector magnetic circuit.

The magnetic circuit shown in Fig. 2 satisfies the following basic relationship: $$\begin{equation*} {\boldsymbol{\mathcal F}} = {\boldsymbol{\mathcal F}}{_R} + {\boldsymbol{\mathcal F}}_L + {\boldsymbol{\mathcal F}}{_C} \tag{4} \end{equation*}$$ where, ${\boldsymbol{\mathcal F}}$_R, ${\boldsymbol{\mathcal F}}$_L, and ${\boldsymbol{\mathcal F}}$_C are the MMF on the reluctance, magductance and hysteretance components, respectively.

1. MMF source: The left integral of (4) is the MMF ${\boldsymbol{\mathcal F}}$ generated by the excitation source. $$\begin{equation*} {\boldsymbol{\mathcal F}} = \oint_{l} {{\bm{H}} \cdot \mathrm{d}{\bm l}} \tag{5} \end{equation*}$$

where, l is the length of the magnetic circuit.

2. Reluctance component: The first term on the right side of (4) represents the MMF drop on the reluctance component, so the port characteristics of the reluctance can be expressed as: $$\begin{equation*}R = \frac{{{\boldsymbol{\mathcal F}}{_R}}}{\Phi } = \frac{l}{{\mu S}} \tag{6} \end{equation*}$$

where, S is the cross-sectional area of the magnetic circuit; μ is the magnetic permeability.

3. Magductance component: The second term on the right side of (4) represents the MMF generated by the eddy-current, which can be expressed by a new magnetic component of magductance. Therefore, the port characteristics of the magductance can be expressed as: $$\begin{equation*}{\mathcal{L}} = \frac{{{\boldsymbol{\mathcal F}}_{\mathcal{L}}}}{{\frac{{d\Phi }}{{dt}}}} \tag{7} \end{equation*}$$

4. Hysteretance component: The third term on the right side of (4) represents the MMF generated by the hysteresis effect of the magnetic core, which can be expressed by a new magnetic component of hysteretance. Therefore, the port characteristics of the hysteretance can be expressed as: $$\begin{equation*}{{C}_{\mathrm{h}}} = - \frac{{\int{{\Phi dt}}}}{{{\boldsymbol{\mathcal F}}{_C}}} \tag{8} \end{equation*}$$

5. Magnetoelectric power law: When the magnetic circuit is excited by a steady-state sinusoidal magnetic electromotive force, the following steady-state expression can be obtained using the phasor method: $$\begin{equation*} \dot{\mathcal{F}} = {\mathcal{R}}\dot{\Phi } + j \omega {\mathcal{L}}_{\mathrm{e}}\dot{\Phi } + j\frac{1}{{\omega {{C}_{\mathrm{h}}}}}\dot{\Phi } \tag{9} \end{equation*}$$

According to the Poynting’ vector [1], for steady state, the energy related to the magnetic field can be represented by complex electrical power as follows: $$\begin{equation*} {{\dot{S}}_V} = \frac{1}{2}j\omega \int_{V}{{({{{\dot{\bm H}}}^ * } \cdot \dot{\bm B})\mathrm{d}V}} \tag{10} \end{equation*}$$ where, S_V (W) is complex electrical power, B is the magnetic flux density. Under the traditional scalar magnetic circuit theory, due to the presence of only a single reluctance component, the active power of the magnetic field cannot be described. If the relationship between B and H is simply reflected by B = μH, it will be mistakenly assumed that all energy related to the magnetic field is stored by the magnetic field. However, in fact, due to hysteresis and eddy current effects, the magnetic field will also consume some energy. This is a problem that cannot be explained by traditional magnetic circuit theory and Poynting's vector. This article still starts from the initial equation of the Poynting's vector, but does not derive it based on the traditional scalar magnetic circuit theory. Instead, combining (4)–​(10), the active power P and reactive power Q generated by magnetic circuit can be expressed as: $$\begin{align*} P + jQ & = {{\bm S}_V} = \frac{1}{2}j\omega \int_{V}{{({{{\dot{\bm H}}}^ * } \cdot \dot{\bm B})\mathrm{d}V}}\\ &= \frac{1}{2}j\omega \iint\limits_{\bm S} {\dot{\bm B} \cdot \mathrm{d}\bm S} \cdot \oint_{\bm l} {{{{\dot{\bm H}}}^*} \cdot \mathrm{d}{\bm l}} \\ &= \frac{1}{2}j\omega \dot{\Phi } \cdot \left(\mathcal{R}\dot{\Phi } - j\omega {{\mathcal{L}}_{\mathrm{e}}}\dot{\Phi } - j\frac{1}{{\omega {{C}_{\mathrm{h}}}}}\dot{\Phi }\right)\\ & = {{\omega }^2}{{\mathcal{L}}_{\mathrm{e}}}{{\left(\frac{{{{\Phi }_{\mathrm{m}}}}}{{\sqrt 2 }}\right)}^2} + \frac{1}{{{{C}_{\mathrm{h}}}}}{{\left(\frac{{{{\Phi }_{\mathrm{m}}}}}{{\sqrt 2 }}\right)}^2} + j\omega R{{\left(\frac{{{{\Phi }_{\mathrm{m}}}}}{{\sqrt 2 }}\right)}^2} \tag{11} \end{align*}$$ where, Φ_m is the amplitude of magnetic flux. The calculation equations for ${\mathcal{L}}_{\mathrm{e}}$ and ${\mathcal{C}}_{\mathrm{h}}$ for specific magnetic cores are as follows [38], [39]: $$\begin{align*} &{{\mathcal{L}}_{\mathrm{e}}} = \frac{{{{N}^2}}}{R} \tag{12}\\ &{{C}_{\mathrm{h}}} = \frac{\mu }{{\omega \sin \gamma }}\frac{S}{l} \tag{13} \end{align*}$$ where, R represents the resistance of the magductance component. γ is the hysteresis angle.

According to (11)–​(13), the eddy-current loss and hysteresis loss of the magnetic circuit can be calculated, and the active power is related to the magductance and hysteretance, while reactive power of the magnetic circuit is related to reluctance.

SECTION III. Analytical Loss Model

A. Analytical Equation for Magnetic Circuit Parameters

As shown in Fig. 3, it is a rectangular magnetic core with a length of 2a₀ and a width of 2b₀. The magnetic core is assumed to be uniform without airgap. The space with length l and thickness of da and db is called a layer. Assuming that the direction of the sinusoidal alternating magnetic field intensity at this time is vertical upwards. According to the right-hand screw rule, eddy currents only flow in a plane perpendicular to the magnetic flux. To simplify the analysis, it is assumed that the flowing of the eddy-currents is also rectangular, and the direction is indicated by the arrow. The length of the eddy-current field is 2a, the width is 2b.

FIGURE 3.

Rectangular core.

Without considering the skin effect, according to Faraday's law: $$\begin{equation*} E(a,t) = N\frac{{\mathrm{d}\Phi (t)}}{{\mathrm{d}t}} = N4ab\frac{{\mathrm{d}B(t)}}{{\mathrm{d}t}} = N4k{{a}^2}\frac{{\mathrm{d}B(t)}}{{\mathrm{d}t}} \tag{14} \end{equation*}$$ where, k is the proportional coefficient, and k = b₀/a₀.

The resistance R_da of the layer is: $$\begin{equation*} {{R}_{\mathrm{d}a}} = \frac{{{{l}_{\mathrm{d}a}}}}{{\sigma {{S}_{\mathrm{d}a}}}} = \frac{{8a}}{{\sigma l\mathrm{d}a}} \tag{15} \end{equation*}$$ where, S_da is the area of the layer, and l_da is the length of the layer.

The instantaneous eddy-current power loss in the layer is: $$\begin{equation*} \mathrm{d}{{p}_{\mathrm{e}}}(a,t) = \frac{{{{E}^2}(a,t)}}{{{{R}_{\mathrm{d}a}}}} = {{N}^2}2\sigma l{{k}^2}{{a}^3}\mathrm{d}a{{\left(\frac{{\mathrm{d}B(t)}}{{\mathrm{d}t}}\right)}^2} \tag{16} \end{equation*}$$

Assuming N = 1 [1], the instantaneous eddy-current power loss in the entire core is: $$\begin{equation*} {{p}_{\mathrm{e}}}(t) = \int_{0}^{{{{a}_0}}}{{\mathrm{d}{{p}_{\mathrm{e}}}(a,t)\mathrm{d}a = }}\frac{{\sigma l{{k}^2}{{a}_0}^4}}{2}{{\left(\frac{{\mathrm{d}B(t)}}{{\mathrm{d}t}}\right)}^2} \tag{17} \end{equation*}$$

The time-average eddy-current power loss P_e in the core is: $$\begin{equation*} {{P}_{\mathrm{e}}} = \frac{1}{T}\int_{0}^{T}{{{{p}_{\mathrm{e}}}(t)\mathrm{d}t}} = \frac{{\sigma l{{k}^2}{{a}_0}^4}}{2}\frac{1}{T}\int_{0}^{T}{{{{{\left(\frac{{\mathrm{d}B(t)}}{{\mathrm{d}t}}\right)}}^2}\mathrm{d}t}} \tag{18} \end{equation*}$$

For a sinusoidal changing magnetic field, the angular frequency is ω, then (18) can become: $$\begin{equation*} {{P}_{\mathrm{e}}} = {{\omega }^2}\frac{{\sigma l}}{{32}}{{\left(\frac{{{{\Phi }_{\mathrm{m}}}}}{{\sqrt 2 }}\right)}^2} \tag{19} \end{equation*}$$

Combining (6), (12) and (13), for the rectangular magnetic core shown in Fig. 3, the analytical expressions for parameters of each magnetic circuit component are given as: $$\begin{align*} &{\mathcal{R}} = \frac{l}{{4\mu {{a}_0}{{b}_0}}} \tag{20}\\ &{{\mathcal{L}}_{\mathrm{e}}} = \frac{{\sigma l}}{{32}} \tag{21}\\ &{{C}_{\mathrm{h}}} = \frac{{4\mu {{a}_0}{{b}_0}}}{{\omega \sin \gamma l}} \tag{22} \end{align*}$$ where, σ is the conductivity of the magnetic core. It should be noted that the derivation of (20)–​(22) is based on the fact that the magnetic core material is uniform everywhere and the magnetic flux density is equal everywhere. Thus the core loss P can be calculated as: $$\begin{align*} P& = {{P}_{\mathrm{e}}} + {{P}_{\mathrm{h}}}\\ &= {{\omega }^2}{{\mathcal{L}}_{\mathrm{e}}}{{\left(\frac{{{{\Phi }_{\mathrm{m}}}}}{{\sqrt 2 }}\right)}^2} + \frac{1}{{{{C}_{\mathrm{h}}}}}{{\left(\frac{{{{\Phi }_{\mathrm{m}}}}}{{\sqrt 2 }}\right)}^2} \tag{23} \end{align*}$$

However, the magnetic flux inside the magnetic core is usually unevenly distributed and needs to be considered in different zones.

B. Equivalent Model of Parallel Vector Magnetic Circuit for Magnetic Cores

If the rectangular cross-section magnetic core shown in Fig. 3 is connected end-to-end, a toroidal magnetic core as shown in Fig. 4 can be obtained. The inner radius of the core is r, the outer radius is R, and the height is h. It is still assumed that the toroidal magnetic core is uniform and has no air gap.

FIGURE 4.

Toroidal core.

If the toroidal magnetic core shown in Fig. 4 is treated as a whole, it is usually assumed that the magnetic permeability inside the core is equal everywhere and the magnetic flux distribution is uniform, but this obviously does not match the real situation. Similar to the methods presented in [33], [34], [35], the toroidal magnetic core shown in Fig. 4 can also considered as a combination of a series of zones. However, the difference is that this series of zones are represented by parallel vector magnetic circuits in this article, as shown in Figs. 5 and 6.

FIGURE 5.

Physical layered zone of toroidal magnetic core.

FIGURE 6.

Vector magnetic circuit of toroidal magnetic core.

It should be noted that the actual eddy current in its cross-section is shown in Fig. 7(a), and cannot be layered as in Fig. 7(b). That is, this layering is a fictional analytical method and cannot change the actual eddy-current inside the core. Therefore, the value of ${\mathcal{L}}$_ek of k-th layer is the average distribution value of the total ${\mathcal{L}}$_e. The rectangular area in Fig. 7 represents the cross-section of the magnetic core, the dashed line represents the eddy-current path, and the arrow represents the direction of the eddy-current.

FIGURE 7.

Eddy current in the cross-section of the magnetic core. (a) cross -section of the magnetic core. (b) Cross section of layered magnetic core.

The magnetic circuit shown in Fig. 6 can also be equivalent to the ladder magnetic circuit shown in Fig. 8. Here, ${\mathcal{L}}_e$ represents the total magductance value of the magnetic core, which can be calculated by the effective length.

FIGURE 8.

Ladder vector magnetic circuit of toroidal magnetic core.

For the k-th layer magnetic circuit in Fig. 6, $$\begin{align*} &{{\mathcal{R}}_k} = \frac{{{{l}_k}}}{{{{\mu }_k}{{S}_k}}} \tag{24}\\ &{{\mathcal{L}}_{\mathrm{e}k}} = \frac{{n{{\sigma }_k}{{l}_k}}}{{32}} \tag{25}\\ &{{C}_{\mathrm{h}k}} = \frac{{{{\mu }_k}{{S}_k}}}{{\omega \sin {{\gamma }_k}{{l}_k}}} \tag{26} \end{align*}$$ where, l_k is the length of the k-th layer magnetic circuit, σ_k is the conductivity of k-th layer magnetic circuit, which is considered to be uniform within the magnetic core in this article, i.e., σ_k = σ. S_k is the cross-sectional area of the k-th layer magnetic circuit. γ_k is the hysteresis angle of the k-th layer magnetic circuit. μ_k is the magnetic permeability of the k-th layer magnetic circuit. All layered magnetic circuits can be considered as parallel connections, therefore the following conditions need to be met. $$\begin{equation*} \left\{ {\begin{array}{l} \dot{F} = {{{\dot{H}}}_k}{{l}_k}\\ {{{\dot{B}}}_k} = {{\mu }_k}{{{\dot{H}}}_k}\\ {{{\dot{\Phi }}}_k} = {{{\dot{B}}}_k}{{S}_k} \end{array}} \right. \tag{27} \end{equation*}$$ where, ${{\dot{H}_k}}$ is the magnetic field strength of the k-th layer magnetic circuit generated by the primary winding in the core. ${{{{\dot{\Phi }}}_k}}$ is the magnetic flux of the k-th layer magnetic circuit. The reluctance, magductance, and hysteretance of each magnetic circuit are affected by the value k, which in turn affects the value of magnetic flux in the magnetic circuit. For toroidal magnetic cores, the length of the inner magnetic circuit is smaller than that of the outer magnetic circuit, resulting in a smaller inner magnetic impedance and more magnetic flux concentration in the inner layer. In addition, different magnetic flux densities correspond to different magnetic permeability and γ, which in turn affect the magnetic circuit parameters. Therefore, the calculation of magnetic circuit parameters after layering is an iterative process. This phenomenon is referred to as the "concentration effect" in this article. The uneven distribution of magnetic permeability in space caused by the uneven distribution of magnetic flux has not been considered in [33], [34], [35]. Experimental measurements or material data manuals usually offer the average magnetic permeability of the magnetic core, and it is difficult to obtain the magnetic permeability of different spatial positions. Therefore, to quantitatively reflect the phenomenon of uneven magnetic permeability inside the magnetic core, assuming that the change of magnetic permeability with position is as follows: $$\begin{equation*} \mu (f,x) = \mu (f) + g(f) \times \left(x - \frac{{R + r}}{2}\right) \tag{28} \end{equation*}$$ where, μ(f) represents the relationship between μ and f; x represents the distance from the center of the magnetic core along the radial direction; g(f) needs to know the partial loss data of the specific magnetic core within the studied frequency range, including at least the data at both ends of the range. Based on the loss results and the loss calculation method of layered magnetic core, the specific value of g(f) can be reversibly derived, and then the expression of g(f) can be obtained by fitting the values at different frequencies through polynomials.

Given the specific geometric shape, conductivity, and permeability of the magnetic circuit, the magnetic circuit components can be calculated using (11), (24)–​(28). In consequence, the core loss P can be calculated as: $$\begin{align*} P & = \sum\limits_{k = 1}^n {{{P}_{\mathrm{e}k}} + {{P}_{\mathrm{h}k}}} \\ &= \sum\limits_{k = 1}^n {\left[{{\omega }^2}{\mathcal{L}_{\mathrm{e}k}}{{{\left(\frac{{{{\Phi }_{k\mathrm{m}}}}}{{\sqrt 2 }}\right)}}^2} + \frac{1}{{{{C}_{\mathrm{h}k}}}}{{{\left(\frac{{{{\Phi }_{k\mathrm{m}}}}}{{\sqrt 2 }}\right)}}^2}\right]} \tag{29} \end{align*}$$

According to (29), the eddy-current loss P_e and hysteresis loss P_h of the toroidal core in Fig. 4 can be obtained. In actual situations, the shapes of magnetic cores are diverse, but most magnetic cores can be obtained by rotating or folding a rectangular magnetic core. For more complex magnetic cores, similar equations can be obtained with the same principle.

Unlike SE and loss separation model, the analytical model proposed in this article starts from the Maxwell equations and separates the hysteresis loss and eddy-current loss inside the magnetic core from the source. A vector magnetic circuit model was constructed by adding magductance and hysteretance components to the traditional scalar magnetic circuit model. By layering the magnetic core and constructing a parallel vector magnetic circuit, the uneven distribution of magnetic flux inside the core is truly reflected, and the loss distribution inside the core and the total power loss of the core are truly expressed.

SECTION IV. Calculation of Model Parameters

In practical applications, accurate calculation of magnetic circuit parameters is a new challenge. Reference [40] has identified various factors that influence magnetic core loss, including magnetic core material, size, peak magnetic flux density, frequency, excitation type, duty cycle, temperature and others. It is difficult to consider all factors in one article. Therefore, this article takes two materials as an example and considers the two most important factors, frequency and magnetic flux density, under sinusoidal excitation.

A. Conventional Parameters of Magnetic Cores

The specifications of the magnetic cores studied in this article are listed in Table 1.

TABLE 1 Specifications of Magnetic Cores

For most of this article, a toroidal core of 22-mm outer diameter, 14-mm inner diameter, and 6.4-mm height (designated R22 × 14 × 6.4 by the manufacturer) of the material 3E6 is studied. To demonstrate the universality of the method, another toroidal magnetic core N87 is also used to assist in verification. The three-dimensional diagram of the magnetic cores is shown in Fig. 4. The resistivity is considered to be a fixed value.

B. The Hysteresis Angle γ of Magnetic Core

For specific ferrite core, material manufacturers usually only provide some general parameters of magnetic cores, such as size, magnetic permeability, conductivity, and core loss per unit volume curves at some excitation frequencies and temperatures under sinusoidal excitation without dc bias [41]. According to (22), the calculation of C_h also requires the value of hysteresis angle γ. However, γ is usually not provided by material manufacturers and cannot be directly obtained from high-frequency measurement data. In order to minimize the impact of eddy-current on the measurement, γ is measured at low frequencies under quasi-static conditions. After multiple measurements, it was found that its values did not change with frequency, which corresponds to [39]. The result of γ measured under 500 Hz condition is shown in Fig. 9. The measurement method of γ can be found in the appendix.

FIGURE 9.

The variation of hysteresis angle γ with magnetic flux density under quasi-static condition of 3E6 material.

From Fig. 9, it can be observed that the hysteresis angle γ undergoes nonlinear changes with magnetic flux density. Fig. 9 can serve as a material characteristic of the 3E6 material introduced in this article.

C. The Magnetic Permeability of Layered Magnetic Core

For specific application of magnetic cores, magnetic permeability will be affected by frequency and magnetic density. It is difficult for material manufacturers to provide the value of magnetic permeability under multiple conditions. Therefore, the B-H curve data of the magnetic core in specific application is required. In this article, the B-H curve data can be obtained by the same test cores under sine wave excitation in an open-source database [20], [40], and [42]. The B-H curve data includes a frequency range of 50 kHz∼500 kHz and an average magnetic flux density amplitude range of 0∼0.2T. Data close to saturation is not included. The B-H curve of partial data of 3E6 is shown in Fig. 10.

FIGURE 10.

The B-H curve of partial data. (a) For different flux density amplitudes at 200 kHz. The dashed line represents the central axis of the B-H curve at B_m = 39 mT to help to show the change in permeability. (b) For different frequencies at B_m = 0.1T.

From Fig. 10, it can be seen that for material 3E6 in this article, the amplitude of magnetic flux density has a significant impact on the loop area, and also has a certain non-linear effect on permeability. A higher frequency not only makes the loop more elliptical, but also has a significant impact on permeability.

According to Fig. 10, the variation pattern of the relative permeability of the magnetic core with frequency can be obtained, as shown in Fig. 11, where μ₀ is permeability of vacuum. The value of relative permeability is obtained from the relationship between the amplitude of magnetic flux density B_m and the amplitude of magnetic field strength H_m.

FIGURE 11.

The variation of magnetic permeability with frequency of 3E6.

For the data range of 3E6 magnetic cores studied in this article, according to the method described in Section III, the expression g (f) used to represent the non-uniform distribution of magnetic permeability is obtained as follows. $$\begin{equation*} g(f) = 0.062{{f}^3} - 61.051{{f}^2} + 17887f - 7220 \tag{30} \end{equation*}$$

The application of (30) requires the specific number of layers. In this article, the magnetic core is divided into 10 layers, i.e., n = 10. Theoretically, the more layers the magnetic core is divided into, the better, but the computational complexity will increase. In addition, for the 3E6 magnetic core used in this article, dividing it into 10 layers has almost the same effect as dividing it into even more layers.

Based on the equations in Sections III and IV, Table 1, Figs. 9–​11, the model parameters described in this article can be accurately calculated.

SECTION V. Validation and Discussion

For the magnetic core 3E6, when the frequency is 200 kHz and the average magnetic flux density is 0.19T, the comparison among the case of uniform distribution of magnetic flux inside the magnetic core, the case of uneven distribution of magnetic flux caused by magnetic circuit length, and the case of considering both magnetic core layering and uneven distribution of magnetic permeability is shown in Fig. 12.

FIGURE 12.

The comparison of magnetic flux density distribution inside magnetic cores under different conditions. (a) Uniform distribution of magnetic flux. (b) Uneven distribution of magnetic flux caused by magnetic circuit length. (c) Uneven distribution of magnetic flux caused by uneven distribution of magnetic permeability and magnetic circuit length.

From Fig. 12, it can be seen that considering both the influence of magnetic circuit layering and the uneven magnetic permeability, the magnetic flux is more concentrated towards the inner magnetic circuit. This is far from the situation where the magnetic flux inside the magnetic core is uniform everywhere and the magnetic permeability is uniform everywhere. By combining (24)–​(29) and (30), the comparison between experimental and calculated losses of magnetic cores under different magnetic flux densities and frequency ranges is shown in Fig. 13, where “Test_P_V” represents experimental volumetric loss, which is the magnetic core loss per unit volume. The prefix “cal” represents the calculated value. The experimental data Test_P_V is sourced from B-H curve data in the open source database described in Section IV-C.

FIGURE 13.

Comparison between experimental and calculated volumetric losses of magnetic core 3E6 considering concentration effect under different magnetic density. (a) B_m = 0.097 T. (b) B_m = 0.121 T. (c) B_m = 0.151 T. (d) B_m = 0.19 T.

From Fig. 13, it can be seen that the calculated values are consistent with the experimental values, and the average error in all cases in Fig. 13 is only 5.9%. In addition, the method shown in this article can also directly calculate the specific values of eddy-current loss and hysteresis loss.

To verify the accuracy of the method presented in this article, the same experimental data Test_P_V is compared with the calculated data SE_P_V of the SE (1), as shown in Table 2. The SE parameters are sourced from open-source databases [40]. The frequency range considered for the coefficients of SE shown in Table 2 is 50 kHz∼500 kHz.

TABLE 2 Comparison of Experimental Data Test_$P_{V}$ and Calculated Data Based on Steinmetz Equation SE_$P_{V}$ of 3E6 (W/m3)

The average error in Table 2 is 13.4%, much larger than 5.9%. In addition, when the frequency is above 300 kHz, the average error calculated using SE is 22.7%, which is also much larger than 8.8% calculated by the method in this article. Similarly, when the frequency is below 100 kHz, the error calculated by SE is 13.9%, also much larger than 2.2% of the method proposed in this paper. This is a common problem of using power functions to fit losses. By comparing Table 2 and Fig. 13, it can be seen that the method proposed in this article outperforms SE in terms of accuracy.

When using the loss separation model, i.e., (2) or (3), to calculate the same loss data in Fig. 13(b), the resulting eddy-current loss, hysteresis loss and excess loss are shown in Fig. 14. The prefix “(2)” and “(3)” represent the values which were calculated by (2) or (3).

FIGURE 14.

Comparison of experimental data and calculated data based on loss separation model. (a) k_e = 0.0055, k_h = 1.5199. (b) k_e = 0.001, k_h = 0.0133, k_ex = 0.1666.

According to Fig. 14, it can be seen that there is a significant difference in the eddy-current loss and hysteresis loss calculated using (2) and (3). When introducing excess loss, this loss accounts for a large part of the loss, resulting in underestimation of eddy-current loss and hysteresis loss. The reason why there are two different results for the same model is that the loss calculation method is essentially the same as SE, which calculates the magnetic core loss through fixed empirical parameters obtained through fitting. However, SE is a power function, and the loss separation model is a polynomial function.

To verify the generality of the method shown in this article, a frequency loss relationship at 0.04T was obtained using the analytical method on another magnetic core N87, as shown in Fig. 15. The same data is also calculated using (2), as shown in Fig. 16.

FIGURE 15.

Comparison of experimental data and calculated data based on the proposed analytical model of N87.

FIGURE 16.

Comparison of experimental data and calculated data based on loss separation model of N87. k_e = 0.0005, k_h = 0.1076.

By comparing Figs. 13 and 16, it can be found that compared to 3E6 material, the eddy-current loss of N87 material accounts for a relatively low proportion. It is not difficult to find from the material properties in Table 1 that the conductivity of 3E6 material is 100 times higher than that of N87 material, which is the main reason for the high proportion of eddy-current losses in 3E6 material. However, the eddy-current loss calculated by (2) is still very large, and such a large eddy current loss is difficult to generate in a magnetic core with a conductivity of only 0.1(S/m). The parameters of the loss separation model in Figs. 14 and 16 are obtained using the least squares method. Comparing Figs. 16 and 14(a), the proportion of eddy-current loss within the material is the same for two materials with different electrical conductivity. Therefore, the loss separation model is difficult to reflect the actual eddy-current loss and hysteresis loss inside the magnetic core.

By verifying the N87 magnetic core, the universality of the proposed method for magnetic cores with different materials and characteristics is further demonstrated.

SECTION VI. Conclusion

In this paper, the vector magnetic circuit theory has been introduced, which provides a different prospective and modeling method for predicting power loss in magnetic circuits. Thus, an analytical loss model for magnetic cores has been proposed. The uneven distribution of magnetic flux is characterized by constructing a parallel vector magnetic circuit. In order to apply the analytical loss model, the variation of hysteresis angle of the magnetic cores 3E6 and N87 are obtained by measuring the B-H curve under quasi-static conditions.

To verify the accuracy of the analytical loss model, a comparison between the analytical loss model and the existing Steinmetz equation has been made, proving that the proposed analytical loss model offers much higher accuracy. By using the loss separation model to calculate the same experimental data, it was demonstrated that the loss separation model cannot truly display the eddy-current loss and hysteresis loss inside the magnetic core. More importantly, as compared with the traditional scalar magnetic circuit theory and existing core loss modeling methods, the vector magnetic circuit theory and the analytical loss model offer following features:

1. The vector magnetic circuit theory, which contains three components: reluctance, magductance, and hysteretance, can not only characterize the amplitude and phase relationship between MMF and magnetic flux, but also separate eddy-current loss and hysteresis loss, so that they can be analyzed separately. Thus, the influence of eddy-current effect and hysteresis effect on magnetic flux can be clearly revealed.

2. It presents an analytical solution to evaluate the distribution of magnetic flux and loss for magnetic cores by introducing the components of magductance and hysteretance in parallel vector magnetic circuits. Given the complete geometrical and physical parameters of magnetic cores, the magductance and hysteretance parameters can be directly predicted, in turn the eddy-current loss and hysteresis loss can be calculated analytically with magnetoelectric power law.

3. The proposed method offers the clear physical descriptions, i.e., magductance for eddy-current loss, hysteretance for hysteresis loss, and parallel vector magnetic circuits for uneven loss distribution, which are not available by the existing loss modeling methods with abstract loss coefficients.

The analytical loss model of magnetic cores proposed in this article for sinusoidal excitation lays a foundation for the calculation of core losses under non sinusoidal excitation, which is undergoing and will be reported in the near future.

Appendix Measurement of Model Parameters

As shown in Fig. 17, an experimental platform has been built to measure the hysteresis angle γ of the magnetic cores 3E6 and N87 based on the two-winding method, which is the most common procedure for B-H loop and core loss characterization. The schematic diagram of the wiring between different equipment is shown in Fig. 18. The measurement process is carried out according to the guidelines in [43], and the measurement frequency 500 Hz under quasi-static conditions was low. For low-frequency measurements, the parasitic effect in the test circuit is small, so the phase discrepancy is small and won't cause a significant error [43], [44]. The capacitance on the primary side is used to eliminate the impact of DC bias caused by the instrument. A voltage excitation is applied to the primary and its current is measured to obtain H. The voltage across the secondary winding is measured to obtain B.

FIGURE 17.

The experimental platform for core loss measurement.

FIGURE 18.

The schematic diagram of the wiring between different equipment.

A more detailed description of the experimental setup is shown below:

1. Excitation: The 3E6 and N87 are excited with sinusoidal voltage, leading to sinusoidal B waveforms in this article. Sinusoidal waveforms are obtained using a power amplifier (Aigtek ATA-4052) taking the reference from a function generator (RIGOL DG1022U).

2. Data acquisition: The measurements for the voltage and current waveform are acquired with the oscilloscope (Tektronix TBS2000B SERIES). Each signal samples 20000 data points, including 3 sine wave periods. High frequency current probe (Tektronix TCPA300) is employed for current measurement. The measurement frequency range of TCPA300 includes DC-10MHz.

Under sinusoidal excitation without dc bias, the relationship between the no-load voltage of the secondary winding and the magnetic flux density inside the magnetic core is as follows: $$\begin{equation*} {{U}_2} = 2\pi f{{N}_2}{{B}_{\mathrm{m}}}S \tag{A1} \end{equation*}$$ where, U₂ is the amplitude of the no-load voltage on the secondary side, B_m is the peak value of the sinusoidal flux density, and N₂ is the number of turns of the secondary winding.

According to Table 1, the size of the measured magnetic cores is relatively small, so the number of turns that can be wound on the primary and secondary sides is limited. At the same time, the working magnetic density value of the ferrite core is relatively low. In order to avoid instrument measurement errors and reading errors caused by low applied voltage, the frequency should not be too low. Therefore, the B-H curve of the 3E6 material magnetic core at 500 Hz was measured in this article. In addition, under the condition of 500 Hz, according to (20) the proportion of eddy-current loss in the total loss is very low and can be ignored. Therefore, it can be considered that this situation satisfies the quasi-static condition. In this case, the hysteresis power loss per unit volume p_h (W/m³) of the magnetic core can be expressed by [1]: $$\begin{align*} {{p}_{\mathrm{h}}}& = f\oint_l {\dot{\bm H} \cdot \mathrm{d}\dot{\bm B}} \\ &= \pi f{{H}_{\mathrm{m}}}{{B}_{\mathrm{m}}}\sin \gamma \tag{A2} \end{align*}$$ where, B_m is the amplitude of magnetic flux density, H_m is the amplitude of magnetic field strength, f is frequency. The values of $\dot{\bm H}$ and $\dot{\bm B}$ can be obtained from the primary current and secondary voltage. Therefore, the variation of hysteresis angle γ with magnetic flux density under quasi-static condition of 3E6 material can be measured.