High-Frequency Modeling and Analysis of Single-Layer NiZn Ferrite Inductors for EMI Filtering in Power Electronics Applications

Abstract

In the high-frequency (HF) region, specifically within the 150 kHz to 30 MHz range for conducting electromagnetic interference (EMI) modeling, NiZn inductors exhibit enhanced efficiency due to their low core losses and stable permeability. Consequently, the accurate modeling of NiZn ferrite toroidal inductors is essential, given their widespread applications in the HF domain, with the aim of addressing existing knowledge gaps. Previous inductor models often relied on the perfect electric conductor (PEC) assumption, which simplifies the analysis but does not fully represent the electromagnetic behavior of the cores to which the PEC assumption cannot be applied. This study investigates the actual electromagnetic behavior of NiZn cores, treating them as dielectrics, which diverges from the traditional PEC-based models. Furthermore, this research fully considers the actual geometry of the inductors, proposing a comprehensive and precise analytical model for NiZn ferrite toroidal inductors. The impact of various winding methods on core capacitance is also explored. The paper provides a detailed explanation of the physical significance underlying the proposed model. A comparative analysis of both modeling methods is presented, and the efficacy of the suggested approach is validated through simulations and experimental results in several distinct scenarios.

1. Introduction

The inductor, as a critical passive component extensively utilized in power electronics systems [1,2], plays a pivotal role in filtering applications to meet power quality standards, minimizing losses and current spikes, and safeguarding device insulation [3]. The rapid advancement in power electronics, driven by demands for device miniaturization and reduced weight, has led to the increased switching of frequencies [4]. Operating at high frequencies (HFs), inductors exhibit parasitic capacitances related to windings and magnetic cores, along with leakage inductance and materials of magnetic cores, all contributing to substantial deviations from the ideal inductor model. These deviations amplify electromagnetic interference (EMI) levels and induce current ringing. Consequently, accurately modeling HF inductors, which constitute a significant portion of a device’s mass, remains a crucial challenge to the optimization of converter performance without increasing weight and volume.

In recent years, a variety of techniques have been employed to characterize and model the HF characteristics of inductors, which are broadly categorized into physical and behavioral models [5,6,7,8,9,10,11,12,13,14]. Among physical modeling techniques, the Finite Element Method (FEM) is noted for its accuracy but critiqued for its complexity, extensive hardware requirements, and significant time consumption [15,16]. This has led to the development of other analytical models such as one using transmission line theory proposed in prior studies, which, while accurate and widely applicable, is limited to single-turn inductors [17]. The landmark study by Marinko et al. is renowned for linking physical parameters with the actual internal geometry of HF inductors [9], thus facilitating optimal design modifications for improved HF performance. However, this model falls short when applied to materials with high permeability and low permittivity, such as nanocrystals. Our recent enhancements have refined this approach by integrating the actual geometry and time-varying fields of the core, thereby extending its accuracy across a broader range of turns and materials and highlighting the importance of material properties in the modeling process. This paper focuses specifically on ferrite cores, given their prevalent use in power electronics, aiming to provide a robust framework for accurately predicting their behavior in HF scenarios.

MnZn and NiZn are the core materials most commonly used in ferrite inductors, each serving specific frequency applications due to their unique properties [18]. MnZn cores are ideal for HFs of several megahertz, offering high impedance due to their high initial permeability, whereas NiZn cores are more effective at higher frequencies because of their low core loss and stable permeability (μ). While MnZn cores have been extensively studied and are often modeled as perfect electric conductors (PECs) due to their high μ or ε [19,20,21,22], which prevents electrical lines from penetrating the core, NiZn cores do not conform to PEC assumptions because of their low μ and ε. Consequently, HF models suitable for MnZn cannot be directly applied to NiZn cores, leading to significant errors in impedance and parasitic capacitance modeling (severely low impedance and high parasitic capacitance). Despite the widespread use of NiZn inductors, there is a notable lack of effective modeling techniques for them. A recent study introduced a model focusing on the equivalent capacitance of NiZn cores [23], offering new research directions for low μ, low ε materials like NiZn. However, this model is limited by the specific core types and geometries it can represent. Therefore, there is an urgent need for a comprehensive, analytical model of NiZn ferrite inductors based on their electromagnetic behavior, to enhance the design and efficiency of HF applications utilizing these inductors.

The remaining sections of this paper are organized as follows: In Section 2, the analytical model of NiZn inductors with different winding methods is introduced in detail based on electromagnetic analysis. Building on previous efforts, in Section 3, the acquisition of physical parameters mentioned above are discussed critically. Finally, verification and discussion of the results are conducted among the proposed model, PEC model, FEM results, and experimental data, reaching a verdict that the proposed model has a higher accuracy.

2. Analysis of NiZn Inductors and Circuit Models

2.1. Electromagnetic Behavior of NiZn Inductor

Figure 1 presents a schematic of a wound toroidal inductor, highlighting its critical geometric parameters. This paper thoroughly considers these actual geometric elements, resulting in more precise modeling outcomes. Furthermore, it is emphasized that the traditional assumption of the PEC is not applicable to NiZn cores; instead, these cores are innovatively treated as dielectrics. This approach allows for the accurate determination of the capacitance for each part of the inductor, thanks to the clear physical meaning of the proposed model, facilitating the optimization of NiZn toroidal inductors based on their geometric structure. The subsequent sections of this paper will introduce an equivalent circuit for modeling NiZn ferrite inductors, primarily discussing NiZn toroidal cores, although the model is also relevant to other shapes of NiZn cores. Additionally, the physical significance of the parameters in the proposed circuit model will be explored, along with detailed calculations for inductance, space capacitance correction, and corresponding core capacitance, enhancing the understanding and application of these models in practical settings.

As an inductor, the most important physical parameter is inductance (L), which is equivalent to the magnetic field energy, as presented in Figure 2a,c. In addition, under HF scenarios, there will be electric field energy between the turns and the turns and core, corresponding to the displacement current in Maxwell’s equations. The electric field energy between turns can be further equated as turn-to-turn capacitance (C_tt) while turn-to-core capacitance (C_tc) occurs in the region between the turns and core. Many studies have explained the mechanism as well as the calculation of C_tc from different perspectives [24,25,26]. The derivation of C_tc in these previous models is based on the assumption of a PEC, and the electric field lines generated from windings tend to terminate at the surface of the core and do not penetrate within, as shown in Figure 2d. It should be noted that the assumption of a PEC does not apply to NiZn inductors, as shown in Figure 2b. Also, it is obvious that there is a static energy stored inside the core, which is called static core capacitance (C_cs). In addition, under a time-varying electromagnetic field, it is known from Faraday’s law that an alternating magnetic flux produces an alternating electric field out of the cross-section inside the core, corresponding to the dynamic core capacitance (C_cd). However, it is known that the electromagnetic wavelength and permeability are related to the dielectric constant according to the electromagnetic wavelength formula. Since NiZn cores have low permeability and permittivity, the wavelength length within 30 MHz is much larger than the maximum size of the core, which makes it unnecessary to consider time-varying effects.

The physical parameters derived from the electromagnetic field equivalents for modeling NiZn inductors introduce two critical concerns that need consideration:

(1) The electromagnetic behavior of NiZn inductors, which cannot be modeled as PECs, necessitates treating NiZn cores as dielectrics. This shift in modeling approach significantly alters the calculation of C_tc and introduces a new parameter C_cs, complicating the calculation of the inductor’s parasitic capacitance.

(2) Since NiZn cores are treated as dielectrics, the energy in the core varies with different winding configurations, necessitating a unique approach to calculating core capacitance. This paper classifies windings into three categories: complete uniform, sub-complete uniform, and incomplete uniform, as illustrated in Figure 3, highlighting the need for distinct calculations for each type.

To further validate the concerns raised, this paper delves into the electromagnetic behavior of NiZn inductors in the HF range. The electromagnetic wavelengths in the air (9.99 m at 30 MHz) and within NiZn cores (1.09 m at 3 MHz) are significantly larger than the inductor’s maximum size, allowing these regions to be treated as electrostatic fields. Conversely, for inductors with MnZn or nanocrystal cores, the wavelengths are comparable to the inductor size, necessitating the consideration of time-varying fields inside these cores. As demonstrated by the FEM results shown in Figure 2b,d, unlike MnZn and nanocrystal cores that can be modeled as PECs due to their high μ or ε, NiZn cores do not support the PEC assumption. This difference underscores the need for a distinct modeling approach for NiZn-based inductors.

From Figure 4a, it is clear that electric field lines do not terminate perpendicularly on the NiZn core’s surface but diverge into the core, leading us to treat the NiZn core as a dielectric. As these field lines penetrate the core, they align along the core’s circumference, perpendicular to the cross-section, influenced by dielectric boundary conditions and opposing electric fields. Additionally, due to Ampère’s law, an alternating current in the windings generates an alternating magnetic flux perpendicular to the core cross-section, which, according to Faraday’s law, induces a time-varying electric field parallel to the core boundary. As illustrated in Figure 4b,c, the static electric field lines are perpendicular to the core cross-section, whereas the time-varying electric field lines are parallel, both oriented orthogonally to each other and not interacting. Thus, the capacitance C_cd associated with these fields can be connected in parallel at both ends of the winding. However, due to the low permittivity ε of the NiZn core, C_cd can be neglected.

The discussion on the electromagnetic behavior of NiZn cores indicates that these cores cannot be treated as PECs. Consequently, electric lines can easily penetrate the core, allowing electric field lines to flow toward the direction of the largest electric field gradient and terminate at the points of lowest potential. This flow is influenced by the presence and interaction of other electric field lines, leading to a specific distribution of electric field lines within the NiZn core. Additionally, the way the windings are executed can significantly influence the trajectory of the electric field lines through the core. Based on these insights, this paper will further explore the electromagnetic behavior of the core under different winding configurations, analyzing how each influences the overall field distribution.

Figure 3 and Figure 5 illustrate various winding configurations for inductors, categorized into complete uniform, sub-complete uniform, and incomplete uniform windings. In incomplete uniform windings, as shown in Figure 5a, electric field energy primarily accumulates in the inner region of the winding because the electric field lines take the shortest path to the lowest potential point, moving directly from high potential turns to low potential areas within the winding. In contrast, sub-complete uniform windings, depicted in Figure 5b, feature electric field lines flowing directly from a few turns with high potential at one end to a few turns with low potential at the other, particularly as the distance between the start and end turns decreases. Meanwhile, in complete uniform windings, shown in Figure 3d and Figure 5d, the proximity of the head turns to the end turns allows more electric field lines to connect directly, resulting in the greatest electric field energy concentration within the core among these scenarios, which means that this configuration results in the greatest static core capacitance, advancing the resonance point significantly.

2.2. Derivation of HF Circuit Model

2.2.1. Case 1: Incomplete Uniform Windings

Based in the discussion from subsection A, the circuit model for the incomplete uniform winding scenario is detailed in Figure 6, clearly showing the interconnections among various physical parameters. In this model, each turn of the inductor is treated as a unit, consisting of self-inductance (L_s), and series-connected eddy current and hysteresis losses in the magnetic core, denoted by (R_s). Parallel to these is the turn-to-turn capacitance (C_tt). One end of the turn-to-core capacitance (C_tc) connects to the left side of the unit, while the other end links to the static core capacitance (C_cs), which in turn connects to the C_tc at the end of the last unit. However, the C_cs of turns near the last turn are neglected due to the minimal impact of electric field lines reaching the low potential region. Dynamic capacitance (C_cd) is connected in parallel with the first and last turns of the winding. Additionally, since the first and last turns lack adjacent turns, the C_tt between the first and second turns is divided into two equal parts, each paralleled to the cells of the first and last turns, representing half of C_tt. This configuration is typical for multiple windings. However, in the case of a single winding where the core is fully utilized, the distance between the first and last turns decreases, eliminating the need to divide C_tt between these turns. Consequently, the C_tt parallel to the last unit becomes the capacitance between the last and first turn (C_N,1). In addition, the impedance curve can be determined using circuit analysis methods at each frequency.

2.2.2. Case 2: Sub-Complete or Complete Uniform Windings

In the scenarios of sub-complete and complete uniform windings, the proximity of the head turns to the end turns significantly influences the circuit model due to the direct flow of electric field lines to the end turns outside the winding region. Given the core’s geometrical constraints, the core capacitance generated from the turns into a specific region becomes considerable, peaking between the first and last turns. The electric field lines emanating from the second turn tend to flow towards the next-to-last turn, influenced by the repulsion from the field lines of the first turn. This dynamic needs to be integrated into the circuit model for accuracy. Thus, all turns that enter this region are modeled to reflect this effect, while the turns outside this specific region are modeled as previously described. This adapted circuit model is depicted in Figure 7, showcasing how the changes in turn proximity affect the overall circuit behavior and core capacitance.

Having developed the circuit model for NiZn inductors and analyzed the underlying physical fields, the next series of sections will focus on acquiring and quantifying the physical parameters. In the subsection on static capacitance modeling, the true geometry of the inductor, along with its electromagnetic characteristics, will be thoroughly considered. Building on this foundation, the acquisition of dynamic capacitance will be explored, extending the insights from earlier analyses. Finally, NiZn inductor samples with different numbers of turns and different winding methods are fabricated. Also, the accuracy of the proposed model is verified by comparing it with the measured data, which demonstrates that the PEC model is not applicable in modelling NiZn inductors.

3. Physical Parameter Acquisition

3.1. Complex Inductance Modeling

For magnetic materials, relative permeability (μ_r) is one of the most important physical parameters. In HF applications, it is necessary to use complex relative permeability (μ*) to model inductance due to the complex impedance from core loss, even if the winding loss is neglected. The real part of the relative permeability is represented as (μ_r′) while the imaginary part is (μ_r″).

Then, according to Ampère’s circuital law, the impedance (Z) of the single-turn inductor can be presented as:

$$Z=j\omega L_s+R_s=j\omega {\mu }_0({\mu }_r^{\prime }-j{\mu }_r^{″}){\displaystyle \frac{h_c}{2\pi }}\ln ({\displaystyle \frac{R_o}{R_{in}}})$$

(1)

where μ₀ is the permeability of vacuum. In an inductor, the self-inductance of a single-turn coil can be expressed as (L_ii), while the mutual inductance and the leakage inductance between turn I and turn j can be expressed as (M_ij) and (L_σij). The relationship among L_ii, M_ij and L_σij can be expressed as

$$L_{ii}=M_{ij}+L_{\sigma ij}$$

(2)

The L_σij is typically several orders of magnitude smaller than L_ii and can be ignored in the modeling of the NiZn inductor.

3.2. Static Capacitance Modeling

In the previous discussion, it was noted that the permeability (λ) of air and the NiZn core is substantial, allowing for the assumption that the energy within the air and core regions remains static. Consequently, this energy can be characterized by static capacitances, specifically C_tt, C_tc, and C_cs. Taking into account the actual geometry and the distribution of the electric field, expressions for the static capacitance in these regions have been derived. As illustrated in Figure 8, the static capacitance for different regions can be analytically calculated using the following approach:

$$dC={\epsilon }_{0}dS/{\displaystyle \sum _i(x_i/{\epsilon }_{ri})}$$

(3)

where dS denotes the differential area, x_i represents the length of electric line, and ε_ri is the permittivity of the medium. As presented in [9] and the model proposed previously in [8], during the calculation of static capacitance, there are three specific regions that can be classified, which are the inner region (C_ttin, C_tcin, C_csin), outer region (C_tto, C_tco, C_cso) and two different lateral regions (Front side: C_ttl, C_tcl, C_csl. Back side: C_ttl′w, C_tcl′w, C_csl′w). The regions classified are illustrated in Figure 9.

3.2.1. Calculation of Turn-to-Turn Capacitance C_tt

As shown in Figure 8a, since the conditions are the same, C_tt can be derived directly by using our previously proposed formula [8]:

$$C_{tt}=C_{ttin}+C_{tto}+C_{ttl}+C_{ttl\prime w}$$

(4)

However, in the case of MnZn, where the core can be approximated as a conductor, the energy is predominantly concentrated between the turns, as well as between the turns and the core. In contrast, in the NiZn scenario, the magnetic core cannot be approximated as a conductor, resulting in the majority of the energy being concentrated within the turns. Consequently, the value of C_tt in the NiZn case requires enhancement. Through comparative analysis, the C_tt in the NiZn scenario can be approximately expressed as twice that in the MnZn case.

Then, C_ttin and C_tto can be presented directly as

$$\left\{\begin{array}{c}{C}_{tto}={\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{2{\epsilon }_0{\epsilon }_i{\epsilon }_a{h}_w(d_c/2)\sin \alpha d\alpha }{2w_i{\epsilon }_a\sin \alpha +\alpha {\epsilon }_i[{\delta }_o+(d_c+2w_i)(1-\cos \alpha )]}}\\ C_{ttin}={\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{2{\epsilon }_0{\epsilon }_i{\epsilon }_a{h}_w(d_c/2)\sin \alpha d\alpha }{2w_i{\epsilon }_a\sin \alpha +\alpha {\epsilon }_i[{\delta }_{in}+(d_c+2w_i)(1-\cos \alpha )]}}\end{array}\right.$$

(5)

In addition, C_ttl can also be presented as

$$\begin{array}{l}{C}_{ttlw}={\displaystyle {\int }_{R_{in}}^{R_o}{\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}}}\hfill \\ {\displaystyle \frac{2{\epsilon }_0{\epsilon }_i{\epsilon }_a(d_c/2)\sin \alpha d\alpha dr}{\alpha {\epsilon }_i[\frac{{\delta }_o-{\delta }_{in}}{R_o-R_{in}}(r-R_{in})+{\delta }_l+(d_c+2w_i)(1-\cos \alpha )]+2w_i{\epsilon }_a}}\hfill \end{array}$$

(6)

However, during the winding process, the coil length in the parallel region does not directly correspond to half of the difference between the inner and outer diameters of the inductor. The length of l′_w can be calculated using the cosine theorem as

$$\begin{array}{l}{l}_w^{\prime 2}=2R_{in}^2(1-\cos \phi )+{(R_o-R_{in})}^2\hfill \\ -2\sqrt{2(1-\cos \phi )}{R}_{in}(R_o-R_{in})\cos ((\pi +\phi )/2)\hfill \end{array}$$

(7)

Therefore, C_ttl′w can be expressed as

$$\begin{array}{l}{C}_{ttl\prime w}={\displaystyle {\int }_{R_{in}}^{R_{in}+l_w^{\prime }}{\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}}}\hfill \\ {\displaystyle \frac{2{\epsilon }_0{\epsilon }_i{\epsilon }_a(d_c/2)\sin \alpha d\alpha dr}{\alpha {\epsilon }_i[\frac{{\delta }_o-{\delta }_{in}}{l_w^{\prime }}(r-R_{in})+{\delta }_l+(d_c+2w_i)(1-\cos \alpha )]+2w_i{\epsilon }_a}}\hfill \end{array}$$

(8)

3.2.2. Calculation of Turn-to-Core Capacitance C_tc

As presented in our previous study [8], accounting for winding transitions, the derivation of an equivalent turn-to-core distance can be calculated as

$$w_s^{\prime }=2\sqrt{w_c(w_c-w_e)}/\log ({\displaystyle \frac{\sqrt{w_c}+\sqrt{w_c-w_e}}{\sqrt{w_c}-\sqrt{w_c-w_e}}})$$

(9)

C_tc can also be derived as

$$C_{tc}=C_{tcin}+C_{tco}+C_{tcl}+C_{tcl\prime w}$$

(10)

As illustrated in Figure 10, there are two situations in the calculation of C_cs. The first one is the case of a partially wound inductor. Since NiZn cores cannot be applied with the assumption of a PEC, the electric field lines between the turns and the core are evanescent. In fact, since the electric field lines are continuous in the region between the turns and the core, as well as in the internal region of the core, this capacitance should be counted as one capacitance. However, for ease of representation, this capacitance is here split into C_tc and C_cs geometrically. Here, C_tc is calculated first. The length of the electric field lines in the winding insulation region needs to be determined first, which can be presented as

$$x_{tc,wi}=w_i$$

(11)

Then, due to the boundary condition of the interface between the two mediums, we can further obtain the length of electric field lines in the air and core insulation regions. First of all, because there are no free surface charges at the interface of the two different mediums, the electric field lines distribution in the air can be obtained, which can be expressed as

$$\left\{\begin{array}{l}{E}_{t,air}=E_{t,i}\hfill \\ E_{n,air}={\displaystyle \frac{{\epsilon }_i}{{\epsilon }_a}}{E}_{n,i}\hfill \end{array}\right.$$

(12)

where E_t,air and E_t,i are the tangential components of the electric field lines in the region of the air and insulation and E_n,air and E_n,i are the normal components of the electric field lines in the region of the air and insulation. Then, the deflection angle of the electric field lines in the air can be presented as:

$${\alpha }_{air}=\arctan ({\displaystyle \frac{{\epsilon }_a}{{\epsilon }_i}}\tan \alpha )$$

(13)

Furthermore, the length of the electric field lines in the air can be obtained as follows:

$$x_{tc,air}={\displaystyle \frac{w_s+(d_c/2+w_i)(1-\cos \alpha )}{\cos {\alpha }_{air}}}$$

(14)

Then, due to a similar derivation process, the length of the electric field lines in the core insulation can be directly obtained, which can be expressed as

$$x_{tc,ci}=w_{ci}/\cos {\alpha }_{ci}$$

(15)

where α_ci can be further presented as

$${\alpha }_{ci}=\arctan ({\displaystyle \frac{{\epsilon }_{ci}}{{\epsilon }_a}}\tan {\alpha }_{air})$$

(16)

Therefore, C_tco can be expressed as

$$\begin{gathered} C_{tco}={\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}} \\ {\displaystyle \frac{{\epsilon }_0{\epsilon }_i{\epsilon }_a{\epsilon }_{ci}{h}_w(d_c/2)\cos {\alpha }_{air}\cos {\alpha }_{ci}d\alpha }{{\epsilon }_a\cos {\alpha }_{air}({\epsilon }_{ci}\cos {\alpha }_{ci}{w}_i+{\epsilon }_i{w}_{ci})+{\epsilon }_i{\epsilon }_{ci}\cos {\alpha }_{ci}[w_{so}+(d_c/2+w_i)(1-\cos \alpha )]}} \end{gathered}$$

(17)

C_tcin can be expressed as

$$\begin{gathered} C_{tcin}={\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}} \\ {\displaystyle \frac{{\epsilon }_0{\epsilon }_i{\epsilon }_a{\epsilon }_{ci}{h}_w(d_c/2)\cos {\alpha }_{air}\cos {\alpha }_{ci}d\alpha }{{\epsilon }_a\cos {\alpha }_{air}({\epsilon }_{ci}\cos {\alpha }_{ci}{w}_i+{\epsilon }_i{w}_{ci})+{\epsilon }_i{\epsilon }_{ci}\cos {\alpha }_{ci}[w_{sin}+(d_c/2+w_i)(1-\cos \alpha )]}} \end{gathered}$$

(18)

In addition, using the same process as in the derivation of C_tco and C_tcin, C_tclw can be derived as

$$\begin{gathered} C_{tclw}={\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}} \\ {\displaystyle \frac{{\epsilon }_0{\epsilon }_i{\epsilon }_a{\epsilon }_{ci}(R_o-R_{in})(d_c/2)\cos {\alpha }_{air}\cos {\alpha }_{ci}d\alpha }{{\epsilon }_a\cos {\alpha }_{air}({\epsilon }_{ci}\cos {\alpha }_{ci}{w}_i+{\epsilon }_i{w}_{ci})+{\epsilon }_i{\epsilon }_{ci}\cos {\alpha }_{ci}[w_{sl}+(d_c/2+w_i)(1-\cos \alpha )]}} \end{gathered}$$

(19)

C_tcl′w can also be expressed as

$$\begin{gathered} C_{tcl\prime w}={\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}} \\ {\displaystyle \frac{{\epsilon }_0{\epsilon }_i{\epsilon }_a{\epsilon }_{ci}{l}_w^{\prime }(d_c/2)\cos {\alpha }_{air}\cos {\alpha }_{ci}d\alpha }{{\epsilon }_a\cos {\alpha }_{air}({\epsilon }_{ci}\cos {\alpha }_{ci}{w}_i+{\epsilon }_i{w}_{ci})+{\epsilon }_i{\epsilon }_{ci}\cos {\alpha }_{ci}[w_{sl}+(d_c/2+w_i)(1-\cos \alpha )]}} \end{gathered}$$

(20)

3.2.3. Calculation of Static Core Capacitance C_cs

Since the NiZn core is treated as a dielectric, the electric field lines are also divergent across the dielectric boundary. The electric field lines, as they gradually approach the axis of the core section, will end up in the region with the lowest potential. The lines are assumed to be x_cs. The derivation of x_cs also applies the boundary conditions of the medium, which are

$$\left\{\begin{array}{l}{E}_{t,c}=E_{t,ci}\hfill \\ E_{n,c}={\displaystyle \frac{{\epsilon }_{ci}}{{\epsilon }_c}}{E}_{n,ci}\hfill \end{array}\right.$$

(21)

where E_t,c and E_t,ci are the tangential components of electric field lines in the regions of the core and core insulation and E_n,c and E_n,ci are the normal components of the electric field lines in the region of the core and core insulation. The deflection angle of the electric field lines in the core can be presented as

$${\alpha }_c=\arctan ({\displaystyle \frac{{\epsilon }_c}{{\epsilon }_{ci}}}\tan {\alpha }_{ci})$$

(22)

As shown in Figure 10, then, x_cs can be obtained as follows:

$$x_{cs}=R_t/\cos {\alpha }_c+l_{arc}$$

(23)

where R_t is 1/2 the thickness of the core, which is different in different regions. In the inner and outer regions, R_t can be expressed as

$$R_t=(R_o-R_{in})/4$$

(24)

while in the lateral region, R_t can be presented as

$$R_t=h_w/2$$

(25)

Similar to with C_tt and C_tc, C_cs can be obtained as follows:

$$C_{cs}=C_{csin}+C_{cso}+C_{cslw}+C_{csl\prime w}$$

(26)

Among these regions, C_csin and C_cso are the same, presented as

$$C_{csin}+C_{cso}={\displaystyle \underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{2{\epsilon }_0{\epsilon }_c{h}_w(d_c/2)\cos {\alpha }_{c}d\alpha }{(R_o-R_{in})/4+l_{arc}}}$$

(27)

C_cslw is given by

$$C_{cslw}={\displaystyle \underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{{\epsilon }_0{\epsilon }_c(R_o-R_{in})(d_c/2)\cos {\alpha }_{c}d\alpha }{h_w/2+l_{arc}}}$$

(28)

Also, C_csl′w can be given by

$$C_{csl\prime w}={\displaystyle \underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{{\epsilon }_0{\epsilon }_c{l}_w^{\prime }(d_c/2)\cos {\alpha }_{c}d\alpha }{h_w/2+l_{arc}}}$$

(29)

3.2.4. Calculation of Static Core Capacitance C_he Between Head and End Turns

In the situation of a complete uniform wound, as shown in Figure 11, the C_cs between the head turns and end turns is indispensable. For ease of calculation, it is necessary to first calculate the equivalent permittivity of the air, the insulation, and the inter-core. Since the electric field lines do not exceed half of the core, the proposed model takes one quarter of the difference between the inner and outer diameters for the core length:

$${\epsilon }_e=d_{all}/{\displaystyle \sum _i(d_i/{\epsilon }_i)}$$

(30)

where ε_e is the equivalent permittivity of the region, d_all is the total thickness of the region, di is the thickness of medium, and ε_i is the permittivity of medium.

Then, C_he can be expressed as

$$C_{he}=C_{hein}+C_{heo}+C_{helw}+C_{hel\prime w}$$

(31)

where the specific expressions are as follows:

$$\left\{\begin{array}{c}{C}_{heo}={\displaystyle {\int }_{-\pi }^0\frac{{\epsilon }_0{\epsilon }_e{h}_w(d_c/2)\sin \alpha d\alpha }{\alpha [{\delta }_o+d_c(1-\cos \alpha )]}}\\ C_{hein}={\displaystyle {\int }_{-\pi }^0\frac{{\epsilon }_0{\epsilon }_e{h}_w(d_c/2)\sin \alpha d\alpha }{\alpha [{\delta }_{in}+d_c(1-\cos \alpha )]}}\end{array}\right.$$

(32)

In addition, C_helw and C_hel’w can also presented as

$$C_{helw}={\displaystyle {\int }_{R_{in}}^{R_o}{\displaystyle {\int }_{-\pi }^0\frac{{\epsilon }_0{\epsilon }_e(d_c/2)\sin \alpha d\alpha dr}{\alpha [\frac{{\delta }_o-{\delta }_{in}}{R_o-R_{in}}(r-R_{in})+{\delta }_l+d_c(1-\cos \alpha )]}}}$$

(33)

$$C_{hel\prime w}={\displaystyle {\int }_{R_{in}}^{R_{in}+l_w^{\prime }}{\displaystyle {\int }_{-\pi }^0\frac{{\epsilon }_0{\epsilon }_e(d_c/2)\sin \alpha d\alpha dr}{\alpha [\frac{{\delta }_o-{\delta }_{in}}{l_w^{\prime }}(r-R_{in})+{\delta }_l+d_c(1-\cos \alpha )]}}}$$

(34)

4. Experiment Verification

To further evaluate the impedance characteristics of NiZn material inductors and to validate the accuracy of the proposed model, several prototype inductors were fabricated with varying numbers of turns and winding conditions, as detailed in

Table 1. Parameters of the different NiZn inductor cases (TDK k10 [27]).

	Case 1	Case 2	Case 3
Winding form	Partial wound	Partial wound	Fully wound
Number of turns	37	27	27
Dimension	50/30/20(mm)
Core permeability	8 × 102–2 × 102
Core permittivity	12
Core insulation	0 mm
Winding insulation	0.1 mm
Equivalent turn-to-core distance	0.2 mm
Diameter of winding	1 mm

and illustrated in Figure 12. The parameters specified in the table were utilized to construct a 3D model for a Finite Element Method (FEM) analysis using Ansys HFSS 2021. The impedance testing was conducted using a TONGHUI TH2851—130 impedance analyzer manufactured by Changzhou Tonghui Electronic Co., Ltd., Changzhou, China, with the frequency range for testing set between 20 kHz and 30 MHz. This comprehensive approach allowed for a detailed assessment of the model’s performance across a broad spectrum of operational conditions.

The comparison among the proposed NiZn model, the PEC model, measured data, and FEM data is illustrated in Figure 13, Figure 14 and Figure 15. It is obvious that the proposed NiZn model has a relatively higher accuracy, whose resonant point and the value of the impedance curves in wide frequencies precisely match the measured results. However, when applying the PEC model, the impedance curve shows a huge deviation with the measured data of the NiZn case, whose resonant point deviation is 3.1 MHz in the case of 37 turns, a partially wound inductor, 6 MHz in the case of 27 turns, a partially wound inductor, and 1.2 MHz in the case of 27 turns, a fully wound inductor, compared to the measured data. Therefore, for NiZn inductors, the PEC inductor model is not applicable; instead, the NiZn model proposed in this paper has a much higher accuracy.

In addition, by comparing the results from 37 turns, a partially wound inductor, and 27 turns, a partially wound inductor, which are shown in Figure 13 and Figure 14, it is obvious that the proposed model for the NiZn inductor has a higher accuracy when varying the number of turns, while the PEC inductor model shows huge errors when there are different numbers of turns. Furthermore, by comparing the results from the 27 turns, partially wound inductor, and the 27 turns, fully wound inductor, which are illustrated in Figure 14 and Figure 15, it is shown that the proposed NiZn inductor model is also suitable for different winding methods with a higher accuracy. It can be seen that the PEC model is not applicable for any of the cases with NiZn cores and leads to huge errors.

5. Conclusions

This paper introduces a comprehensive analytical model for single-layer NiZn ferrite inductors, designed to be applicable across various constructions of NiZn inductors. The model aims to overcome the limitations of traditional assumptions by providing an in-depth discussion of the physical significance underlying the model. It addresses the actual electromagnetic behavior of NiZn materials and proposes a new set of assumptions, integrating advanced techniques for deriving internal lumped parameters. Additionally, the model considers the actual geometric characteristics of the inductors to enhance accuracy. The study also explores the impact of different winding methods on core capacitance and incorporates these effects into the model. The validity and high modeling accuracy of the proposed model are convincingly demonstrated through simulations and experimental results, which cover a range of scenarios involving different numbers of turns and winding conditions. This robust model offers significant improvements over traditional modeling approaches, providing a valuable tool for the design and optimization of NiZn ferrite inductors.