An overview of power loss calculation methods for high-frequency litz wires

Abstract

Accurate calculation of power losses has always been important for the design and optimization of magnetic components consisting of windings and magnetic cores, such as inductors and transformers. Focusing on windings, litz wire outperforms its counterparts in terms of eddy current losses, making it particularly suitable for high-frequency (HF) applications. The power loss calculation of litz wire is however challenging due to its complex structure of thin strands twisted in multiple levels. This article therefore aims to review various kinds of litz wire power loss calculation methods from analytical methods, numerical methods to coupling ones. In addition to, the principles and applications of each method, their inherent correlation and differences are also highlighted in this article. On this basis, a comprehensive review and comparison of different calculation methods of litz-wire power loss are provided. Finally, future challenges and directions are then summarized, whose ultimate goal is to calculate the power loss of litz wire accurately and efficiently.

Keywords

Calculation methods eddy current losses high frequency litz wires overview

1. Introduction

POWER electronic devices are switching progressively faster due to advancements of wide-bandgap semiconductor materials, like silicon carbide (SiC) and gallium nitride (GaN) [1–4]. By increasing its switching frequency, the required passive components can be greatly reduced, increasing the power density of power electronic systems [5–7]. Among passive components, the magnetic components, like inductors and transformers, can functionally provide energy storage, filtering, isolation, or transformer [8,9]. The high-frequency (HF) magnetic components are therefore promising and have been adopted in many applications, such as photovoltaic systems, electric locomotive traction, and wireless power transfer [10–12].

The HF operation, on the other hand, will inevitably increase the power losses of magnetic components, complicating its design and optimization [13,14]. For a magnetic component, its indispensable windings are usually made of copper foils or solid round conductors [15–17]. Unfortunately, the winding losses increase significantly with frequency due to the eddy-current effect (skin effect and proximity effect). Compared with traditional copper conductors, the litz wires experience reduced eddy-current losses, making them more suitable for HF applications under certain conditions [18–21]. They also offer the possibility of optimization and design of windings in inductors, transformers, motors, or wireless power transfer systems [22–24]. However, the power loss calculation of litz wires is difficult because of its complex structure of thin strands twisted in multiple levels, which imposes challenges to the design and optimization of magnetic components.

There are many efforts made by scholars in the study of litz-wire loss calculation as described in Fig. 1. Different analytical methods have been proposed to calculate the losses of litz wires [25–45]. Generally, they can be divided into two categories based on whether the 3-D twisting structure is involved. Without considering the twisting structure, they are assumed that the twisting works perfectly to balance current between strands, and the expressions of these are generally derived from an ideal litz-wire structure, making it only available in specific cases. Therefore, they are not helpful in choosing the details of the twisting configuration. Instead, considering the 3-D twisting structure can explicitly analyze the effect of the twisting structure on losses, which is based on the inclination angle of strands, coordinate transformation, or probabilistic model.

Fig. 1.

Development history of the various calculation methods for high-frequency litz wires.

Numerical methods seem to have more potential to solve these kinds of problems [46–60]. Among them, the finite-element method (FEM) is most commonly used to simulate the litz-wire losses. However, it requires fine mesh in the domain of coppers and air [61,62], which will result in complex mesh and a long computational time. To avoid the above-mentioned problems, homogenization-based 2-D FEM models are presented. However, the twisting characteristics of realistic litz wires are neglected, which hinders the loss evaluation for realistic litz wires. The partial-element equivalent circuit (PEEC) method, based upon an integral formulation of Maxwell’s equations [63–65], avoids air discretization, so it can achieve both a realistic twisting structure and the reduction of the computational time. Compared with the PEEC method, the strand element equivalent circuit (SEEC) method, analogous to the PEEC method, can further improve the computational speed [66].

In addition, the coupling method, such as combining FEM with the analytical method [67] or combining FEM with the PEEC method [68], can integrate the advantages of different methods. An explicit extraction scheme for external magnetic fields is presented by using the coupling method. At present, these coupling methods have been applied in some high-frequency transformer designs.

To calculate the power losses of HF litz wires accurately and efficiently, more efforts are still needed. However, until now, there is a lack of systematic review of the aforementioned calculation methods. To cover this gap, this article provides a comprehensive review of existing power loss calculation methods for HF litz wires. The advantages and disadvantages of these methods for dealing with the complex structure of the litz wires are summarized. More importantly, the inherent correlation and developing procedures of these methods are highlighted to provide readers with a better understanding of existing calculation methods. Meanwhile, some applications of the loss calculation methods for new types of litz wires are briefly described.

Although the development of calculation methods for litz-wire loss is illustrated in Fig. 1, this article does not follow a temporal order but is organized according to the commonalities and characteristics between the different methods. Section 2 describes the basic characteristics of the litz wires and the classification of their HF losses. Section 3 introduces several classic analytical methods to calculate the losses of litz wires. Section 4 presents the 2-D and 3-D FEM to calculate the losses of litz wires, and mainly focuses on the homogenization of 2-D FEM. Furthermore, the PEEC method and SEEC method are provided to consider the realistic structure of litz wires. Section 5 describes the coupling methods that combine the advantages of analytical methods and numerical methods. Section 6 summarizes some challenges and suggested feasible solutions, and provides an outlook on the future development direction of calculation methods for litz wires.

2. Fundamentals of litz wire

The term “litz wire” is derived from the German word “Litzendraht” meaning woven wire. It consists of a bundle of fine films insulated and enameled, magnet wires, which are bunched, stranded together, twisted, or braided together into a uniform pattern and are externally connected in parallel. Litz wires are often wrapped with nylon textile or yarn for added strength and protection. The design of the litz-wire multi-level structure can minimize power losses exhibited in solid wires due to the skin effect and proximity effect [69].

Fig. 2.

The structural characteristics of a litz wire. (a) The 3-D model of a litz wire. (b) The contours of the bundles are visible in the real litz wire.

2.1. Structure of litz wire

In order to easily distinguish the different parts of litz wires that have been named according to the multi-level structure. The structural characteristics of a litz wire are shown in Fig. 2.

Strand: The magnet wires are used as a starting point for litz-wire construction. The strand is a single insulated filament in litz wires.

Bundle: A higher-level structure consisting of strands can be further classified as sub-bundles or top-bundle depending on the structure of the litz wires.

Pitch: Pitch is defined as the axial distance of one rotation that any point on a strand or bundle moves along the same spiral [70].

Perfect twisting: The strands are twisted together in both radial and azimuthal directions. These strands are braided together in such a way that each single strand occupies every position in the total cross-section for periodic length intervals along the bundle [69].

Imperfect twisting: When more than six strands are twisted, they are only twisted along the azimuthal direction. Some strands placed in the center of the wire always stay in the center and are not twisted [71].

Fig. 3.

Case (a) shows the current density in the dc case in a 7 × 0.2 mm litz wire. Case (b) shows only skin effect at 200 kHz. Case (c) the internal proximity effect and skin effect is considered. Case (d) additional the external magnetic field is considered.

2.2. Classification of litz-wire losses

When the skin depth is less than the strand diameter of a single strand, due to the complex structure of litz wires, the losses caused by eddy currents can be divided into four categories [57,72]. Figure 3 depicts the different types of losses.

DC loss: the loss is independent of eddy currents and is solely determined by the cross-sectional area and length of the conductor.

Skin loss: eddy current losses originate from the magnetic field of the same strand.

Internal proximity loss: eddy current losses in a single strand originate from the magnetic field of other strands in the same litz wires.

External proximity loss: eddy current losses in a single strand originate from the external magnetic field (nearby wires and the core including the air gaps).

3. Analytical methods

In the field of electromagnetics, the formulation of Maxwell’s equation plays an important role. It describes all interactions between currents, electric and magnetic fields. However, solving full 3-D electromagnetic field systems by analytical methods is very complex and only available for some theoretical examples [73]. Consequently, there are no exact analytical methods to solve real 3-D electromagnetic systems. Instead, some specific approximation formulas or 2-D simplifications are used [74,75]. Especially, the complex 3-D structures like litz wires, the exact solution of the electromagnetic field problem is almost impossible. Therefore, reasonable simplification is the key to solving complex 3-D electromagnetic field problems and ensuring solution accuracy of the litz-wire losses.

3.1. Without considering the twisting structure

3.1.1. Based on Dowell’s formula

Wen-Jian Gu et al. used the area-equivalence method to evaluate the litz-wire losses [27]. In the beginning, the circular cross-section litz wires are transformed into square cross-sections with equal areas. Subsequently, the square litz wires are treated as copper foil windings with a certain packing factor. Finally, Dowell’s formula [76] is used to calculate the AC resistance factor of equivalent copper foils. Figure 4 shows the equivalent procedure of litz wires to foil conductor $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F_{r,n}={\rm\Delta}^{\prime }\left[{\varphi}_{1}^{\prime }+\frac{2}{3}(m^{2}N_{s}-1){\varphi}_{2}^{\prime }\right]\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle {\varphi}_{1}^{\prime }=\frac{\sinh (2{\rm\Delta}^{\prime })+\sin (2{\rm\Delta}^{\prime })}{\cosh (2{\rm\Delta}^{\prime })-\cos (2{\rm\Delta}^{\prime })}\quad \\[12.0pt] \displaystyle {\varphi}_{2}^{\prime }=\frac{\sinh {\rm\Delta}^{\prime }-\sin {\rm\Delta}^{\prime }}{\cosh {\rm\Delta}^{\prime }+\cos {\rm\Delta}^{\prime }}\quad \end{array}\right.\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle {\rm\Delta}^{\prime }=\sqrt{n}\frac{d_{s}}{2{\delta}}\cdot \sqrt{{\pi}{\eta}}\quad \\[12.0pt] \displaystyle {\eta}=\frac{\displaystyle \sqrt{n}\frac{d_{s}}{2{\delta}}\cdot \sqrt{{\pi}{\eta}}}{2(D_{L}+d_{L})h_{c}}\quad \end{array}\right.\end{eqnarray}$ (3) where N _S is the number of strands in litz wires, D _L is the external diameter of litz wires, d _s is the diameter of a single strand, d _L is the distance between litz wires and m is the number of layers. The solution procedure of this method requires meeting the assumptions of Dowell’s model. However, in actual transformer or inductor design, these assumptions are difficult to be realized.

Fig. 4.

The equivalent procedure of litz-wire windings to foil conductor.

3.1.2. Ferreira method

Ferreira used the Bessel function solution derived from an isolated cylinder to calculate the proximity loss and skin loss in the individual strand of litz wire in [28,29]. The power dissipation per unit length of the cylindrical wire and strip conductors can be expressed $\begin{eqnarray}\displaystyle P_{u}=\sum _{i=1}^{\infty }(P_{usi}+P_{upi}) & & \displaystyle\end{eqnarray}$ (4) where P _usi is the skin effect loss at the ith harmonic and P _upi is the proximity effect loss at the ith harmonic. Among them, P _us and P _up are calculated as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{us}=\mathit{FI}^{2}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{up}=\mathit{GH}_{e}^{2}\end{eqnarray}$ (6) where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F=\frac{R_{dc}}{4}\frac{[ber({\gamma})bei^{\prime }({\gamma})-bei({\gamma})bei^{\prime }({\gamma})]}{[ber^{\prime 2}({\gamma})+bei^{\prime 2}({\gamma})]}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle G=\frac{-2{\pi}r}{{\sigma}}\frac{[ber_{2}({\gamma})ber^{\prime }({\gamma})+bei_{2}({\gamma})bei^{\prime }({\gamma})]}{[ber^{2}({\gamma})+ber_{2}({\gamma})]}\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\gamma}=\frac{d_{s}}{\sqrt{2}{\delta}}\end{eqnarray}$ (9) where I is the peak value of the current in the conductor and H _e is the peak value of the external magnetic field. d _s and 𝛿 denote the diameter of strands and skin depth, respectively. ber and bei are the real and imaginary parts of the Bessel function, respectively. Ferreira method is a classic analytical method for calculating the litz-wire losses, which is improved and modified by the following methods.

3.1.3. Bartoli method

The reference [30] based on the reference [31] takes into account the one-dimensional approach and uses the orthogonality principle [77,78] to derive the AC resistance of skin effect, internal and external magnetic field, respectively. Among them, the calculation method of skin effect loss is the same as Ferreira’s, and internal proximity loss is modified with an internal porosity factor (Eqs (11), (12)). The expression for the AC resistance factor of its winding is written as: $\begin{eqnarray}\displaystyle F_{r,n}=\frac{{\gamma}}{2}\left\{{\psi}_{1}({\gamma})-2{\pi}\left[\frac{4(m^{2}-1)}{3}+1\right]\cdot N_{s}\left({\eta}_{1}^{2}+{\eta}_{2}^{2}\frac{p}{2{\pi}N_{s}}\right){\psi}_{2}({\gamma})\right\} & & \displaystyle\end{eqnarray}$ (10) where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\eta}_{1}=\frac{d_{s}}{t_{0}}\sqrt{\frac{{\pi}}{4}}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\eta}_{2}=\frac{d_{s}}{t_{s}}\sqrt{\frac{{\pi}}{4}}\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\beta}=\frac{N_{s}r_{s}^{2}}{r_{0}^{2}}\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\psi}_{1}({\gamma})=\frac{ber({\gamma})bei^{\prime }({\gamma})-bei({\gamma})ber^{\prime }({\gamma})}{ber^{\prime 2}({\gamma})+bei^{\prime 2}({\gamma})}\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\psi}_{2}({\gamma})=\frac{ber_{2}({\gamma})ber^{\prime }({\gamma})-bei_{2}({\gamma})bei^{\prime }({\gamma})}{ber^{2}({\gamma})+bei^{2}({\gamma})}\end{eqnarray}$ (15) where p is the filling factor of a litz wire, which can be written as $\begin{eqnarray}\displaystyle p=\frac{N{\pi}r_{s}^{2}}{{\pi}r_{0}^{2}} & & \displaystyle\end{eqnarray}$ (16) where 𝛾 is defined in the same way as the Eq. (9). d _s = 2r _s is the diameter of the single strand and t ₀ is the distance between the centers of two adjacent litz wires. t _s and r ₀ represent the distance between the centers of two adjacent strands, the litz-wire radius excluding the insulating layer, and N _s is the number of strands in litz-wire winding. m is the number of the litz-wire layers. These analytical expressions are derived from the following assumptions:

Based on the one-dimensional approach proposed in [79], a winding layer is treated as a foil conductor with the magnetic field assumed to be parallel to the conductor.

The core material has a high magnetic permeability, and the winding layer wires fill the entire breadth of the bobbin.

The curvature of the winding is neglected when the magnetic field distribution is calculated.

These assumptions may limit the scope of application of the analytic formulas.

3.1.4. Tourkhani method

In 2001, Tourkhani et al. based on Ferreira’s model proposed a loss analytical model of round litz-wire winding [32]. The eddy current loss density expression of the round conductor is based on the internal and external leakage magnetic field distribution of a single strand. Each bundle of litz wires is treated as a round conductor considering a packing factor, and the loss density is integrated with a two-dimensional polar coordinate system. The leakage field H _ext is supposed one-dimension (y direction). The magnetic field distribution of a litz wire is shown in Fig. 5. The parameters m, k, m ^′, j are the number of layers in windings. H _int and H _ext are the internal and external leakage magnetic fields, respectively. r _c is the radius of a litz-wire bundle, (r, 𝜃) is the position of a strand in the litz-wire conductor, and Δx is the distance of a strand along the x direction. Based on this model, a closed-form formula of the AC resistance of round litz-wire winding is developed. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R_{\text{AC}}=\frac{\sqrt{2}N{\rho}}{{\pi}{\delta}N_{0}d_{0}}\left({\psi}_{1}({\zeta})-\frac{{\pi}^{2}N_{0}{\beta}}{24}\left(16m^{2}-1+\frac{24}{{\pi}^{2}}\right){\psi}_{2}({\zeta})\right)\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\psi}_{1}({\zeta})=\frac{ber({\zeta})bei^{\prime }({\zeta})-bei({\zeta})ber^{\prime }({\zeta})}{ber^{\prime 2}({\zeta})+bei^{\prime 2}({\zeta})}\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\psi}_{2}({\zeta})=\frac{ber_{2}({\zeta})bei^{\prime }({\zeta})+bei_{2}({\zeta})ber^{\prime }({\zeta})}{ber^{2}({\zeta})+bei^{2}({\zeta})}\end{eqnarray}$ (19) where ber and bei denote the real and imaginary parts of the Bessel function. The packing factor 𝛽, the number of strands in a conductor N ₀, the diameter of a strand d ₀, and the number of conductors in the winding N can be easily obtained from the parameters of the winding structure. 𝜁 = d ₀∕𝛿 is the normalized value of the diameter of a strand, 𝛿 is the skin depth, m is the number of layers and 𝜌 is the copper resistivity. However, the author neglects the components of the external magnetic field along the x and z directions, which is the main reason for the error of the proximity loss.

Fig. 5.

Schematic of the leakage magnetic field distribution of a litz wire.

Since the estimation of proximity loss depends on the distribution of the magnetic field in litz wire, when AC resistance is considered, the above four methods share a commonality in that their calculation of the proximity loss is based on a 1-D magnetic field assumption. The losses can be evaluated relatively accurately only when the winding height is close to the window height.

3.1.5. Based on the squared-field-derivative method

The scholar Sullivan introduced a squared-field-derivative method for only calculating proximity effect losses in litz-wires or round-wire winding of transformers or inductors [33]. The method can analyze losses caused by 2-D or 3-D fields in multiple windings with arbitrary waveforms, the expression for total AC losses in all conductors written as $\begin{eqnarray}\displaystyle P_{e,j}=\frac{{\pi}\ell _{t,j}N_{j}d_{c,j}^{4}}{64{\rho}_{c}}\left\langle \overline{\left(\frac{\mathit{dB}}{dt}\right)^{2}}\right\rangle _{j} & & \displaystyle\end{eqnarray}$ (20) where N _j and ℓ _{t, j} are the total numbers of strands in the winding, and the average length of the turn, respectively. 𝜌_c is the resistivity of the wire, and d _c is the diameter of the strands. This calculation method neglects circulation current between strands, because of considering the ideal bundle-level twisting structure. In [34], the formula (20) is improved and used in the analytical calculation of proximity-effect loss of the planar litz-wire coil in inductive power transfer, which is simplified as $\begin{eqnarray}\displaystyle P_{e}=\frac{{\pi}\ell _{t}Nd_{c}^{4}{\omega}^{2}}{128{\rho}_{c}}(\langle B_{\text{internal}}^{2}\rangle +\langle B_{\text{external}}^{2}\rangle ) & & \displaystyle\end{eqnarray}$ (21) where the B _internal and B _external are the fields generated by the winding itself and all the other turns, respectively. 𝜔 is the radial frequency of the inverter’s frequency.

In summary, the aforementioned methods most based on ideal litz-wire geometry structure neglect the effect of twisting characteristics on the litz-wire losses.

3.2. Considering the twisting structure

3.2.1. Based on coordinate transformation

To analyze the effect of twisting structure on the losses of litz wires. An analytical model considering bundle-level twisting structure is proposed in [35]. The skin loss and proximity loss are divided into strand level and bundle level, respectively. The skin effect factor for each successive level can be approximated by using the average conductivity and diameter of the bundle. The total skin loss can be calculated by multiplying the skin-effect factors at each level. The total skin loss is given by $\begin{eqnarray}\displaystyle P_{skin}=R_{dc}I_{\text{rms}}^{2}F_{0}F_{1}F_{2}\ldots & & \displaystyle\end{eqnarray}$ (22) where R _dc and F are the DC resistance and the skin effect factor at each level, respectively and I _rms is the current in the wires. When the bundle-level proximity effect is considered, this can be analyzed by using local coordinates that twist with the bundles. In these coordinates, a uniform field is converted into a twisted field rather than the bundle is twisted as shown in Fig. 6, and the expression of the proximity loss can be obtained by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{proximity}=\frac{G}{l}\left(\int _{0}^{l}(\cos (kz)H_{x}(z)+\sin (kz)H_{y}(z))dz\right)^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \qquad \hspace{5.0pt}+\frac{G}{l}\left(\int _{0}^{l}(\cos (kz)H_{y}(z)+\sin (kz)H_{x}(z))dz\right)^{2}\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle G=\frac{{\pi}d_{i}^{4}w^{2}{\mu}_{0}^{2}}{128{\rho}_{\text{eff},i}}\end{eqnarray}$ (24) where H _x and H _y are the x and y-direction components of H _ext, respectively. G is the proximity effect factor at each level, l is the length of the litz wire, k = 2𝜋∕r denotes the length of pitch, and d _i is the diameter of a bundle or strand at twisting step i. 𝜌_{eff, i} is the effective resistivity, 𝜇₀ is the vacuum permeability, and w is the radian frequency of a sinusoidal current. This coordinate transformation is not only used to calculate the proximity loss of a single litz-wire conductor but also considered to apply in air-core helical litz-wire windings [36].

Fig. 6.

Illustration of using a coordinate rotation analyzes the effect of a field on a twisted bundle.

Fig. 7.

Magnetic field distribution in litz wire generated as a result of twisting.

3.2.2. Based on inclination angle 𝜃

The effect of the twisting structure on the proximity loss is also considered in [37]. A 3-D calculation model that considers the effect of the inclination angle 𝜃 of the bundle on the litz-wire losses is proposed. In addition, the expressions for the external and internal magnetic fields related to inclination angle 𝜃 are derived. Among them, the internal magnetic field H _int can be expressed as the sum of the magnetic field H _int ⊥ perpendicular to the litz wire and the magnetic field H _int∥ in parallel to the litz wire shown in Fig. 7. Based on the [37 ], some corresponding additions have been made in [38 ] by the same research team. Firstly, the packing factor, length ratio of the 1st level bundle to the litz wire, and the radius of the 1st level bundle are specified. Then, the extraction of the external magnetic field is also briefly explained. Finally, the copper loss of litz wires can be given by expression (25). Besides, more experimental data for evaluating the proposed model is added. $\begin{eqnarray}\displaystyle P_{L}=R_{L}I_{L}^{2}+G_{L}H_{L}^{2} & & \displaystyle\end{eqnarray}$ (25) where I _L and H _L are the AC flowing through the litz wire and the RMS value of the external magnetic field, respectively. R _L and G _L are the copper loss coefficients that can be written by $\begin{eqnarray}\displaystyle R_{L} & = & \displaystyle \frac{m{\rho}}{{\pi}{\alpha}_{s}^{2}n_{\text{total}}}F({\gamma}_{s})F({\gamma}_{b})+\frac{4{\pi}{\rho}n_{\text{total}}K({\gamma}_{s})}{8{\pi}^{2}{\alpha}_{L}^{2}}\left(\frac{4m^{3}}{3}-\frac{13m}{6}+\frac{11}{6m}\right)\end{eqnarray}$ (26) $\begin{eqnarray}\displaystyle G_{L} & = & \displaystyle 4{\pi}{\rho}n_{\text{total}}K({\gamma}_{s})\left(\frac{3m}{4}+\frac{1}{4m}\right)\end{eqnarray}$ (27) where 𝛾_s is defined in (28), 𝛾_b is defined in (29) and n _total is the total number of strands. 𝛼_s, 𝛼_b, and 𝛼_L represent the strand radius, the radius of the 1st level bundle, and the radius of the litz wire. m = 1∕cos𝜃 is the length ratio of the strand to the litz wire. F and K are functions of the real number x defined as (30) and (31), respectively. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\gamma}_{s}\equiv {\alpha}_{s}\sqrt{\frac{{\omega}{\mu}_{0}}{{\rho}}}\end{eqnarray}$ (28) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\gamma}_{b}={\alpha}_{b}\sqrt{\frac{{\omega}{\mu}_{0}}{{\rho}_{\text{eff}}}}\end{eqnarray}$ (29) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F(x)\equiv \frac{x}{2}\frac{berxbei^{\prime }x-ber^{\prime }beix}{(ber^{\prime }x)^{2}+(bei^{\prime }x)^{2}}\end{eqnarray}$ (30) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K(x)\equiv -x\frac{ber_{2}xber^{\prime }x+bei_{2}xbei^{\prime }x}{(berx)^{2}+(beix)^{2}}\end{eqnarray}$ (31) where 𝜔 is the angular frequency, and 𝜇₀ is the permeability of the air. ber and bei are the special functions related to the Bessel function [80 ]. 𝜌 is the resistivity of the copper and 𝜌_eff is the effective resistivity of the bundle-level skin effect. Furthermore, this research team also proposed a calculation method for the rectangular cross-section litz wires [39 ]. A factor M is introduced to modify the Eqs (26) and (27), which is dependent on the cross-section geometry and is defined as the ratio of $H_{\text{int}}^{2}$ averaged values for the litz wire with a rectangular cross-section to that with a circular cross-section of the same area. It can be expressed as $\begin{eqnarray}\displaystyle M=\frac{\displaystyle \int _{S_{\text{rect}}}H_{\text{int}\_\text{rect}}^{2}\mathit{dS}_{\text{rect}}}{\displaystyle \int _{S_{\text{circ}}}H_{\text{int}\_\text{circ}}^{2}\mathit{dS}_{\text{circ}}} & & \displaystyle\end{eqnarray}$ (32) where S _rect and S _circ are the cross-section region of the rectangular and circular cross-section litz wire, respectively; H _{int_rect} and H _{int_circ} are the root-mean-square internal magnetic field of the rectangular and circular cross-section litz wire, respectively. In [37–39] the two-level twisting bundle structure of rectangular or circular litz wire is considered and the expressions for the inclination angle 𝜃 between the bundle level and the litz wire are derived.

3.2.3. Based on the probabilistic model

In an ideal litz wire, each strand changes its position with each other in a way that the entire current is equally distributed across all of the strands, so the imperfect twisting strands in the litz wires can cause additional losses. Moreover, due to the inhomogeneous current distribution, the connectors soldered by both ends of the litz wires can also result in additional losses. In [40], an analytical calculation method of the frequency-dependent litz wires resistance considering the wire connectors is presented. To achieve non-ideal twisting strands, authors used probability theory to describe the radial position of the strands and according to the probabilistic positions of strands provided the skin effect factor of the connector. The skin effect factor can be expressed as $\begin{eqnarray}\displaystyle F_{\text{con}} & = & \displaystyle \frac{\text{Var}(J_{R_{a}+R_{b}})+E[J_{R_{a}+R_{b}}]^{2}}{E[J_{R_{a}+R_{b}}]^{2}}\nonumber\\ \displaystyle & = & \displaystyle \int _{-\infty }^{\infty }\left|\frac{J_{z,R_{a}+R_{b}}(r)}{E[J_{R_{a}+R_{b}}]}-1\right|^{2}\cdot f_{R_{a}+R_{b}}(r)dr+1\end{eqnarray}$ (33) which represents the resistance increase due to the unequal current distribution among the strands. Where f _{R
_a+R
_b}(r) is the probability density function of the sum of random variables R _a and R _b, and J _{R
_a+R
_b} represents current density depending on both connector positions. This probabilistic method to determine the strand position is also mentioned in [41]. The research objective of this literature is wireless power transmission coils whose influence of external fields is not taken into account. So this article mainly focuses on loss calculation of bundle-level skin effect.

3.2.4. Considering bunched litz-wire model

The losses calculation model of a bunched litz wire which is a common type in practice is proposed [42]. This model based on analytical equations not only considers the twisting structure but also analyzes the impaction of the direction of pitch on litz-wire AC resistance. The analytical equations are written as $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}cc@{}}(\mathbf{R}+j{\omega}\mathbf{L}) & \mathbf{Q}\\ \mathbf{Q}^{T} & 0\end{array}\right]\left[\begin{array}{@{}c@{}}\hat{\mathbf{I}}\\ {\varphi}\end{array}\right]=\left[\begin{array}{@{}c@{}}-j{\omega}{\phi}\\ \hat{\mathbf{I}}_{\boldsymbol{ S}}\end{array}\right]. & & \displaystyle\end{eqnarray}$ (34)

The matrices $\mathbf{R}\in \mathbb{R}^{n\times n}$ , $\mathbf{L}\in \mathbb{R}^{n\times n}$ , and vector ${\phi}\in \mathbb{R}^{n}$ can be calculated using $\begin{eqnarray}\displaystyle R_{k,l} & = & \displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle \frac{l_{z}}{{\sigma}A_{k}},\quad k=l\quad \\[12.0pt] 0,\quad k\neq l\quad \end{array}\right.\end{eqnarray}$ (35) $\begin{eqnarray}\displaystyle L_{k,l} & {\approx} & \displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle -\frac{{\mu}_{0}l_{z}}{8{\pi}},\quad k=l\quad \\[12.0pt] \displaystyle -\frac{{\mu}_{0}l_{z}}{4{\pi}}\left(\ln \left(\frac{|\mathbf{r}_{k}-\mathbf{r}_{l}|}{r_{c}^{2}}\right)+1\right),\quad k=l\quad \end{array}\right.\end{eqnarray}$ (36) $\begin{eqnarray}\displaystyle {\phi}_{k} & {\approx} & \displaystyle l_{z}A_{z,e}(\mathbf{r}_{k})\end{eqnarray}$ (37) where 𝜔 is the electrical angular frequency, and Q is the reduced incidence matrix. $\hat{\mathbf{I}}$ is the vector with the unknown current in the strands, and 𝜑 is the vector with the unknown potentials. $\hat{\mathbf{I}}_{\mathbf{S}}$ is the vector with the source currents, and l _z is the length in the z-direction. r _k is the position vector and A _{z, e} is the external vector potential. 𝜎 is the conductivity and 𝜇₀ is the permeability in vacuum. Although this model focuses on the losses of realistic litz wires, the measurement only using a simple structure litz wire (35 × 5 strands) validates the result of the calculation and the experimental content is not enough.

3.3. Optimization of the litz-wire losses

The aforementioned analytical methods focus on how to accurately evaluate the litz-wire losses but do not mention how to optimize the litz-wire losses. Therefore, an analytical model based on one-dimensional Dowell’s equation considering the optimization of litz-wire strand diameter at the conditions of sinusoidal and multi-harmonic currents is proposed [43]. Not only the AC-to-DC windings resistance ratios F _R of litz wire at low, medium, and high frequencies are specified but also the expressions for the diameter of a single strand that corresponds to the local minimum losses for sinusoidal and multi-harmonic currents conditions are derived. The optimum strand diameter at the local minimum of the litz-wire winding losses for sinusoidal current is calculated as $\begin{eqnarray}\displaystyle d_{\text{str}\text{-}\text{opt}}=\sqrt[6]{\frac{1440{\delta}_{{\omega}}^{4}p^{2}}{{\pi}^{3}(5N_{ll}^{2}-1)}}. & & \displaystyle\end{eqnarray}$ (38) The optimum strand diameter at the local minimum of litz-wire winding losses for multi-harmonic current is calculated as $\begin{eqnarray}\displaystyle d_{\text{strn}\text{-}\text{opt}}=\sqrt[6]{\frac{5760{\delta}_{{\omega}}^{4}p^{2}}{{\pi}^{3}(5N_{ll}^{2}-1)\displaystyle \mathop{\sum }_{n=1}^{\infty }n^{2}}} & & \displaystyle\end{eqnarray}$ (39) where 𝛿_𝜔 is the skin depth, and N _ll is the number of winding sub-layers. p describes the distance between the center of the strand, and n represents the number of harmonics. This analytical model has been used in the optimization and design of high-frequency transformers [81].

In [44], the analytical method based on coordinate transformation is also used to design wireless power transmission systems in electric vehicles. The optimal diameter parameters of strands or bundles in litz wire are derived. $\begin{eqnarray}\displaystyle d_{i}=\sqrt{\frac{n_{i}d_{i-1}^{2}}{K_{i}}\left(1+\frac{{\pi}^{2}n_{i}d_{i-1}^{2}}{4K_{i}p_{i}}\right)} & & \displaystyle\end{eqnarray}$ (40) where n is the number of strands or bundles, K is the packing factor, d is the diameter of the strand or bundle and p is the pitch. When i = 1 represents the bundle level and i = 0 is the strand. It is worth mentioning that the overall losses can be reduced with the aid of replacing copper litz wires with aluminum and the aluminum wire offers better performance than copper in some conditions. The aluminum litz wire coil-wrapped magnetic tape is also used to reduce losses and improve transmission efficiency in the same field [45].

In summary, the analytical methods are more suitable for application in the optimization design procedure of high-frequency transformers or inductors. However, they are mostly based on the ideal structure of litz wires or subject to different degrees of simplification. So they can achieve better calculation accuracy only under certain conditions. To better evaluate the losses of realistic litz wires, another calculation method, the numerical method is introduced in the next section.

4. Numerical methods

Apart from these analytical methods, some numerical methods are often used to solve the litz-wire loss problems. Numerical solutions to electromagnetic problems started in the mid-1960s with the availability of modern high-speed digital computers. The numerical approach has the advantage of allowing the actual work to be carried out by operators without knowledge of higher mathematics or physics, with a resulting economy of labor on the part of the highly trained personnel [82]. So it is widely used in various fields, also including the calculation of litz-wire losses.

4.1. Finite element method (FEM)

The FEM originated in the field of structural analysis and this method was not applied to solve electromagnetic problems until 1968. Since then, the FEM has been gradually applied in various fields, such as electric motors, machinery, and electromagnetic radiation. Because of the improvement in accuracy and implementation of more and more physical effects, the simulation tools of FEM have been commonly accepted. Some commercial software like ANSYS, COMSOL, or ABAQUS is widely used in the industry field and scientific research field.

4.1.1. Full 3-D FEM

At present, many researchers have used the FEM to calculate the litz-wire losses. In [46], authors used a supercomputer to analyze the losses of three kinds of coils made by parallel wire, bunch stranded wire, and rope lay wire, respectively. All coils are composed of 16 strands and 8 turns, as shown in Fig. 8. The losses of the three kinds of coils are simulated by 3-D FEM, and the losses of rope lay coil are smaller than wound coil and bunch stranded coil in the range of frequency between 100 kHz and 1 MHz. The calculation time, number of elements, and nodes of the FEM simulation for three different ropes are recorded in Table 1.

Fig. 8.

Three kinds of coils: (a) parallel wire, (b) bunch stranded wire, (c) rope lay wire.

Table 1

Discretization data and calculation time

Model		Wound	Bunch stranded	Rope lay
Number of elements		95,599,939	96,615,183	98,519,695
Number of nodes		16,567,571	16,204,298	17,092,821
	100 kHz	34.50	24.30	26.06
Calculation time (min.)	600 kHz	67.91	36.50	51.11
	1 MHz	87.64	43.11	88.48

With the help of a supercomputer, such a simple structure of stranded wire takes dozens of minutes. For more complex bundles and twisting structures of realistic litz wires, the time consumption of calculation losses may increase exponentially, which will lead to slower simulations and larger memory requirements. Therefore, more and more researchers are interested in how to reduce the 3-D FEM computational time and costs.

4.1.2. Simplified 3-D FEM

In [47], a set of the detailedly simplified procedure of the litz-wire structure model based on commercial FEM soft Ansoft Maxwell 16.0 is presented. The details of the simplification are shown in Fig. 9. Firstly, the circular cross-section of the conductor is substituted by a polygon with 20 sides. The continuous winded flat litz-wire coil is simplified to a concentric circle structure placed in the same plane. Then, the effect of the connector soldering both endings of a litz wire is considered. The result of the FEM simulation shows that the current distribution in the litz wire caused by the connectors is analogous to the behavior of AC in a solid wire. Moreover, the twisting structure of litz wires is considered in the 2-D simulation. Finally, two simulation schemes are proposed based on whether the 2-D twisting characteristic is considered. Although this method already has a better-simplified procedure of FEM simulation and better accuracy, the computational costs are still very high.

Fig. 9.

Simplified procedure of litz-wire FEM simulation.

4.1.3. Homogenization-based 2-D FEM

To further reduce the costs of calculation, another method based on homogenization 2-D FEM is proposed. It can accelerate the calculation performance of FEM because it avoids discretization of the components in fine-structured material into small finite elements [83]. In [48,49], Nan et al. derived the complex permeability of packing patterns of rectangular and hexagonal. And a comparison of the high-frequency proximity-effect loss factor in hexagonally packed wires with different rectangularly packed wires. To calculate the complex permeability of the litz-wire windings, the homogenization of litz-wire bundles and the spaces between bundles are discussed. The homogenization modeling procedure for litz-wire is shown in Fig. 10.

Fig. 10.

The homogenization modeling procedure of litz wire.

The homogenization method is also used in [50]. Meeker argued that the impact of the skin effect is neglected in reference [48]. Expressions for effective conductivity (related to the skin effect) and effective permeability (related to the proximity effect) are derived. The results of this work are more consistent with finite-element results than Nan’s. Igarashi further proposed a semi-analytical approach for FEM of multi-turn considering skin and proximity effect [51]. Different from the aforementioned two kinds of homogenization methods, in Igarashi’s method, the complex permeability 𝜇 of a round wire is expressed in a closed form. Then the cross-section of a multi-turn coil is evaluated using the Ollendorff formula [84]. The Ollendorff formula is given by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \langle {\mu}_{r}\rangle =1+\frac{{\eta}({\mu}_{r}-1)}{1+N(1-{\eta})({\mu}_{r}-1)}\end{eqnarray}$ (41) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\mu}_{r}={\mu}/{\mu}_{0}\end{eqnarray}$ (42) where 𝜇 is permeability, 𝜇₀ is the permeability of the vacuum, and 𝜂 is the fill factor. N denotes the diamagnetic constant. The homogenization complex permeability of the multi-turn round coil is written as $\begin{eqnarray}\displaystyle \langle \dot{{\mu}_{r}}\rangle =1+\frac{2{\eta}\left({\mu}_{r}\displaystyle \frac{J_{1}(z)}{zJ_{1}^{\prime }(z)}-1\right)}{2+(1-{\eta})\left({\mu}_{r}\displaystyle \frac{J_{1}(z)}{zJ_{1}^{\prime }(z)}-1\right)} & & \displaystyle\end{eqnarray}$ (43) where J _n is the nth order Bessel function. The P _prox can be calculated by homogenization complex permeability and P _skin can be evaluated by introducing the AC resistance.

As the studies in [51], the Ollendorff formula is chosen to fit the proximity-effect reference solutions from the 2-D FE problems [52]. A corrected complex fill factor $\boldsymbol{\eta}$ is introduced in the closed expression (43) to better match the results of the 2-D FE model. The expression of $\boldsymbol{\eta}$ is given as $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle \eta={\lambda}(1+\boldsymbol{{\Theta}}({\lambda})\boldsymbol{{\Gamma}}(z)) \end{eqnarray}$ (44) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{{\Theta}}({\lambda})=\sum _{0}^{n_{{\Theta}}}\boldsymbol{h}_{j}{\lambda}^{j}\end{eqnarray}$ (45) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{{\Gamma}}(z)=\sum _{0}^{n_{{\Gamma}}}\boldsymbol{g}_{j}z^{j}\end{eqnarray}$ (46) Expressions (45) and (46) are complex polynomials with order n _𝛩 and n _𝛤, respectively. h _j and g _j are coefficients to fit, and 𝜆 is the fill factor. Then, the results of (43) corrected by complex fill factor (44) are compared with both Meeker’s [50] and Igarashi’s [51] approximations. With 𝜆 = 0.7, the error is only 0.42%, which is lower than Meeker’s and Igarashi’s. The mesh details of the litz-wire coil are shown in Fig. 11.

Fig. 11.

The mesh details of the litz-wire cross section.

4.1.4. Considering twisting structure homogenization 2-D FEM

Although homogenization of 2-D FEM further reduces the computational costs compared with 3-D FEM, the twisting structure of real litz wire is not considered or only treated as perfectly twisted strands. Some researchers are aware of the problems mentioned above. Therefore, based on [51], a new method is introduced to evaluate the eddy current of litz wires using homogenization-based FEM in conjunction with integral equations [53]. Because the stranded wires have a periodic structure in the direction of the wire axis, the magnetization satisfies $\begin{eqnarray}\displaystyle \boldsymbol{M}(x+n\boldsymbol{d})=\boldsymbol{M}(x) & & \displaystyle\end{eqnarray}$ (47) where M is the complex magnetization vector, n denotes the integer and d is the distance vector between the adjacent unit cells. The integral equations for the wire with periodic structure can be expressed as $\begin{eqnarray}\displaystyle \frac{\boldsymbol{M}(x)}{\dot{{\mu}_{r}}-1}={\tau}\times \left[\left(\mathop{\sum }_{n\in \mathbb{Z}}\int _{{\Omega}_{\text{unit}}}{\mathcal{G}}\boldsymbol{M}_{n}dv^{\prime }+\boldsymbol{H}_{0}(x)\right)\times {\tau}\right]. & & \displaystyle\end{eqnarray}$ (48)

The macroscopic permeability considering the twisting structure is calculated by the following equation: $\begin{eqnarray}\displaystyle \langle \dot{{\mu}_{r}}\rangle =\frac{j{\omega}\displaystyle \int _{{\Omega}_{\text{unit}}}{\displaystyle \frac{|\boldsymbol{B}_{\text{ave}}|^{2}}{{\mu}}}dv}{j{\omega}\displaystyle \int _{{\Omega}_{\text{unit}}}{\displaystyle \frac{|\boldsymbol{B}_{\text{ave}}|^{2}}{{\mu}}}dv+j{\omega}\int _{{\Omega}_{\text{unit}}}{\sigma}|\boldsymbol{E}|^{2}dv}. & & \displaystyle\end{eqnarray}$ (49) Because of the limitation of space, the detailed meaning of the parameters in the formula (48) and (49) can be found in [53]. Furthermore, it is shown that the litz-wire twist has little effect on the complex permeability. However, another factor (circulation current) that can cause additional AC losses is ignored in this paper. In practice, the ideal twisting litz wires are not always achieved and the magnetic field can vary along the wire. Because strands in litz wires are shorted at both terminals of them, the circulation current can flow between the strands and increases the AC losses in wires shown in Fig. 12.

Fig. 12.

The circulation current flows between the strands.

In [54], not only the proximity loss is calculated by homogenization-based FEM using the complex permeability, but also the losses caused by the circulating currents are evaluated by solving the corresponding circuit equation $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathop{\sum }_{j}A_{j}\int _{{\Omega}}(\text{rot}∼\boldsymbol{N}_{i})^{\text{t}}\text{T}^{\text{t}}\boldsymbol{\nu}\text{T}(\text{rot}∼\boldsymbol{N}_{j})d{\Omega}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -\mathop{\sum }_{k}I_{k}\int _{{\Omega}\text{coil}}\boldsymbol{N}_{i}\cdot \boldsymbol{j}_{k}d{\Omega}=0\end{eqnarray}$ (50) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{R_{k}zJ_{0}(z)}{2J_{1}(z)}I_{k}+jw\mathop{\sum }_{j}A_{j}\int _{{\Omega}\text{coil}}\boldsymbol{N}_{j}\cdot \boldsymbol{j}_{k}d{\Omega}=V_{k}\end{eqnarray}$ (51) where N _i and j _k are the vector interpolation function and unit current density j _k = J _k∕I _k, respectively. I _k is the circuit current including the circulating currents, R _k denotes the DC resistance V _k is the input voltage and T^t𝛎T is the reluctance tensor. This method concludes that the inter-strand and inter-bundle circulating current losses are sufficiently smaller than the proximity loss when the bundles and strands are twisted with a pitch smaller than the spatial variation of the magnetic field along the litz wire.

In summary, the method based on homogenization 2-D FEM can significantly save computation costs and take into account computational accuracy, but the twisting structure, length of the pitch, and the bundles’ position of non-ideal litz wire are still rarely discussed. So some researchers use another numerical method named the partial-element equivalent circuit (PEEC) method.

4.2. Partial-element equivalent circuit (PEEC)

In 1972, A. Ruehli proposed the PEEC method based on the integral equation, which was initially applied to the calculation of partial inductance between the complex three-dimensional conductors [85]. The PEEC method initially had strict requirements for the structure of a system that must contain only metal conductors. In this method, the impacts of various electromagnetic effects on the calculation are not considered under high frequency so its applications are limited. Since then, many efforts have been made to improve the PEEC by experts and scholars in various fields [86–92]. The specific mathematical background and theoretical content of the PEEC method can be found in [93]. At present, the scope of the method includes the calculation of all electromagnetic parameters in the time and frequency domain.

To calculate the litz-wire losses, the FEM is sometimes used, but the particular geometry of such kinds of wires, which includes very thin and twisting conductors, makes the meshing become a complicated task [94]. In contrast to FEM, the PEEC method based on the integral equation that exploits the background of Green’s function avoids air discretization. The electromagnetic field problem in the simulation domain is transformed into an equivalent LRC circuit problem [95,96]. A simple electric structure of two connected conductors can be transformed into an equivalent electrical circuit, as shown in Fig. 13. Moreover, in elements of the PEEC method, only circuit variables are stored in memory. The CPU time for the calculation of power loss achieves a significant reduction. Therefore, this method is applied to calculate the power loss of the complex structure of litz wires by some researchers. It can also simulate the 3-D non-ideal structure of litz wires.

Fig. 13.

An equivalent electrical circuit of two connected conductors.

Fig. 14.

Interface of the numerical simulation tool Fastlitz [55].

4.2.1. Fast numerical simulation tool “Fastlizt”

In [55,56], a specialized computational procedure characterizing the losses of realistic litz-wire structure is presented by Richard Y. Zhang with a fast numerical simulation tool named Fastlitz as shown in Fig. 14. The author extends this approach by simulating two loss mechanisms (losses driven by the net current flow and the external magnetic field). The frequency-dependent resistance of the conductor is evaluated and the PEEC model is implemented in MATLAB, and the impedance matrix [Z] can be obtained by solving the core matrix equation $\begin{eqnarray}\displaystyle [M][Z][M]^{T}I_{m}=V_{m} & & \displaystyle\end{eqnarray}$ (52) where $[M]\in \mathbb{R}^{n\times l}$ is the mesh-analysis matrix, which maps n branches to l closed loops of current with order l nonzero entries. V _m and I _m are known as the mesh loop voltage and current vectors respectively. Furthermore, the effect of the twisting construction at each level is discussed. The author summarizes the following points:

An integer number of twists minimizes proximity loss. Avoid using less than 1 twisted wire.

To minimize skin effect factor F in the two-level structure, the inter-level should have integral twists relative to the outer level.

To minimize proximity effect factor G, both levels should have integral twists relative to the global axis.

Zhang’s calculation method provides a reference for other researchers’ subsequent studies. However, some researchers argue that Zhang’s model neglects the impacts of the litz-wires shorted terminal and the circulation current between the strands cannot be calculated. Although the efficiency has been greatly improved, this method still has a significant computational burden when a large number of strands are considered.

4.2.2. Fast 2.5-D PEEC method

To overcome the aforementioned shortcomings, Thomas Guillod et al. further proposed a fast 2.5-D PEEC method to consider the twisting structure of the non-ideal litz wires [57]. The author separated the calculation into two parts: packing and twisting. This fast 2.5-D PEEC method neglects the current flowing in azimuthal and radial directions. In this way, it can simulate a litz wire with thousands of strands. Moreover, this method mainly focuses on analyzing the implications of imperfect twisting on the current distribution. Two kinds of twisting schemes in most litz-wire structures are explained:

Perfect twisting: also called ideal twisting has been mentioned in Section 1., which is difficult to achieve with more than five strands.

Bunched wires: also known as imperfect twisting, refer to strands that are twisted only in an azimuthal direction.

The impacts on imperfect twisting in litz-wire three levels (top-level, mid-level, and strand-level) are analyzed. The results show that for twisting imperfections at the top level, the loss increase is more than 100%; For imperfect twisting at the mid-level, the increase is much smaller (less than 40%); The impacts on loss can be ignored at the strand level. For the number of pitches, similar conclusions are obtained in [55]. The workflow for the computation of losses in HF litz wires is shown in Fig. 15.

Fig. 15.

The work flow for the computation of losses in HF litz wire [57].

4.2.3. Multilevel PEEC method

In both Fastlizt and 2.5-D PEEC methods, a rectangular cell is adopted for the meshing, but the cross-sections of litz wires are commonly circular, which could result in serious errors in loss calculation. In [58], a multilevel PEEC method that significantly improves the efficiency of the calculation of the litz wires is proposed. The mesh is also generated into two types: packing and twisting. The packing level uses an adaptive circular meshing scheme on the cross-section and focuses on the non-uniform meshing in both radial and azimuthal directions, which can greatly improve calculation accuracy. After meshing, the divided cells of the resistance and the inductance are calculated and the cross section of the litz wire is divided into N _t cells. The impedance of each strand in the packing level can be written by $\begin{eqnarray}\displaystyle \boldsymbol{Z}_{p} & = & \displaystyle (\boldsymbol{A}^{T}\boldsymbol{Z}_{c}^{-1}\boldsymbol{A})^{-1}\end{eqnarray}$ (53) $\begin{eqnarray}\displaystyle \boldsymbol{Z}_{c} & = & \displaystyle \left[\begin{array}{@{}cccc@{}}R_{1}+sL_{1} & sM_{12} & \cdots ∼ & sM_{1N_{t}}\\ sM_{12}^{T} & R_{2}+sL_{2} & \cdots ∼ & sM_{2N_{t}}\\ \vdots & \vdots & \vdots & \vdots \\ sM_{1N_{t}}^{T} & sM_{2N_{t}}^{T} & \cdots ∼ & R_{N_{t}+}sL_{N_{t}}\end{array}\right]\end{eqnarray}$ (54) where Z _p represents an admittance matrix of one segment in cross-section, A is a selection matrix and Z _c is the N _t × N _t impedance matrix. R _n denotes the self-resistance of cell n, L _n is the self-inductance and M _mn is the mutual inductance between cell m and cell n. For the twisting level, it is divided into wire level and multi-pole level which is further divided into far and very far fields. A simplified fast multipole method (FMM) is used to speed up the calculation of the far field. Finally, compared with the simulation tool Fastlitz proves that this approach can achieve higher accuracy and efficiency. Furthermore, the same research team used a similar method to calculate the AC resistance of the uninsulated twisted wire. Then the performances of litz wires and uninsulated twisted wires are compared and discussed under various conditions [59]. The author described the prospects of uninsulated twisted wire that can reduce the costs in application areas. For two-level structure wires, common litz wires can be replaced by uninsulated twisted wires when the ratio of bundle radius to skin depth is less than a specific value. From the perspective of estimation accuracy and economic cost, the scheme of using uninsulated twisted wires to replace common litz wires is an application prospect.

Fig. 16.

The schematic work flow of the SEEC method [66].

4.3. Strand element equivalent circuit (SEEC)

A fast numerical method (SEEC: strand element equivalent circuit) analogous to PEEC is introduced to calculate the power loss for high-frequency litz wires [66]. From the author’s point of view, this method is most similar to [57], which is based on the numerical elements of [97]. In this method, each single strand is divided into several numbers of strand elements, and the vector potential for each numerical strand element is derived. The system of equations can be written as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{}\underline{\mathbf{M}}\cdot \text{}\underline{\mathbf{I}}=\text{}\underline{\mathbf{b}}\quad \text{and}\quad \text{}\underline{\mathbf{I}}=[\text{}\underline{i}_{1},\text{}\underline{i}_{2},\ldots ,\text{}\underline{i}_{m},\ldots \text{}\underline{i}_{N}]^{\text{T}}\end{eqnarray}$ (55) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{}\underline{\mathbf{b}}=R_{dc}\mathbf{1}\text{}\underline{i}-\text{}\underline{\mathbf{V }}_{\text{ex}}+\mathbf{1}\text{}\underline{v}_{\text{ex}}\end{eqnarray}$ (56) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{}\underline{v}_{\text{ex}}=\frac{1}{N}\sum _{n=1}^{N}\text{}\underline{v}_{n,\text{ex}}\end{eqnarray}$ (57) where I is the vector of strand currents and M is a quadratic matrix of any terms linked with the strand currents. The voltage v _n is dependent on the magnetic vector potential, $\text{}\underline{\mathbf{V}}$ is the vector of induced voltages in the strands, and 1 is a unit vector of order N. The complex twisting structure is also considered by using vector potential to introduce an equivalent circuit. Then, the internal field and external field are expressed by vector potential to solve the currents flowing through each strand. The schematic workflow of the SEEC method is shown in Fig. 16. Realistic litz-wire skin effect factor and proximity effect factor are explicitly compared with ideal litz wires at the condition of different twisting directions, the position of bundles, and the lengths of the pitch. The results show that pitch lengths and pitch directions have a minor effect on skin loss. when the twisting bundles are more than five and a sub-bundle stays in the center of the bundle level, the skin loss is significantly increased. For the proximity effect, a change in the twisting direction does not cause excessive proximity loss, as long as the absolute pitch lengths of sub-bundles are not too long. The calculation results of the SEEC method are close to the measured values and have lower computational costs compared with the PEEC method.

Compared with the FEM, the PEEC and SEEC methods not only take the details of the complex twisting structure of realistic litz wire into account but also reduce the computational time and costs. In summary, both the PEEC method and SEEC method can provide an efficient alternative approach for solving the power loss of high-frequency litz wires.

5. Coupling methods

In many practical engineering problems, multiple physical domains are involved, so using only a single computational method may not achieve the desired results. In this case, combining the advantages of multiple approaches to analyze these problems is necessary, such as numerical and analytical methods. In the domain of electricity, the first approaches combining different numerical and analytical methods have been investigated to solve multi-scale problems at the end of the last century [98]. At present, there are two coupling methods proposed for litz-wire loss calculation: coupling the numerical method with the analytical method or with the PEEC method.

5.1. Coupling analytical and numerical methods

In reference [67], a coupling process of numerical simulation and analytical method is introduced, which enables efficient calculation of the full 3-D structure of litz wires. The solving procedure can be split into two parts:

The magnetic field distribution of the 3-D system (transformer, coils, etc.) is simulated by numerical method and the litz wire is assumed to be a solid wire.

Analytical calculation of the skin and proximity losses is based on the 2-D litz-wire structure.

For the 3-D numerical simulation part, the detailed procedure of extracting external magnetic distribution is shown in Fig. 17. This circular cross-section is simplified and sketched with four points P _i on the surface. The total magnetic field intensity H _tot is calculated by numerical simulation tools (ANSYS 14.5). Then, the external magnetic field H _ext of each cross-section of the conductors can be obtained by equation $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{H}_{\text{ext}}=\boldsymbol{H}_{\text{tot}}-\boldsymbol{H}(r)\end{eqnarray}$ (58) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{H}(r)=\frac{I}{2{\pi}r}\end{eqnarray}$ (59) where H (r) is the internal magnetic field of the cross-section of a conductor, and r is the radius of a circle on the cross-section. Then, an analytical method is utilized to calculate the overall losses (Eq. (60)) of one cross-section. The analytical method can achieve a very fast calculation and provide the possibility of the optimization of electrical devices, such as transformers and inductors. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{\text{loss}}=n\cdot P_{\text{skin}}+\frac{l}{{\kappa}}\sum _{i=1}^{n}(\boldsymbol{H}_{\text{ext}}+\boldsymbol{H}_{\text{int}})^{2}\cdot D_{s}\end{eqnarray}$ (60) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{H}_{\text{int}}(d)=d\cdot \frac{I}{2{\pi}r^{2}}\end{eqnarray}$ (61) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{\text{skin}}=I_{\text{RMS}}R_{0}\cdot F_{s}\end{eqnarray}$ (62) where D _S and n are the proximity effect factor and the number of strands, respectively. d refers to the distance of a strand to the middle of the conductor. r is the radius of a strand and 𝜅 is a specific conductance. R ₀ is the DC resistance of the litz wire. and I _RMS is the RMS value of the current flowing through the litz wire. The skin effect factor F _s can be obtained from analytical methods in Section 3. H _ext is the external magnetic field of the litz wire, which can be extracted according to the solution procedure as shown in Fig. 18.

Fig. 17.

The detailed procedure of extracting external magnetic fields.

Fig. 18.

The simulation procedure combines FEM with analytic or PEEC [68].

This kind of coupling method combining FEM with analytical formula has been applied in the optimization procedure of medium frequency transformer design [99]. While this coupling method offers higher accuracy than utilizing an analytical method alone, its reliance on running the FEM simulation for each calculation poses challenges when applied to a large number of iterative optimizations.

5.2. Coupling FEM and PEEC method

A novel approach for the simulation of litz wires systems is presented, which combines the specific benefits of the FEM with PEEC methods [68]. The advantages and disadvantages of the FEM and PEEC method is summarized as follows [100–102]:

(a) FEM

Advantages: Calculation of electromagnetic field distribution for materials with inhomogeneous conductivity or magnetic permeability can be achieved. It can be perfectly compatible with CAD software and development programs.

Disadvantages: In the case of complex structures, the meshing process (entire domain) and a huge number of elements can result in large linear systems of equations.

(b) FEEC Method

Advantages: The electromagnetic field is transformed into an equivalent LRC circuit problem. Time and frequency sweeps can be calculated efficiently to reduce complexity.

Fig. 19.

Summary of the correlation between the various methods of calculating litz-wire power loss.

Disadvantages: Only available for specialized domains focused on reducing the complexity of physical models, such as antennas, circuit boards, and litz wires. The computation of the field and current density distribution requires supplementary postprocessing steps.

Based on this approach, the magnetic field distribution of complex 3-D geometries in the entire system, involving a solid conductor, is simulated using standard FEM method tools. Additionally, the loss calculation for different bundle structures and pitch lengths of the litz-wire is performed using the PEEC method, as shown in Fig. 18. The external magnetic field H _ext is extracted using a method similar to the one described in reference [67]. This extracted field is then utilized as the boundary condition for the PEEC method, which is used to calculate the 3-D structure of realistic litz wires.

In conclusion, both coupling methods have their advantages. The analytical method is fast but may be limited by the perfect twisting structure. The PEEC method can take into account the realistic 3-D twisting characteristics of the common litz wires but has a high computational cost. The appropriate calculation methods can be chosen according to different applications and requirements.

Table 2

Summary of advantages and limitations of different methods

	Methods	Twisting	Advantages	Limitations
Analytical methods	Based on Dowell’s formula [27]	No	Very fast calculation speed; Very simple formulas;	Require to transform the litz wire to copper foil; Based on 1-D magnetic field model.
	Ferreira method [28,29]	No	Bessel functions are introduced to consider circular conductor models; Very fast calculation speed.	Based on 1-D magnetic field model.
	Bartoli method [30]	No	Based on the Ferreira method, the porosity is corrected and the calculation is very fast.	Based on 1-D magnetic field model.
	Tourkhani method [32]	No	Based on the Ferreira method, instead of using the average amplitude of H _e for each layer, the RMS value of H _e is utilized and the calculation is very fast.	Based on 1-D magnetic field model.
	Squared-field derivative method [33,34]	No	Capable of analyzing 2-D and 3-D magnetic field effects.	The circulating current between strands is neglected.
	Based on coordinate transformation method [35,36]	Yes	The effect of the litz-wire twisting structure on the external field can be considered and very fast calculation speed.	Lack of clarity in the characterization of the external magnetic field.
	Twisting inclination angle method [37–39]	Yes	The twisting inclination angle is introduced and it can consider rectangular and circular cross-section litz wire, and very fast calculation	Unclear characterization of the external magnetic field of litz wire.
	Based on probabilistic model [40,41]	Yes	Consideration of the inhomogeneous current distribution in litz wire, caused by the connectors.	The effect of the magnetic field around the litz wire on losses is not considered.
	Bunched litz-wire model [42]	Yes	Consider the circulating currents; Fast calculation speed;	Only focuses on bunch litz wire; Inadequate experimental verification

Numerical methods	Full 3-D FEM [46]	Yes	Very high computational accuracy and full characterization of the 3-D structure of the litz wire.	Extremely high computational time and cost.
	Simplified 3-D FEM [47]	Yes	The calculation speed is faster than the full 3-D FEM model.	Relatively high computational time and cost

Table 2 (Continued).

	Methods	Twisting	Advantages	Limitations
	Homogenization 2-D FEM [48–52]	No	The number of strands is not involved and refinement of the mesh is avoided, making the calculation faster than 3-D FEM.	The effect of the twisting structure of the litz wire on loss is neglected.
	Twisting structure homogenization 2-D FEM [53,54]	Yes	Integral equations are introduced in FE to calculate and the twisting structure is considered.	Integral equations are complicated and poorly generalized.
	Fast numerical simulation tool “Fastlitz” [55,56]	Yes	The 3-D model of the litz wire is built and calculation accuracy is comparable to 3-D FEM, and calculation speed is faster than 3-D FEM.	When the number of strands is high, the calculation time is longer; External magnetic fields need to be extracted using additional method.
	Fast 2.5-D PEEC method [57]	Yes	Consider the twisting structure of the realistic litz wires; Improve calculation efficiency than “Fastlitz”	External magnetic fields need to be extracted additionally.
	Multilevel PEEC method [58,59]	Yes	Use an adaptive circular meshing scheme and improve the accuracy of the FEEC method.	External magnetic fields need to be extracted additionally.
	Strand element equivalent circuit (PEEC) [66]	Yes	Consider the complex twisting structure of realistic litz wire and reduce the computational time and costs.	Extracting the external magnetic field is required by other methods (e.g. FEM)

Coupling methods	Coupling analytical and numerical methods [67,99]	Uncertain (based on analytical method)	Effectively characterize the magnetic field distribution around the litz wire windings and improve accuracy of analytical methods.	The accuracy of the calculation is limited by the selected analytical method.
	Coupling FEM and PEEC methods [68]	Yes	Effectively characterize the magnetic field distribution around the litz wire and improve the accuracy of PEEC methods.	3-D FEM is used to extract the external magnetic field, which lead to relatively high calculation cost.

6. Conclusion

This article presents an overview of power loss calculation for litz wires in recent years and comprehensively analyzes the correlation between different loss calculation methods as shown in Fig. 19. The loss calculation methods are divided into three main categories: analytical method, numerical method, and coupling method. First, several classic analytical methods are introduced and their commonalities and differences are briefly illustrated. Second, several numerical methods are described, according to the computational speed, degree of simplicity, and whether the realistic litz wires structure is considered. Third, two kinds of coupling methods are introduced and the detailed calculation procedure of each method is illustrated with emphasis placed on the extraction method of external magnetic fields. The advantages and limitations of these methods are systematically summarized in Table 2. Finally, the research challenges are summarized from the authors’ point of view. Based on these challenges, the suggested feasible solutions and prospects for the applications of the aforementioned calculation methods are listed in the following.

6.1. Challenges

Although many loss calculation methods for litz wires have been proposed, there are still some challenges to address, concerning their optimization and application, as explained in the following.

For analytical methods, these are more suitable for application in the optimization design of high-frequency transformers or inductors. However, based on the ideal structure of litz wires or different degrees of simplified models may lead to relatively serious calculation errors.

For numerical methods, the detailed structure for litz wires using 3-D FEM is very time-consuming due to their complex bundles and twisting structure. The 2-D FEM can save computational costs and ensures accuracy, but the twisting structure, length of the pitch, and the bundles of position in non-ideal litz wires are still rarely discussed.

The coupling method also provides an efficient way to solve the power loss calculation of high-frequency litz wires. However, the distribution of external fields is mainly extracted by FEM which requires large computational resources and is difficult to be applied in optimization procedures.

6.2. Suggested feasible solutions

Reasonable application of these loss calculation methods under different conditions is the key to facing the above-mentioned challenges.

The analytical methods are preferred to be considered when the optimization and design of transformers or inductor windings. Furthermore, the 2-D FEM based on homogenization is one of the options.

The PEEC method probably becomes the most promising one to solve these problems of realistic litz wires. Then, the SEEC method analogous to the PEEC method shows lower computational costs compared with the common PEEC method.

For coupling methods employed in the optimization procedure, the external magnetic field is extracted using the FEM only once at the beginning of the iterative calculation. Subsequently, the iterative calculation solely relies on the analytical method.

If only the losses of litz wires need to be calculated, without considering the structural optimization of the magnetic components or the litz-wire windings, the PEEC or SEEC method is still an acceptable choice.

6.3. Prospects

In certain specific cases, new types of litz wires, such as aluminum litz wires or uninsulated litz wires, can yield comparable effects to conventional litz wires while reducing the complexity of processing and manufacturing costs. Accurate calculation of losses associated with these new types of litz wires serves as the foundation for their widespread application, allowing for potential reductions in both the weight and price of power electronics equipment.

Footnotes

Acknowledgement

This work was supported in part by the National Natural Science Foundation of China under Grant 52077053 and Grant 52377008.

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