SWITCH mode power supplies (SMPS) are evolving to- wards achieving high switching frequency and high power density. Designing magnetic components in power converters has emerged as the paramount challenge impeding further advancements in both efficiency and power density [1]. The losses within magnetic components comprise both winding loss and core loss, which limit the further improvement of the efficiency of power converters.

Winding loss is predominantly influenced by the skin effect, proximity effect, and the impact of fringing flux. High-permeability cores employed in power inductors are typically designed with air-gap to mitigate magnetic saturation [2]. The presence of a gap in the center limb of a magnetic core can give rise to increased eddy current loss, which stem from the fringing effect [3]-[4]. Consequently, numerous investigations have been focused on the winding loss characteristic of gapped magnetic components.

Developing accurate calculation and simulation techniques for winding loss can offer a quantitative evaluation of the winding loss characteristics of gapped magnetic components, which are essential for the optimization of gapped magnetic components.

In [5], T. Ewald presented a theoretically derived computational method for determining the inductance and winding loss of gapped inductors employing both solid-round-wire and Litz-wire configurations, taking into account the influence of fringing flux stemming from the air-gap. However, this method may not be suitable for gapped foil inductors due to the shielding effect of copper foil [6]. The finite element method (FEM) is also a path to solve the winding loss of gapped magnetic components, but it may be impractical for the components using Litz-wire because the numerous strands occupy too much computational resource. Thus, the use of homogenization methods is imperative [7]-[8]. Different from the computational methods such as analytical, numerical, and semi-empirical approaches proposed in previous studies [9]-[11], a semi-analytical model for calculating the winding loss was introduced in [12]. The model combines a concise analytical method with high-precision numerical techniques, significantly enhancing the accuracy of traditional methods.

Currently, the majority of research efforts have been primarily focused on optimizing and theoretically calculating the winding loss of gapped magnetic components. Nevertheless, all of these schemes should be verified by the experimental measurement. So, developing the measurement techniques for winding loss evaluation in gapped magnetic components is crucial for both theoretical analysis and practical applications. In [13], a measurement method was introduced that aims to capture the winding loss of inductors under the large-signal test. This approach utilizes a reference transformer to offset reactive voltage and the core loss. The winding loss can also be measured by the calorimetric method [14], but it demands stringent requirements for the experimental environment and equipment.

As the relationship between the winding loss and current magnitude exhibits a linear characteristic, a prevalent method for quantifying winding losses is to measure the AC resistance with an impedance analyzer. When assessing the losses in gapped inductors, it is essential to take fully account the impact of the air-gap on the distribution of the magnetic field. Three common misconceptions about the measurement of AC resistance in gapped inductors were summarized in [15], all of which involve the influence of core loss on the measurement results. Hence, when measuring the winding loss of gapped inductors, special attention must be paid to separating the core loss from the measured losses.

The air-gap or other windings can lead to changes in the magnetic field and then cause resistance variations of the measured winding, and this phenomenon is often described as mutual resistance [15]-[17]. For transformers with multiple windings, the winding short-circuit method removes the core loss from the total losses. However, separating the winding loss from the core loss in single-winding inductors can be challenging. Consequently, indirect measurement methods have been developed, involving the construction of auxiliary windings to obtain the core loss of the inductor, which is then subtracted from the total losses [15]-[17]. Due to the influence of high quality factors, the indirect measurement methods can cause large errors in resistance measurement [18]. To solve these problems,[19] proposed a direct measurement scheme. By using a special fixture and employing a three-terminal measurement method, the quality factor can be greatly reduced so that the AC resistance of the winding can be measured directly. However,[17] proved that this method has stringent requirements on the location of the auxiliary winding. A construction method for an effective auxiliary winding was proposed in [17] but the principle of the method was not clearly explained. A method was proposed in [20] to characterize the magnetic field distribution within the core window of the inductor with an air-gap, using a gapless magnetic core and the winding for an equivalent air-gap (WEAG). It is noteworthy that the experimental frequency range in [20] was limited to the range of 10 to ${150}\mathrm{{kHz}}$ and the DC resistance of the Litz-wire was adopted as its AC resistance to remove the winding loss of WEAG. However, as the test frequency rises further, this approach is prone to introduce increasingly significant errors.

In this paper, based on [20], the gapped core of the inductor under test (IUT) was replaced with a gapless core with WEAG. By using the transformer winding short-circuit method, the total AC resistance was obtained, which contains the AC resistance of IUT’s winding $\left({R}_{\mathrm{m}}\right)$ and WEAG $\left({R}_{\mathrm{s}}\right)$. Subsequently, ${R}_{\mathrm{s}}$ was measured based on magnetic field equivalence with the help of auxiliary winding, enabling the extraction of ${R}_{\mathrm{m}}$. Compared with the existing scheme, the advantage of the proposed scheme is that the auxiliary winding is easy to construct. Due to the balance of ampere-turns between WEAG and IUT’s winding, the magnetizing flux inside the core is canceled to lower the quality factor of the inductor. The influence of high quality factors on the measurement result can be reduced. In Section II, the reason why WEAG can be used to replace the air-gap is analyzed in detail. In Section III, the method of how to separate ${R}_{\mathrm{m}}$ and ${R}_{\mathrm{s}}$ is exhibited. Section IV experimentally verifies the proposed scheme. Section V presents an error analysis. Finally, Section VI gives the conclusions of this paper.

In Section II, a transformer was constructed using the winding of IUT and WEAG, and the AC resistance of the transformer winding was obtained under the impedance test. This scheme enables the separation of winding loss and core loss, but the measured resistance is the sum of ${R}_{\mathrm{m}}$ and ${R}_{\mathrm{s}}$. In order to obtain ${R}_{\mathrm{m}}$, it is necessary to remove ${R}_{\mathrm{s}}$ from the measurement results.

To validate the effectiveness of the proposed method, the magnetic components made of EE35 cores as shown in Fig. 10 were created for impedance measurement. The measurement results were compared with the simulation results of the 3-D eddy current field. In order to guarantee the consistency between the 3-D simulation models and the actual inductors, single-layer winding structure was adopted. Detailed parameters are listed in Table I.

Fig. 11 shows the test platform for impedance measurement, where the device employed is LCR-8110G produced by GWINSTEK. Fig. 12 illustrates the 3-D simulation models of the gapped inductor and the inductor with WEAG. To save computing resources, 1/4 models were adopted.

Fig. 13 shows ${R}_{\mathrm{m}}$ obtained from different approaches:

1) FEM-scheme 1: simulated gapped inductor;

2) FEM-scheme 2: simulated inductor with WEAG;

3) Scheme in [20]: measured AC resistance deducts the calculated DC resistance of WEAG;

4) Proposed scheme: measured AC resistance deducts the measured AC resistance of WEAG.

Fig. 14 demonstrates the errors between other schemes and FEM-scheme 1. It can be seen that the error of FEM-scheme 2 increases with frequency in the range of ${10}\mathrm{{kHz}}$ to $1\mathrm{{MHz}}$, yet it consistently remains within 10%.

Scheme in [20] exhibited errors exceeding 20% at frequencies below ${22}\mathrm{{kHz}}$. This is because when the frequency is low, the impedance of the magnetizing inductance is not large enough and its branch cannot be regarded as an open circuit. Consequently, employing (5) to calculate ${R}_{\mathrm{m}}$ introduces significant inaccuracies. Within the range of 22 to ${80}\mathrm{{kHz}}$, the error was less than 20%. However, as the frequency rises the AC resistance of WEAG increases significantly to far larger than its DC resistance, leading to escalating measurement errors. Hence, the scheme has a limited frequency application range.

The proposed scheme also exhibited considerable errors at frequencies below ${53}\mathrm{{kHz}}$, which is also attributed to the insufficient impedance of the magnetizing inductance in the process of using the short-circuit method twice. Nevertheless, the scheme had enough accuracy at higher frequencies. Specifically, the error remained within 20% from 53 to 400 $\mathrm{{kHz}}$ and droped to ${10}\%$ at the range of ${400}\mathrm{{kHz}}$ to $1\mathrm{{MHz}}$.

Fig. 15 shows the error of the proposed scheme for changing the distance between WEAG and IUT’s winding by adding tape between WEAG and the center limb. Situation 1 is one layer and situation 2 refers to two layers of tape added, whose thickness is ${0.2}\mathrm{\;{mm}}$. It can be seen that when only one layer of tape was added, the error of the proposed scheme could still be maintained at less than ${20}\%$ after ${60}\mathrm{{kHz}}$. In this situation, the scheme in [20] achieved the same effect only in the range of 37 to ${95}\mathrm{{kHz}}$. After adding two layers of tape, both schemes became invalid with errors of more than 50% in most of the tested frequencies. But it is worth mentioning that in the two situations, the proposed scheme still had higher accuracy from ${100}\mathrm{{kHz}}$ to $1\mathrm{{MHz}}$ than the scheme in [20].

To further validate the method’s robustness and applicability, the proposed scheme was applied to inductors with cores of different sizes and different structures. Fig. 16 shows the inductors under test and the results are demonstrated in Fig. 17. Detailed parameters of them are listed in Table I. The measurement errors came from comparisons between the two schemes and the 3-D FEM simulation results.

Fig. 17(a) shows the measurement results when the inductor was made of EE28 core. It can be found that from ${230}\mathrm{{kHz}}$ to $1\mathrm{{MHz}}$, the proposed scheme had smaller errors than the scheme in [20]. Besides, the errors of the proposed scheme remained below ${20}\%$ in the range of ${86}\mathrm{{kHz}}$ to $1\mathrm{{MHz}}$. When the inductor was made from the EE16 core, the error of the proposed scheme was less than ${20}\%$ from ${220}\mathrm{{kHz}}$ to $1\mathrm{{MHz}}$ as exhibited in Fig. 17(b). It’s also in this range that the proposed scheme showed better results than the other.

Fig. 17(c) demonstrates the results when the inductor was made of RM10 core. The proposed scheme exhibited superiority over the scheme in [20] from ${85}\mathrm{{kHz}}$ to $1\mathrm{{MHz}}$. The error of the proposed scheme remained below 15% in this range. When the inductor was with the PQ2020 core, as depicted in Fig. 17(d), the two schemes had similar accuracy before ${100}\mathrm{{kHz}}$. However, the accuracy of the proposed scheme was higher than the scheme in [20] in the range from ${150}\mathrm{{kHz}}$ to $1\mathrm{{MHz}}$.

As illustrated in Fig. 13 and Fig. 17， ${R}_{\mathrm{m}}$ has a steep growth tendency after ${100}\mathrm{{kHz}}$, emphasizing the importance of accurately measuring the AC resistance of WEAG in the high-frequency range. Hence, the proposed scheme is particularly suitable for evaluating the winding loss of gapped inductors operating at high switching frequencies.

The analysis and experiments conducted in the previous sections demonstrate the feasibility of the proposed method. However, as a measurement technique, it is imperative to perform an error analysis. The errors of the proposed method are categorized into the following types:

1) Error in replacing the air-gap with WEAG. Both the 2-D simulation in Section II and the 3-D simulation in Section IV indicate that this error maintains within 10%, which is reasonable.

2) Error of substituting AW for IUT’s winding to generate MMF. The 2-D finite element simulation of different AW schemes in Section III shows that the AC resistance error of WEAG is within 10%, which is acceptable.

3) Error caused by the core loss. Although the magnetic flux in the core is canceled by ampere-turn balance, a small portion of the main-flux still exists near WEAG. This introduces an error due to core loss in the measurement results. However, since the winding loss is indirectly obtained by measuring the AC resistance under the small signal, the core loss is further reduced, making this error negligible.

4) Error introduced by the transformer winding short-circuit method. In the process of using this method, the branch of magnetizing inductance is treated as an open circuit. However, this requires that the impedance of magnetizing inductance is much greater than that of the secondary winding transformed to the primary side. Therefore, this method may introduce significant errors at low frequency. Since the proposed scheme uses the method twice, the error is large in the low-frequency range. However, at low frequencies, the change in the $\mathrm{{AC}}$ resistance of ${R}_{\mathrm{m}}$ compared to its DC resistance is very small, so the AC resistance can be equal to the DC resistance.

5) Error caused by using an air-core inductor to measure ${R}_{\mathrm{a}}$. The key to reducing this error lies in the wire gauge selection and position of AW. By minimizing the resistance of AW and choosing an appropriate winding position, this error can be reduced to a reasonable range.

This paper proposes a method for measuring the winding loss in gapped inductors. By magnetic field equivalence twice, this method obtains the total resistance of ${R}_{\mathrm{m}}$ and ${R}_{\mathrm{s}}$, as well as the total resistance of ${R}_{\mathrm{a}}$ and ${R}_{\mathrm{s}}$. After measuring ${R}_{\mathrm{a}}$ using an air-core inductor, the transformer winding short-circuit method is used to extract ${R}_{\mathrm{m}}$. Through finite element simulation and experimental verification, the following conclusions can be drawn:

1) The use of a gapless inductor with WEAG to characterize the magnetic field of the gapped inductor is feasible, with an error of less than 10%.

2) AW can substitute IUT’s winding for generating MMF, effectively characterizing the magnetic field of WEAG excited by IUT’s winding. The margin of error across various AW configurations remained within 10%.

3) Impedance test demonstrated that this method has sufficient measurement accuracy in the high-frequency range. By applying the proposed method to inductors with cores of different sizes and structures, the robustness and applicability of it were verified.

The proposed scheme is currently only applicable to inductors with a single air-gap, and future research will focus on its application in magnetic components with multiple air-gaps.