High-Frequency Eddy Current Measurement of Graphene Doping Content in Silver/Graphene Coatings Using Effective Medium Theory

ABSTRACT

The graphene doping content in silver/graphene coatings is critical for quality control. In this study, a characterization method based on apparent eddy current conductivity is proposed. First, the linear relationship between electrical conductivity and doping amount is developed using the Maxwell–Garnett effective medium theory. Second, a mapping model relating apparent eddy current conductivity to impedance signals is constructed by combining a transformer equivalent circuit. The characteristics of the eddy current field in silver/graphene composite coatings are investigated via finite-element simulation, and high-frequency excitation is used to suppress the influence of coating thickness. An embedded eddy current system based on Zynq SoC is developed, adopting digital quadrature demodulation technology to extract weak impedance signals.

KEYWORDS: silver/graphene composite coatings, eddy currents, apparent conductivity, embedded system

1. Introduction

The reliability of outdoor high-voltage disconnector switches is vital to power grid safety [1]. Their core components, electrical contacts, are susceptible to environmental corrosion and mechanical wear over long service periods, which can easily trigger failures [2]. Silver/graphene composite coatings have become an ideal choice for high-performance contacts due to their excellent electrical conductivity and wear resistance [3]. The performance of these composite coatings primarily depends on the graphene doping content: a concentration that is too low fails to form an effective protective layer, while an excessively high concentration can lead to graphene agglomeration, compromising coating density and reducing surface hardness [4]. Therefore, achieving accurate detection of graphene content in silver/graphene composite coatings is a key step in ensuring the service reliability of contacts and promoting their engineering applications.

Although existing methods such as scanning electron microscopy, energy dispersive spectroscopy, and Raman spectroscopy offer high precision, their destructive nature toward specimens and cumbersome sample preparation make it difficult to meet the demands for efficient online detection of composite coatings during industrial production [5, 6]. Common nondestructive testing (NDT) techniques, such as terahertz imaging and computed tomography, can characterize sample uniformity and microscopic defects [7, 8]. However, these methods suffer from insufficient penetration depth into highly conductive metal substrates and are associated with high equipment costs. In contrast, eddy current testing (ECT), as a mature NDT technology, can not only quantitatively evaluate the structural integrity of conductive workpieces [9–11] but also demonstrates significant advantages in characterizing bearing hardness, metal purity, and thermal aging properties through conductivity measurement [12–15]. Therefore, using ECT to measure the conductivity of silver/graphene oxide (GO) composite coatings to characterize graphene content has a clear physical mechanism and high feasibility.

Current-conductivity detection primarily follows two main paths: parameter inversion based on analytical models and physical-quantity mapping based on signal-feature extraction. Parameter inversion solves inverse problems by constructing forward analytical models of parameters such as impedance, conductivity, thickness, and liftoff, using optimization algorithms [16–18]. By combining swept-frequency eddy current technology with analytical models and inversion algorithms, simultaneous nondestructive detection of parameters such as coating thickness, conductivity, and permeability can be achieved [19–21]. However, the precision of parameter inversion methods depends on the degree of agreement between the model and the experimental detection. In actual measurement processes, hardware systems are affected by parasitic parameter coupling and environmental noise, making it difficult for measured values to match ideal simulation models and limiting the stability of inversion accuracy under complex working conditions [22].

Recently, decoupling methods based on signal features have become a significant research focus. Some researchers have characterized material conductivity using the ratio of the imaginary and real parts of impedance changes, achieving high-linearity conductivity solutions while suppressing liftoff interference [23, 24]. Ansari, Wang, and others used the spectral features of apparent eddy current conductivity under high-frequency asymptotics to suppress liftoff interference, achieving parameter decoupling for liftoff, thickness, and conductivity of nonmagnetic metal coatings [25, 26]. Li et al. introduced a three-winding transformer equivalent model that transformed complex eddy-current field problems into concise circuit-network solutions and confirmed the existence of a clear quantitative mapping mechanism between phase features, coating thickness, and conductivity. Based on this model architecture [27, 28], Guo et al. verified its effectiveness for the precise decoupling and characterization of key coating properties under high-frequency conditions [29]. Additionally, data-driven methods such as neural networks have been applied to construct nonlinear mapping models [30, 31]. However, due to the distribution gap between the training and test samples, such methods exhibit errors in practical detection.

Both the silver-based composite coating and the red copper substrate are highly conductive materials. The coating is extremely thin (~25 μm) and the conductivity difference between the two is minimal, resulting in weak reflection and transmission of the eddy current field at the material interface. Consequently, methods based on interface impedance difference analysis suffer from insufficient sensitivity and fail to simultaneously decouple parameters such as thickness, conductivity, and liftoff. Therefore, establishing the relationship between eddy current impedance signals and the graphene doping content of silver/graphene composite coatings under unknown thickness conditions is of great importance.

To address these issues, this paper proposes a method for detecting graphene doping content in silver/graphene composite coatings based on apparent eddy current conductivity. A quantitative relationship between apparent eddy current conductivity and doping content is established using the Maxwell–Garnett effective medium theory. Furthermore, an apparent eddy current conductivity detection model suitable for silver/graphene composite coatings is developed based on transformer equivalent circuit theory. By analyzing the response mechanism of eddy current impedance to changes in apparent eddy current conductivity, the excitation frequency is optimized to suppress the influence of coating thickness, achieving rapid quantitative characterization of the graphene doping content.

2. Methodology for Graphene Doping Content in Composite Coatings

The proposed measurement chain links graphene doping concentration to a directly measurable electromagnetic signal in two steps. Section 2.1 applies effective medium theory to derive a linear relationship between the doping concentration and the macroscopic electrical conductivity of the silver/graphene coating in the dilute regime. Section 2.2 then connects this coating conductivity to the apparent eddy current conductivity recovered from the detection coil’s impedance phase, using a weighted-conductivity model and a transformer equivalent-circuit analysis. Under high-frequency excitation, the apparent eddy current conductivity becomes a direct proxy for the coating conductivity, allowing the doping concentration to be inferred from the phase change in the coil impedance.

2.1. Characterization of Coating Doping Content via Electrical Conductivity Based on Effective Medium Theory

Microstructurally, the silver/graphene composite coating is a heterogeneous two-phase composite system composed of a continuous silver matrix (continuous phase) and dispersed GO (dispersed phase). To establish a quantitative relationship between graphene doping content and macroscopic electrical conductivity, this paper introduces effective medium theory [32] to characterize the composite material’s conduction mechanism.

Considering the distribution state of graphene within the silver matrix, at low doping concentrations, graphene does not form a macroscopically continuous conductive network but instead exists as isolated scattering centers distributed at the silver grain boundaries. Although single-layer graphene itself has high electrical conductivity, contact resistance occurs at the graphene–silver matrix interface in the composite coating, and grain boundaries increase the probability of electron scattering. Consequently, during electron transport, the dispersed graphene phase acts as a high-impedance region that hinders electron flow [33], thereby decreasing the coating’s electrical conductivity.

The influence of graphene flakes on the local electric field can be corrected by introducing a shape factor β. The modified effective medium model can be expressed as:

(1) (σeff − σAg) / (σeff + βσAg) = ϕ · (σG − σAg) / (σG + βσAg)

where

σeff is the effective electrical conductivity of the composite coating,

σAg is the electrical conductivity of the pure silver matrix,

σG is the equivalent electrical conductivity of the graphene phase after considering the interface scattering effect, and

ϕ is the volume fraction of graphene in the coating.

The graphene doping concentration range investigated in this study is 0 to 1.0 g/L, which lies within the dilute limit below the percolation threshold. Within this range, graphene particles have not yet interconnected to form long-range conductive pathways, and their volume fraction is extremely low (ϕ ≪ 1). Due to significant grain boundary scattering, the equivalent conductivity of the graphene phase is much smaller than that of the silver matrix (σG ≪ σAg). Under these conditions, by expanding the equation at ϕ = 0 using a Taylor series and neglecting higher-order terms, the linear approximation model can be obtained:

(2) σeff ≈ σAg(1 − kϕ)

where

k is the structural sensitivity coefficient related to the dispersion morphology, orientation, and interfacial potential barrier of the graphene.

Under constant electroplating process parameters, the volume fraction ϕ of graphene in the coating is positively correlated with the GO concentration C in the plating solution. Consequently, the relationship between macroscopic electrical conductivity and doping concentration satisfies the following monotonically decreasing relationship:

(3) σeff = σ0 − λ · C

where

σ0 represents the intrinsic electrical conductivity of the undoped pure silver coating, and

λ is a comprehensive coefficient reflecting the codeposition rate of electrodeposition and the intensity of electron scattering.

Equation 3 indicates that a clear negative linear correlation exists between the macroscopic effective electrical conductivity σeff of the composite coating and the graphene doping concentration C, allowing characterization of the doping content from the electrical conductivity.

2.2. Measurement of Coating Doping Content Using Apparent Eddy Current Conductivity

According to electromagnetic field theory, an alternating magnetic field generates a skin effect inside a conductor, and the distribution depth of the eddy current density is determined by the standard penetration depth δ:

(4) δ = 1 / √(πfμσ)

where

f is the excitation frequency,

μ is the magnetic permeability, and

σ is the electrical conductivity.

In silver/graphene composite coatings with micrometer-scale thickness, eddy currents easily penetrate the coating and enter the red copper substrate, thereby influencing the impedance signal. In this case, the conductivity measured by the detection coil is no longer the conductivity of the coating itself, but a weighted average of the coating conductivity σm and the substrate conductivity σs, manifested as the apparent eddy current conductivity σAECC. This physical process can be represented by the following linear weighted mathematical model:

(5) σAECC = α · σm + (1 − α) · σs

where

α is the weight factor, representing the proportion of the eddy current loss distributed within the coating relative to the total loss, which can be derived from the skin effect equation [27, 28]:

(6) α = 1 − e^(−2d/δavg)

where

d is the coating thickness, and

δavg is the average penetration depth after considering the composite system.

When the excitation frequency is low, δavg is large and α approaches 0; at this point, σAECC ≈ σs, and the detection signal is primarily dominated by the substrate. As the frequency increases, δavg decreases. When the penetration depth is equal to or less than the coating thickness, α rapidly approaches 1, and the detection signal is primarily dominated by the coating.

Combined with Equation 3, where the coating conductivity σm and the graphene doping concentration C satisfy a linear relationship, substituting Equation 5 into Equation 3 yields the total mapping model between apparent eddy current conductivity and doping concentration:

(7) σAECC = [ασ0 + (1 − α)σs] − (αλ) · C

Under high-frequency excitation, α approaches a constant near 1, allowing the total model to be simplified as:

(8) σAECC ≈ σm = σ0 − λ · C

In other words, under high-frequency conditions, the measured apparent eddy current conductivity directly reflects the intrinsic properties of the coating, enabling rapid, direct measurement of the graphene doping content.

To accurately convert the coil impedance signal to the apparent eddy current conductivity σAECC, a transformer model is employed. As shown in Figure 1, the transformer model treats the probe coil and the tested conductor system as a dual-winding transformer. Specifically, the excitation coil serves as the primary winding, while the induced eddy current in the tested conductor is equivalent to a single-turn short-circuited secondary winding. The detection coil can be regarded as a series combination of resistance R0 and inductance L0. The secondary circuit of the transformer is composed of the resistance Rx and inductance Lx of the equivalent eddy current loop connected in series. The voltage of the excitation coil is U, and the excitation current and induced current are I1 and I2, respectively.

According to Kirchhoff’s laws [27], the resistance change ΔR and the reactance change ΔX of the coil can be expressed as:

(9) ΔR = R0 − Rair = [ω²M² / (Rx² + (ωLx)²)] · Rx
ΔX = ω(L0 − Lair) = −ω · [ω²M² / (Rx² + (ωLx)²)] · Lx

where

ω denotes the angular frequency of the sinusoidal excitation signal, ω = 2πf, with f representing the excitation frequency of the detection coil,

Rair and Lair are the resistance and inductance of the coil in air, respectively, and

M denotes the mutual inductance between the detection coil and the specimen under test.

The equivalent resistance and inductance of the eddy current loop can be expressed as:

(10) Rx = 2π / [he · σAECC · ln(r2/r1)]
Lx = μ0 · S(r1, r2)

where

Rx is the equivalent eddy current resistance,

Lx is the equivalent eddy current inductance,

he is the equivalent eddy current loop thickness,

σAECC is the apparent eddy current conductivity,

μ0 is the vacuum magnetic permeability, with a value of 4π × 10⁻⁷ H/m, and

S(r1, r2) is a function whose value depends solely on the geometric parameters r1 and r2.

According to Equations 9 and 10, the phase of the impedance change can be expressed as:

(11) ΔP = |ΔX/ΔR| = ωLx/Rx = [μ0 · ω · σAECC · he / (2π)] · S(r1, r2) · ln(r2/r1)

It can be seen that the phase change ΔP of the coil impedance depends only on the material’s apparent eddy current conductivity and the equivalent eddy current thickness.

During high-frequency detection, when the total thickness of the substrate and the coating is greater than 4× the penetration depth, Equation 11 can be further simplified as:

(12) ΔP = G · √(σAECC/f)

where

G is a determined constant.

The phase change depends solely on the apparent eddy current conductivity and frequency [34]. When the excitation frequency is fixed, the relationship between ΔP and σAECC can be established as:

(13) ΔP = a · √(σAECC) + b

where

the coefficients a and b can be determined by calibration with standard specimens.

3. Finite Element Simulation and Factor Analysis

Based on finite element theory and standard modeling procedures, a simulation model was constructed in COMSOL Multiphysics. The model’s parameter settings are shown in Table 1.

TABLE 1 Parameters in simulations

3.1. Excitation Frequency

The excitation frequency is the key parameter determining the penetration depth of the eddy current field, directly influencing the weight distribution between coating and substrate information in the detection signal. To determine the optimal excitation frequency, this section performs a swept-frequency parametric simulation analysis for a silver/graphene composite coating with a thickness of 25 μm.

Figure 2a shows the relationship between the phase change and the coating conductivity at different frequencies. As the excitation frequency increases, the slopes of the curves increase significantly, substantially enhancing the detection system’s sensitivity to variations in coating conductivity. This indicates that high-frequency excitation can effectively enhance the distribution of eddy currents within the coating, thereby overcoming the influence of the highly conductive substrate. Figure 2b shows the error distribution of the detected apparent eddy current conductivity after logarithmic calibration using standard specimens based on the previously proposed method. To determine the optimal detection frequency, average relative error (ARE) and maximum relative error (MRE) are introduced as quantitative evaluation indicators, defined as follows:

(14) ARE = (1/N) · Σ (i = 1 to N) |(σmeas − σtrue) / σtrue| × 100%

(15) MRE = max(|(σmeas − σtrue) / σtrue| × 100%)

where

σmeas and σtrue represent the measured conductivity and the true value of the specimen, respectively, and

N is the number of test samples.

In the low-frequency range (5 to 7 MHz), although the ARE remains low, the MRE consistently exceeds 4%, indicating a substantial numerical discrepancy between the two error metrics. This suggests that in the low-frequency range, due to the large penetration depth, the eddy current field fails to effectively suppress the substrate’s influence. Detection outcomes are therefore mixed with random interference components arising from substrate coupling, an uncertainty that is difficult to eliminate in industrial testing.

In contrast, the system exhibits the best peak value control capability and error stability at 7 MHz and 10 MHz, effectively eliminating the risk of extreme misjudgment. However, as frequency increases beyond 10 MHz, both the ARE and MRE show a rebounding trend. At the simulation level, this is primarily due to numerical discretization errors arising from finite-element mesh partitioning at ultra-thin skin depths. In practical engineering applications, excessively high frequencies also introduce nonlinear interferences, such as coil parasitic capacitance and lead inductance, thereby reducing the system’s actual signal-to-noise ratio (SNR).

Although 7 MHz yields a lower average error, a comprehensive evaluation of MRE and ARE indicates that 10 MHz is more advantageous. The optimal excitation frequency for the system is therefore set at 10 MHz.

3.2. Influence of Coating Thickness

In the actual electroplating production process, the thickness of silver/graphene composite coatings is difficult to maintain at a fixed, ideal value due to process variability, typically ranging from 23 μm to 27 μm. According to the skin effect theory, at a frequency of 10 MHz, the eddy current field does not completely decay within the coating. Since the coating thickness does not meet the requirement of being greater than 4× the penetration depth, thickness variations will inevitably affect the final impedance signal [29].

To evaluate the extent to which the proposed detection method is affected by coating thickness, this section conducts a simulation analysis of the influence of thickness variations on conductivity detection accuracy. At an optimal excitation frequency of 10 MHz, a parametric scan of coating thickness was performed, ranging from 20 μm to 30 μm with a step size of 1 μm.

As shown in Figure 3, despite significant variations in coating thickness, the measured conductivity remains highly stable, with values concentrated between 52.7 and 53.0 MS/m. A thickness fluctuation of 10 μm results in a deviation of only ~0.3 MS/m in the conductivity measurement. Throughout the entire scanning range, the relative measurement error remains stable between −4.2% and −3.8%, meaning the impact of thickness variation on the conductivity detection error is only 0.4%.

In summary, at an excitation frequency of 10 MHz, the influence of coating thickness fluctuations on conductivity detection is highly limited. Consequently, detection accuracy for graphene doping content can be maintained even for composite coating specimens with thickness variations due to the manufacturing process.

4. Experimental Verification and Result Analysis

This section presents experimental validation of the proposed method using silver/graphene composite coating specimens prepared at varying doping concentrations. The specimens, detection system, and measurement results are described in turn.

4.1. Specimen Preparation

To verify the effectiveness of the proposed detection method, this study commissioned China Academy of Machinery Wuhan Research Institute of Materials Protection Co. Ltd. to prepare a series of silver-based composite coating specimens with different graphene doping concentrations. Red copper substrates measuring 60 mm × 60 mm × 5 mm were used. After pretreatment, the substrates were placed in silver-based plating solutions with various GO doping concentrations for electroplating.

The plating system used deionized water as the solvent, with the silver salt content fixed at 40 g/L. Five groups of electrolytes with different compositions were prepared by varying the GO addition (0, 0.1, 0.2, 0.4, and 1.0 g/L, respectively), with the 0 g/L GO group serving as the pure silver control sample. The same types and dosages of auxiliary additives were used in all plating solutions, and the pH, temperature, and stirring speed were strictly controlled to ensure consistency in the electroplating process. The coating thickness of all samples was controlled between 23 μm and 27 μm.

After electroplating, the specimens were cleaned and dried to obtain silver/graphene composite coatings with uniform surfaces and good bonding. The macroscopic morphology and typical cross-sectional metallographic structure of the specimens are shown in Figure 4.

4.2. Eddy Current System

For quality control of silver/graphene composite coatings, an embedded eddy current system was developed. The system architecture is shown in Figure 5.

The system uses Zynq SoC architecture. The PL (programmable logic) side integrates a DDS excitation source and digital IQ demodulation, working in coordination with high-speed AD/DA links to complete signal excitation and feature extraction. Data is transmitted via AXI DMA to the PS (processing system) side for denoising and impedance calculation. Finally, the host computer, in combination with a decoupling algorithm, enables real-time characterization of coating parameters. The key parameter specifications for the hardware and probe are shown in Table 2, and the complete detection system is shown in Figure 6.

TABLE 2 Key parameters of eddy current system and probe

4.3. Experiments and Results

To evaluate the proposed graphene-doping content detection method, a group-based validation strategy is adopted to systematically test the prepared silver/graphene composite coating specimens. Testing covered 10 independent specimens across five gradient concentrations from 0 to 1.0 g/L (two specimens per concentration). Given the potential for local agglomeration and microscopic nonuniform distribution of trace graphene within the composite coating, a multipoint discrete sampling strategy was used. As shown in Figure 7, eight independent test points were selected within the effective testing area of each specimen. By calculating statistical features of multipoint data, random measurement noise introduced by the material’s microscopic nonuniformity was effectively suppressed, ensuring the reliability of the apparent conductivity data.

To evaluate the signal stability and resolution, 1000 continuous sampling tests were conducted on standard specimens in a constant-temperature laboratory environment. Figure 8 illustrates the time-domain fluctuations of the raw voltage amplitude signal for a silver/graphene composite coating with a doping concentration of 0.4 g/L, alongside the results from an 80-point moving-average filter. Experimental results show that the static fluctuation range of the system’s output voltage amplitude is controlled within 1.0 mV, and the SNR reaches 75 dB. This high SNR and stability allow the system to clearly and sensitively capture the response characteristics induced by different doping concentrations.

Using this system, specimens were scanned at an excitation frequency of 10 MHz to obtain the apparent conductivity response characteristics corresponding to different doping concentrations. As shown in Figure 9, increasing the GO concentration in the plating solution results in a significant decrease in the macroscopic conductivity of the composite coating. A linear fit was performed on the measured data using the least-squares method (R² = 0.967), establishing a quantitative mapping relationship between the doping concentration C and the apparent conductivity σ. This high goodness-of-fit indicates that eddy current conductivity can serve as a reliable physical parameter for characterizing graphene content.

Based on this model, tests were performed on the specimens, with results presented in Figure 10 and Table 3.

TABLE 3 Detection results of doping concentration

The data indicate that detection results for doping concentration are closely distributed around the y = x ideal line, and the confidence intervals for each concentration group do not overlap. This demonstrates that the system can precisely identify subtle concentration differences among specimens from different batches, exhibiting a robust linear mapping relationship and high resolution across the full measurement range.

The detection error in the low-concentration range is controlled within 0.03 g/L, while the maximum absolute error in the high-doping concentration range is maintained within 0.072 g/L. The detection results and error statistics indicate that the confidence intervals for each concentration gradient do not overlap. The self-developed instrument can clearly distinguish key process nodes at 0.1, 0.2, 0.4, and 1.0 g/L. This demonstrates that the proposed method is suitable not only for high-precision characterization but also for rapid classification and quality control of composite coatings in industrial settings.

5. Conclusion

To address the challenge of online detection of graphene doping content in silver/graphene composite coatings used in high-voltage disconnector electrical contacts, this paper proposes a quantitative characterization method based on apparent eddy current conductivity. The main contributions are as follows:

1. By introducing effective medium theory and a transformer equivalent circuit model, a quantitative mapping relationship between doping concentration and apparent conductivity was established.

2. Based on transformer equivalent circuit theory, a detection model for apparent eddy current conductivity was developed for silver/graphene composite coatings. High-frequency excitation was used to suppress the interference of coating thickness fluctuations on conductivity characterization, achieving precise decoupling of a single physical parameter.

3. An embedded digital eddy current testing system based on the Zynq SoC architecture was independently developed, achieving high-speed acquisition, orthogonal demodulation, and real-time feature extraction of weak impedance signals.

Experimental results indicate that the system exhibits excellent linearity and resolution over the concentration range of 0 to 1.0 g/L. The detection error does not exceed 0.072 g/L, enabling reliable differentiation of graphene content in composite coatings. This provides a robust engineering solution for optimizing the preparation process and quality control of electrical contact materials.

ACKNOWLEDGMENTS

This work was supported by the Science and Technology Program from State Grid Corporation of China (No. 5500-202455122A-1-1-ZN: Research on Mechanical Property Evaluation of Novel Composite Silver Coating and Application of Terahertz Non-destructive Testing Technology).

REFERENCES

1. Yuan, X., F. Fu, and R. He. 2024. “Graphene-enhanced silver composites for electrical contacts: A review.” Journal of Materials Science 59 (9): 3762–79. https://doi.org/10.1007/s10853-024-09473-z

2. Karslioglu, R., and L. Al-Falahi. 2020. “Electrical contact performance of various electro-deposited graphene reinforced silver-based nanocomposites.” Materials Testing 62 (4): 401–07. https://doi.org/10.3139/120.111498

3. Qiang, H., Q. Sun, X. Zheng, M. Qi, M. Chen, N. Luo, C. Xu, and X. Ye. 2023. “Laser direct fabrication of graphene on silver-based contact as a high-end electrical product.” Journal of Materials Science 58 (19): 8178–88. https://doi.org/10.1007/s10853-023-08509-0

4. Lv, W. Y., K. Q. Zheng, and Z. G. Zhang. 2018. “Performance studies of Ag, Ag-graphite, and Ag-graphene coatings on Cu substrate for high voltage isolation switch.” Materials and Corrosion 69 (12): 1847–53. https://doi.org/10.1002/maco.201810053

5. Yan, J., Q. Wang, Z. Zhang, M. Chen, and H. Yang. 2025. “Optimization of silver/graphene composite coating for GIL touch spring based on cyanide-free electroplating technology.” Materials Research Express 12 (2): 026507. https://doi.org/10.1088/2053-1591/adb092

6. Nguyen, V. H., B. K. Kim, Y. L. Jo, and J. J. Shim. 2012. “Preparation and antibacterial activity of silver nanoparticles-decorated graphene composites.” Journal of Supercritical Fluids 72: 28–35. https://doi.org/10.1016/j.supflu.2012.08.005

7. Dadrasnia, E., F. Garet, D. Lee, J.-L. Coutaz, S. Baik, and H. Lamela. 2014. “Electrical characterization of silver nanowire-graphene hybrid films from terahertz transmission and reflection measurements.” Applied Physics Letters 105 (1): 011101. https://doi.org/10.1063/1.4889091

8. Smith, C. T. G., C. A. Mills, S. Pani, R. Rhodes, J. J. Bailey, S. J. Cooper, T. S. Pathan, V. Stolojan, D. J. L. Brett, P. R. Shearing, and S. R. P. Silva. 2019. “X-ray micro-computed tomography as a non-destructive tool for imaging the uptake of metal nanoparticles by graphene-based 3D carbon structures.” Nanoscale 11 (31): 14734–41. https://doi.org/10.1039/C9NR03056E

9. Zhu, R., S. She, T. Meng, X. Zheng, A. Peyton, and W. Yin. 2025. “Ectar: An integrated augmented reality and eddy current array system for non-destructive testing of metallic plate.” Nondestructive Testing and Evaluation 1–20. https://doi.org/10.1080/10589759.2025.2581208

10. Su, B., Y. Li, H. Wei, W. Gao, Z. Zhao, and Z. Chen. 2025. “Quantitative assessment of back-surface flaws in cladded conductors via near- and remote-field integrated transient eddy current testing.” Materials & Design 260: 115209. https://doi.org/10.1016/j.matdes.2025.115209

11. Li, J., Z. Tong, Q. Yang, J. Wang, C. Jia, W. Guo, Y. Zhang, P. Jiang, W. Qiu, T. Uchimoto, S. Xie, Z. Chen, and T. Wang. 2025. “Non-destructive evaluation of defects in thermal barrier coating system using combined electromagnetic and thermographic signals.” Nature Communications 16: 8697. https://doi.org/10.1038/s41467-025-63717-3

12. Sha, J., H. Zhang, M. Fan, B. Cao, and F. Sun. 2025. “Characterization of heat-related bearing rings via measurement of electromagnetic properties for pulsed eddy current evaluation.” NDT & E International 146: 103268. https://doi.org/10.1016/j.ndteint.2024.103268

13. Shen, W., J. Li, Z. Zhai, X. Wang, W. Feng, and Z. Lei. 2026. “Gold purity detection via pulsed eddy current testing with adaptive compensation residual shrinkage network.” NDT & E International 158: 103563. https://doi.org/10.1016/j.ndteint.2025.103563

14. Yang, G., D. Liu, K. Zeng, Z. Liu, J. Cheng, Y. Zheng, D. Zou, J. Zhou, R. Liu, G. Tian, B. Gao. 2025. “High frequency resonance eddy current multi-parameters fusion sensing method for 12CrMoV steel thermal aging characterization.” IEEE Sensors Journal 25 (6): 10145–54. https://doi.org/10.1109/JSEN.2025.3531785

15. Xie, Y., X. Geng, P. Huang, and Y. Yan. 2026. “A method for conductivity measurement through eddy current testing and closed-loop feedforward control.” NDT & E International 159: 103612. https://doi.org/10.1016/j.ndteint.2025.103612

16. Zhu, S., R. Huang, J. R. Salas Avila, Y. Tao, Z. Zhang, Q. Zhao, A. J. Peyton, and W. Yin. 2020. “Simultaneous measurements of wire diameter and conductivity using a combined inductive and capacitive sensor.” IEEE Sensors Journal 20 (19): 11617–24. https://doi.org/10.1109/JSEN.2020.2997525

17. Xia, Z., J. Yan, R. Huang, M. Lu, Z. Cui, A. Peyton, W. Yin, and W. Yang. 2022. “Fast estimation of metallic pipe properties using simplified analytical solution in eddy current testing.” IEEE Transactions on Instrumentation and Measurement 72: 1–13. https://doi.org/10.1109/TIM.2023.3296814

18. Lee, K. M., B. Hao, M. Li, and K. Bai. 2019. “Multiparameter eddy current sensor design for conductivity estimation and simultaneous distance and thickness measurements.” IEEE Transactions on Industrial Informatics 15 (3): 1647–57. https://doi.org/10.1109/TII.2018.2843319

19. Tai, C. C. 2000. “Characterization of coatings on magnetic metal using the swept-frequency eddy current method.” Review of Scientific Instruments 71 (8): 3161–67. https://doi.org/10.1063/1.1304862

20. Yu, Y., D. Zhang, C. Lai, and G. Tian. 2017. “Quantitative approach for thickness and conductivity measurement of monolayer coating by dual-frequency eddy current technique.” IEEE Transactions on Instrumentation and Measurement 66 (7): 1874–82. https://doi.org/10.1109/TIM.2017.2669843

21. Xu, J., J. Wu, W. Xin, and Z. Ge. 2020. “Fast measurement of the coating thickness and conductivity using eddy currents and plane wave approximation.” IEEE Sensors Journal 21 (1): 306–14. https://doi.org/10.1109/JSEN.2020.3014677

22. Abu-Nabah, B. A., and P. B. Nagy. 2007. “Lift-off effect in high-frequency eddy current conductivity spectroscopy.” NDT & E International 40 (8): 555–65. https://doi.org/10.1016/j.ndteint.2007.06.001

23. Wang, C., M. Fan, B. Cao, B. Ye, and W. Li. 2018. “Novel noncontact eddy current measurement of electrical conductivity.” IEEE Sensors Journal 18 (22): 9352–59. https://doi.org/10.1109/JSEN.2018.2870676

24. Cao, B., L. Lu, M. Fan, and C. Li. 2024. “Conductivity measurement of metal films based on eddy current testing.” Nondestructive Testing and Evaluation 39 (2): 366–83. https://doi.org/10.1080/10589759.2023.2198234

25. Ansari, Z. A., B. A. Abu-Nabah, M. Alkhader, and A. Muhammed. 2020. “Experimental evaluation of nonmagnetic metal clad thicknesses over nonmagnetic metals using apparent eddy current conductivity spectroscopy.” Measurement 164: 108053. https://doi.org/10.1016/j.measurement.2020.108053

26. Wang, D., J. Xu, W. Xin, S. Zhu, and Y. Han. 2024. “Thickness and conductivity measurement of nonferromagnetic metal coating using apparent eddy current conductivity.” IEEE Transactions on Instrumentation and Measurement 73: 1–10. https://doi.org/10.1109/TIM.2024.3427749

27. Li, C., M. Fan, B. Cao, B. Ye, Q. Wang, and J. Jiang. 2024. “Thickness measurement of thermal barrier coating based on mutual inductance of eddy current system.” IEEE Transactions on Industrial Electronics 71 (7): 8099–109. https://doi.org/10.1109/TIE.2023.3312422

28. Li, C., M. Fan, B. Cao, B. Ye, and J. Yan. 2025. “Eddy current instrumentation of bond coating thickness by developing three-winding transformer model.” IEEE/ASME Transactions on Mechatronics 30 (3): 2245–56. https://doi.org/10.1109/TMECH.2024.3427330

29. Guo, J., B. Cao, F. Sun, and B. Ye. 2026. “Highly sensitive and robust eddy current system for thickness measurement of ceramic coating using near-resonance inductance.” IEEE/ASME Transactions on Mechatronics 31 (1): 26–37. https://doi.org/10.1109/TMECH.2025.3576920

30. Aldbaisi, A., B. A. Abu-Nabah, M. Alkhader, and M. A. Jaradat. 2025. “Eddy current conductive coating layer assessment on conductive substrate: A machine learning approach.” Nondestructive Testing and Evaluation 40 (11): 5290–319. https://doi.org/10.1080/10589759.2024.2440642

31. Tesfalem, H., A. J. Peyton, A. D. Fletcher, M. Brown, and B. Chapman. 2020. “Conductivity Profiling of Graphite Moderator Bricks From Multifrequency Eddy Current Measurements.” IEEE Sensors Journal 20 (9): 4840–49. https://doi.org/10.1109/JSEN.2020.2965201

32. Nan, C. W., R. Birringer, D. R. Clarke, and H. Gleiter. 1997. “Effective thermal conductivity of particulate composites with interfacial thermal resistance.” Journal of Applied Physics 81 (10): 6692–99. https://doi.org/10.1063/1.365209

33. Pietrak, K., and T. S. Wiśniewski. 2014. “A review of models for effective thermal conductivity of composite materials.” Journal of Power Technologies 95 (1): 14–24. https://papers.itc.pw.edu.pl/index.php/JPT/article/view/463

34. Wang, H., W. Li, and Z. Feng. 2015. “Noncontact thickness measurement of metal films using eddy current sensors immune to distance variation.” IEEE Transactions on Instrumentation and Measurement 64 (9): 2557–64. https://doi.org/10.1109/TIM.2015.2406053