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This paper presents a modelling procedure to take into account the capacitive effect at high frequencies, in Eddy Current NonDestructive Characterization (ECNDC) of Unidirectional Carbon Fiber Reinforcement Composite (UDCFRC) rods. To simulate the complete ECNDC systems, first, the multilayer circular air coil is physically modeled by a finite element (FE) axisymmetric eddy current model coupled to equivalent RL circuit. Each layer of the coil is represented by an equivalent resistance (R) in series with the equivalent inductance (L). Secondly, R and L of the coil layers are computed for several frequencies up to 5Mhz, and then introduced into the equivalent RLC circuit with considering interturn and interlayer capacitances. Then the inversion problem is solved in order to identify all inner capacitances of the coil. Finally, the UDCFRC rod is introduced into the FE eddy current axisymmetric model coupled to an equivalent RLC circuit, as a homogenized conductive material with an equivalent transverse conductivity. The coil with the presence of the homogenized UDCFRC rod is then modeled as a transformer with a secondary connected to a capacitor in parallel with a resistance in order to evaluate the inner capacitor of the UDCFRC. All evaluated parameters are then introduced in the last model. The comparison between the computed impedance parts and the measured ones shows a mean error less than 2% and a maximum one of 5% according to the frequency.
composite materials, eddy currents, finite element analysis, inverse problem, nondestructive characterization, parameter identification, parasitic capacitance
Unidirectional carbon fiber reinforcement composite rods, are used in an important number of industrial applications, particularly in civil engineering [1]. These materials offer a low density, high strength and stiffness, relatively low maintenance costs and noncorrosive behavior as compared to traditional materials [1, 2]. However, these materials, like traditional ones, are subject to degradation and ageing due to environmental factors and accidents. In order to maintain the safety needed by the composite structures, the use of NonDestructive Testing (NDT) techniques is required. Different NDT techniques can be used to detect the defects in CFRC. Eddy Current NDT (ECNDT) technique is widely used for CFRC in recent years [3].
Usually, according to the form of the studied structure, the ECNDT working frequency must respect the condition: "skin depth need to be less or equal to the rod radius or the plate thickness". This frequency is very high in the case of CFRC target, it can be up to 2MHz. This is due to the low conductivity of these materials and the low diameter or thickness of the studied structure forms. Therefore, design and analysis of the ECNDT systems of CRFC, require the precision models where the high frequency phenomena must be taken into account. It should be noted that these phenomena appear in both eddy current probe and target CFRC.
The electromagnetic properties of CFRC target, like as electrical conductivity tensor, are very important input parameters of the ECNDT model. It must be determined accurately. It should be pointed out that that UDCFRC rod materials have a longitudinal electrical conductivity less than 50000 S/m [46] and a transversal one less than 200 S/m [69].
Due to highly anisotropic properties of the UDCFRC materials and the capacitive effect, new methods and models must be carried on to understand their behavior and to evaluate all important parameters. This is the main purpose of this paper.
2.1 Experimental setup
The experimental setup (Figure 1) contain:
Figure 1. Experimental setup performed with a precision LCRmeter
Table 1. Dimensions and electrical parameters of CFRC rod and coil 2
UDCFRC rod Height 
300 mm 
UDCFRC rod diameter 
15.9 mm 
Height of coil 
31.7 mm 
Coil inner diameter 
16.5 mm 
Coil external diameter 
21.7 mm 
Copper wire diameter 
0.55 mm 
Coil turn number 
138 
Coil layer number 
3 
UDCFRC rod transverse conductivity s_{t} 
81 S/m 
Copper conductivity 
58 MS/m 
Figure 2. Geometric definition of ECNDE of CFRC rod
2.2 Global computation procedure
To illustrate the high frequency behavior of air coils, the measuring of real and imaginary impedance parts according to frequency up to 5MHz is done using a precision LCRmeter.
As appears in Figure 3, no resonance point is observed for coil 1 up to the maximum frequency, but for the coil 2 a first resonant point is detected at 1.2MHz. This is due to the fact that the number of turns in coil 2 is higher compared to the coil 1. Therefore, the equivalent capacitor of the coil 2 is more important than the coil 1 one.
The capacitive effect inside the coil 1 could be neglected when the effect of equivalent inner capacitor of UDCFRC rod is ignored. This capacitor is outcome from the displacement currents between the fibers through the polymer matrix. Authors [810] proposed the equivalent electric circuit of the laminated UDCFRC materials applied to the contact voltamperemeter conductivity measurement. Figure 4 shows the electrical diagram (Figure 4a) inside a UDCFRP and equivalent circuit proposed by Jones et al. [11]. There is only the conduction current type in the parallel direction of the fibers that expressed by an inductance in series of a resistance. But, the two current types (displacement and conduction) arise in the transversal direction. The conduction current is due to the contacts of the adjacent fibers modeled by a resistance. The displacement current appears for a high frequency, is due to the dielectric nature of the media between fibers, it is modeled by a capacitor.
The impedance variation of the coil is computed by the difference between the impedances with and without presence of the UDCFRC rod. During the ECNDC of these materials the impedance variation of the coil 1 is used to evaluate transversal electrical conductivity [6].
Due to the important number of carbon fibers impregnated in the rod and their size compared to global dimensions, the UDCFRC rod is replaced in the FE axisymmetric model by the homogenized one with an equivalent tensor conductivity according [46].
Figure 3. Measured of real and imaginary part of ECprobe impedances without presence of CFRC rod target
Figure 4. Electrical diagram inside a UDCFRP a) Simplified electrical diagram of UDCFRP, b) Possible equivalent electrical circuit of UDCFRP
FE eddy current model coupled to RLC equivalent circuit can be used to evaluate the capacitors of the coil and the UDCFRC rod inner capacitor. This evaluation requires the multifrequency analysis. The inversion of the model takes a lot of time, due to the important number of iterations. To reduce the computation time, a quick procedure given by the successive steps below is proposed.
Figure 5. Aircoil equivalent circuit with and without UDCFRC rod
Figure 6. Inverse problem algorithm to compute
The complete equivalent circuit model can be represented as a transformer model, with the aircoil in the primary and the homogenized UDCFRC rod in the secondary shortcircuited to capacitor in parallel to resistance.
The equivalent inner capacitor Cr of the UDCFRC rod can be obtained manually using the FE axisymmetric model coupled to RLC equivalent circuit, by the verification of the experimental first resonant point of the coil.
All evaluated parameters are introduced into the FE eddy current model coupled to the equivalent RLC circuit, to calculate impedances as a function of frequency. Then, they are compared with the measured results.
The FE models coupled to equivalent circuits and the inversion algorithm are implemented in both GNU software, FEMM [12] and Octave [13].
3.1 FE axisymmetric model coupled to RL equivalent circuit
Figure 2 show the geometry of the experimental setup to be modeled. The FE axisymmetric eddy current model coupled to RL equivalent circuit is expressed using magnetic vector and electrical scalar potentials Av. The FE formulation in the sub regions k_{e} of the coil turn with the same imposed voltage $U_{k_{e}}^{\prime}$ can be written [1416] as bellow:
$\begin{array}{c}
\frac{\partial}{\partial r}\left(v^{\prime} \frac{\partial A^{\prime}}{\partial r}\right)\frac{\partial}{\partial z}\left(v^{\prime} \frac{\partial A^{\prime}}{\partial z}\right)j \omega \sigma_{c o}^{\prime} A^{\prime}+\sigma_{c o}^{\prime} \frac{U_{k_{e}}^{\prime}}{2 \pi} \\
=0
\end{array}$ (1)
$\begin{aligned}
U_{k_{e}}^{\prime}=2 \pi r \cdot \operatorname{grad} & V_{k_{e}} \\
&=R_{k_{e}}^{\prime} I_{k_{e}}+j \omega R_{k_{e}}^{\prime} \int_{S_{k_{e}}}^{} \sigma_{c o}^{\prime} A^{\prime} d s
\end{aligned}$ (2)
$R_{k_{e}}^{\prime}=2 \pi / \int_{s_{k_{e}}}^{} \sigma_{c o}^{\prime} d s$ (3)
All coil turns through which the total current I flows, are in series. The current I_{k} in each turn k_{c} of the coil layer k_{l} according to the currents $I_{k_{e}}$ in the conductor sub region, is given by:
$I_{k_{c}}=\sum_{k_{e}=1}^{N_{e k c}} I_{k_{e}}=I_{k_{l}}=I$ (4)
The coil applied voltage according to the layer voltage $U_{k_{l}}$ can be expressed as:
$U_{i n}=\sum_{k_{l}=1}^{N_{L}} U_{k_{l}}$ (5)
with,
$U_{k_{l}}=\sum_{k_{c}=1}^{N_{c}} U_{k_{c}}$ (6)
where,
A': Modified magnetic vector potential (T·m3), A'=(0,$r A_{\varphi}$,0);
$\sigma_{c o}^{\prime}=\frac{\sigma_{c o}}{r}$ with σ_{co} is the copper conductivity [S/m], σ_{co}=57MS/m;
ω: pulsation of current I (rad/s); ω=$2 \pi f$;
f: is the working frequency;
ν'=$\frac{1}{r \mu_{0}}$ with μ0 is the Magnetic permeability of air,
μ_{0}=4$\pi$107 H/m;
$R_{k_{e}}{ }^{\prime}$: DC resistance of the conductor sub region;
$I_{k_{l}} \text { and } U_{k_{l}}$: Electrical current and voltage on the layer $k_{l}$;
$I_{1}=I_{2}=I_{3}=\ldots=I_{k_{c}}=I_{k_{l}}=I$;
Nc: Turn number in each layer;
NL: Layer number of the coil.
The formulation in air domain, is given as follow:
$\iint_{\Omega_{a}}^{} \frac{\partial}{\partial r}\left(v^{\prime} \frac{\partial A^{\prime}}{\partial r}\right)\frac{\partial}{\partial z}\left(v^{\prime} \frac{\partial A^{\prime}}{\partial z}\right)=0$ (7)
3.2 RLC equivalent circuit
The proposed equivalent electrical circuit of the three layers of coil without presence of the UDCFRC rod is given by Figure 5. Where, the inductances L1, L2, L3 and the resistances R1, R2, R3 of the layer i are given by the FE axisymmetric eddy current model coupled to RL equivalent circuit as a real and imaginary parts of the layer impedance that computed as:
$R_{k_{l}}=r e\left(\frac{U_{k_{l}}}{I_{k_{l}}}\right) L_{k_{l}}=\frac{\operatorname{Im}\left(\frac{U_{k_{l}}}{I_{k_{l}}}\right)}{2 \pi f} k_{l}=1,2,3$ (8)
The equivalent impedance $Z_{e k_{l}}$ of the layer $k_{l}$ with considering the equivalent capacitor $C_{k_{l}}\left(Z_{C k_{l}}=\frac{R_{C k_{l}}}{1+j C_{k_{l}} \omega R_{C k_{l}}}\right)$ of layer i, is expressed as:
$Z_{e k_{l}}=\frac{Z_{k_{l}} Z_{C k_{l}}}{Z_{k_{l}}+Z_{C k_{l}}} k_{l}=1,2,3, \ldots, N_{L}$ (9)
The total impedance Zcal of the three layers considering the equivalent capacitors C12, C23 (Figure 5) between layers is expressed, by using the delta/star transformation [17], as:
$\begin{array}{c}
Z_{c a l}=\frac{Z_{e 1} Z_{C 12}}{Z_{e 1}+Z_{e 2}+Z_{C 12}}+\cdots \\
\frac{\left(\frac{Z_{e 2} Z_{C 12}}{Z_{e 1}+Z_{e 2}+Z_{C 12}}+Z_{e 3}\right)\left(\frac{Z_{e 1} Z_{e 2}}{Z_{e 1}+Z_{e 2}+Z_{C 12}}+Z_{C 23}\right)}{\frac{Z_{e 2} Z_{C 12}+Z_{e 1} Z_{e 2}}{Z_{e 1}+Z_{e 2}+Z_{C 12}}+Z_{C 23}+Z_{e 3}}
\end{array}$ (10)
where, ZC12 and ZC23 are the equivalent impedances of C12 and C23 with considering the dielectric dissipation resistances.
All capacitors of the coil are evaluated using the inverse problem algorithm illustrated in Figure 6. The difference between the measured and computed impedances, expressed by the goal function FG, is minimized until verification of the convergence criterion. The inverse problem goal function FG is written as:
$F_{G}=\sqrt{\frac{1}{N_{f}} \sum_{k=1}^{N_{f}} \frac{1}{2}\left[\left(\frac{R_{m_{k}}R_{c_{k}}}{R_{m_{k}}}\right)^{2}+\left(\frac{X_{m e s_{k}}X_{c_{k}}}{X_{m_{k}}}\right)^{2}\right]}$ (11)
where, R_{mk} and X_{mk} are successively real and imaginary parts of the measured impedance at the frequency point k.
R_{ck} and X_{ck} are successively real and imaginary parts of the computed impedance at the frequency point k.
Nf: is the number of measured points.
The inner capacitor of the UDCFRC rod will be evaluated manually using the FE axisymmetric model coupled to RLC equivalent circuit given in next sub section.
Bat algorithm optimization [18] implemented in Octave® is used to minimizing the goal function.
3.3 FE axisymmetric model coupled to RLC equivalent circuit
To take into account the capacitive effect in the UDCFRC, it will be modeled by a solid conductor in parallel with a capacitor and a dielectric losses resistor. The coil layers are also replaced by the solid conductors in parallel to capacitors and resistors.
The complete equivalent circuit model can be represented as a transformer model.
Eqns. (1) and (2) that written in the sub domain turn coil remain unchanged. Eq. (4) becomes:
$I_{k_{c}}=I_{k_{l}}=\sum_{k_{e}=1}^{N_{e k c}} I_{k_{e}}=\frac{U_{k_{l}}}{Z_{k_{l}}}$ (12)
The total current I, is obtained by dividing the input voltage by the computed impedance given by Eq. (9):
$I=\frac{U_{i n}}{Z_{c a l}}$ (13)
Eq. (6) remains unchanged but (5) is rewritten by several equation using meshes law of Kirchhoff applied to the electrical equivalent circuit illustrated in Figure 5.
To write the FE formulation in the homogenized UDCFRC rod target we replace $\sigma_{C O}^{\prime}$ will be replaced by $\sigma_{t}^{\prime}$ in Eq. (1). $\sigma_{t}^{\prime}=\frac{\sigma_{t}}{r}$ with σ_{t} is a transverse conductivity of the UDCFRC rod [S/m];
Eq. (2) is also written for the UDCFRC by replacing the conductivity as above.
The equivalent inner capacitor Cr of the UDCFRC rod is be obtained manually using the presented FE axisymmetric model coupled to RLC equivalent circuit, by the verification of the experimental first resonant point of the coil.
Figure 7 shows both the real and imaginary parts of the coil impedance measured with the LCRmeter and computed using the classical model where the capacitive effect is neglected. The both cases, coil only and coil with UDCFRC rod are considered.
Figure 7. Computed and measured impedance without considering capacitors
We can see that the classical model diverges after the frequency of 0.5MHz. Consequently, from this frequency the capacitive effect is very significant for the considered coil. Thereby, the classical model is not convenient for the frequency greater than 0.5MHz. The proposed approach in this article, allows this effect to be taken into account for all frequencies.
On the other hand, the measurements show that the presence of the UDCFRC rod affects the resonant point of the coil. Where, the resonance frequency increases and the amplitude of the impedance decreases. For the considered coil, the frequency at the resonant point for coil only is 1219 kHz and become 1223 kHz with presence of UDCFRC rod. At the resonant point, the amplitude of the impedance for coil only is 14.5 kW and become 13.8 kW with presence of UDCFRC rod.
These observed phenomena are exploited to evaluate the equivalent inner capacitor C_{r}of the UDCFRC rod, using the proposed model.
Table 2 show the equivalent capacitors evaluated using the procedure that illustrated in subsection 2.2.
After introducing all evaluated equivalent capacitors to the FE axisymmetric model coupled to RLC equivalent circuit with the presence of the UDCFRC rod, the impedance is computed for several frequencies up to 5MHz and then it is compared to the measured ones (Figure 8). We can see that the computed impedances obtained with proposed model follow exactly the measured ones. A mean error less than 2% and a maximum one of 5% are observed.
Table 2. Identified parameters of the model
C_{1} 
C_{2} 
C_{3} 
C_{12} 
C_{23} 
C_{r} 
55pF 
65pF 
75pF 
100pF 
120pF 
50uF 
R_{C1} 
R_{C2} 
R_{C3} 
R_{C12} 
R_{C23} 
R_{C23} 
21.4$\mathrm{k} \Omega$ 
21.4$\mathrm{k} \Omega$ 
21.4$\mathrm{k} \Omega$ 
50$\mathrm{k} \Omega$ 
50$\mathrm{k} \Omega$ 
$\infty$ 
Figure 8. Computed and measured impedance with considering capacitive
In order to taking into account of the capacitive effect in eddy current NDC/NDT model of UDCFRC rod, a new fast procedure is proposed. The interspire and interlayer equivalent capacitors of the air coil are firstly evaluated using the RLC equivalent circuit and the FE axisymmetric model coupled to RL equivalent circuit. Then the FE axisymmetric model coupled to RLC equivalent circuit is applied to identify the inner capacitor of the UCCRFC, by adjusting the resonant point.
To show the effectiveness of the proposed approach, the computed results are confronted to the measured ones, with and without considering capacitive effect. The comparison between the experimental and simulation results, shows a good concordance with a mean error less than 2% according to frequency.
The proposed approach can also be used in the case of the stratified CFRC plate, but the 3D model is required.
A 
magnetic vector potential, T.m^{2} 
C 
capacitance, F 
f 
frequency, Hertz 
I 
electrical current, A 
r 
radial coordinate, m 
R 
electrical resistance, Ohm 
U 
voltage, V 
v 
electrical potential, V 
X 
reactance, Ohm 
Z 
electrical impedance, Ohm 
Greek symbols 

$\sigma_{t}$ 
transverse conductivity of the UDCFRC rod, S/m 
µ 
magnetic permeability, H/m 
Subscripts 

$Z_{e k_{l}}$ 
equivalent impedance of the coil layer k_{l} 
F_{G} 
objective function or goal function 
[1] Bakis, C.E., Bank, L.C., Brown, V., Cosenza, E., Davalos, J.F., Lesko, J.J., Triantafillou, T.C. (2002). Fiberreinforced polymer composites for construction—Stateoftheart review. Journal of Composites for Construction, 6(2): 7387. https://doi.org/10.1061/(ASCE)10900268(2002)6:2(73)
[2] Basim, S., Hejazi, F., Rashid, R.S.B.M. (2019). Embedded carbon fiberreinforced polymer rod in reinforced concrete frame and ultrahighperformance concrete frame joints. International Journal of Advanced Structural Engineering, 11(1): 3551. https://doi.org/10.1007/s40091019002537
[3] Koyama, K., Hoshikawa, H., Kojima, G. (2013). Eddy current nondestructive testing for carbon fiberreinforced composites. Journal of Pressure Vessel Technology, 135(4): 041501. https://doi.org/10.1115/1.4023253
[4] Bensaid, S., Trichet, D., Fouladgar, J. (2009). Electrical conductivity identification of composite materials using a 3D anisotropic shell element model. IEEE Transactions on Magnetics, 45(3): 18591862. https://doi.org/10.1109/TMAG.2009.2012833
[5] Bensaid, S., Trichet, D., Fouladgar, J. (2015). Optimal design of a rotating eddycurrent probe—Application to characterization of anisotropic conductive materials. IEEE Transactions on Magnetics, 51(3): 14. https://doi.org/10.1109/TMAG.2014.2363174
[6] Safer, O.A., Bensaid, S., Trichet, D., Wasselynck, G. (2017). Transverse electrical resistivity evaluation of rod unidirectional carbon fiberreinforced composite using eddy current method. IEEE Transactions on Magnetics, 54(3): 14. https://doi.org/10.1109/TMAG.2017.2751962
[7] Wasselynck, G., Trichet, D., Ramdane, B., Fouladgar, J. (2010). Microscopic and macroscopic electromagnetic and thermal modeling of carbon fiber reinforced polymer composites. IEEE Transactions on Magnetics, 47(5): 11141117. https://doi.org/10.1109/TMAG.2010.2073456
[8] Wasselynck, G., Trichet, D., Fouladgar, J. (2013). Determination of the electrical conductivity tensor of a CFRP composite using a 3D percolation model. IEEE Transactions on Magnetics, 49(5): 18251828. https://doi.org/10.1109/TMAG.2013.2241039
[9] Bui, H.K., Senghor, F. D., Wasselynck, G., Trichet, D., Fouladgar, J., Lee, K., Berthiau, G. (2017). Characterization of electrical conductivity of anisotropic CFRP materials by means of induction thermography technique. IEEE Transactions on Magnetics, 54(3): 14. https://doi.org/10.1109/TMAG.2017.2742979
[10] Piche, A., Revel, I., Peres, G., Pons, F., Lazorthes, B., Gauthier, B., Cadaux, P.H. (2009). Numerical modeling support for the understanding of current distribution in carbon fibers composites. In 2009 International Symposium on Electromagnetic CompatibilityEMC Europe, pp. 14. https://doi.org/10.1109/EMCEUROPE.2009.5189695
[11] Jones, C.E., Norman, P.J., Galloway, S.J., Burt, G.M., Kawashita, L.F., Jones, M.I., Hallett, S.R. (2016). Electrical model of carbon fibre reinforced polymers for the development of electrical protection systems for moreelectric aircraft. In 2016 18th European Conference on Power Electronics and Applications (EPE'16 ECCE Europe), pp. 110. https://doi.org/10.1109/EPE.2016.7695300
[12] Meeker, D. (2010). Finite element method magnetics. FEMM, 4, 32. https://www.femm.info/.
[13] Eaton, J.W., Bateman, D., Hauberg, S. (1997). Gnu octave (p. 42). London: Network theory. https://www.gnu.org/software/octave/
[14] Bastos, J.P.A., Sadowski, N. (2003). Electromagnetic Modeling by Finite Element Methods. CRC press.
[15] Lombard, P., Meunier, G. (1993). A general purpose method for electric and magnetic combined problems for 2D, axisymmetric and transient systems. IEEE Transactions on Magnetics, 29(2): 17371740. https://doi.org/10.1109/20.250741
[16] Bensaid, S. (2006). Contribution à la caractérisation et à la modélisation électromagnétique et thermique des matériaux composites anisotropes. Ph.D. Dissertation, Nantes University.
[17] Kennelly, A.E. (1899). The equivalence of triangles and threepointed stars in conducting networks. Electrical World and Engineer, 34(12): 413414.
[18] Yang, X.S., Gandomi, A.H. (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations, 29(5): 464483. https://doi.org/10.1108/02644401211235834