Active Damped Oscillation Calibration Method for Receiving Coil Transition Process Based on Early Acquisition of Pulsed Eddy Current Testing Signal

Abstract

As a common signal sensing device in pulsed eddy current detection, coil sensors often have parameter offset problems in practical applications. The error in the receiving coil parameters will have a great impact on the early signal. In order to ensure the accuracy of the early signal, this paper first analyzes the response characteristics of the receiving coil and the influence of the coil parameters on the accuracy of signal deconvolution and establishes the mathematical relationship between the response signal and the characteristic parameters, and between the characteristic parameters and the receiving coil parameters under active underdamped oscillation. Subsequently, the parameter feature extraction errors under different state switching capacitors were compared through simulation analysis, the state switching capacitor value was determined, and the receiving coil parameter solution method based on the Levenberg–Marquardt (LM) algorithm was determined based on the parameter feature extraction results. The experimental results demonstrate that the proposed method achieves a capacitance estimation error of just 0.0159% and an inductance error of 0.158%, effectively minimizing early signal distortion and enabling precise identification of receiving coil parameters.

1. Introduction

As a magnetic field sensing instrument, the coil sensor is a common pulse eddy current response signal detection device. The pulse eddy current method is an effective method for detecting underground low-resistance conductors using the principle of electromagnetic induction [1,2,3]. Due to its wide detection spectrum, no need to contact the object to be detected, and rich time domain characteristics contained in the signal, it is widely used in the detection of minerals, pipelines, grounding bodies of power equipment, and underground spaces in cities [4,5,6,7].

During eddy current detection, the secondary magnetic field signal excited by the target object is captured by the receiving coil sensor [8]. For pulsed eddy current shallow detection, when performing time domain analysis on the signal, the shallow characteristics are often reflected in the early signals, so it is crucial to ensure the early accuracy of the response signal [9,10,11,12]. However, the electromagnetic coupling path of the receiving system will introduce signal distortion—the collected signal can be regarded as the output response of the original induced signal after the transfer function is applied. In order to restore the true magnetic field change characteristics, the collected signal needs to be deconvolved to eliminate the transient response characteristics of the receiving system [13]. This process is called receiving coil system calibration [14]. The deconvolution accuracy is determined by the transfer function of the system itself. When the system parameters are offset, the deconvolved signal will change because of the error associated with the system parameters [15].

Signal deconvolution calculation is an important process to remove the transient process of the receiving coil and ensure signal accuracy [15,16]. The frequency response method serves as a fundamental approach for coil system characterization. This technique employs sinusoidal excitation signals across discrete frequencies within a target spectrum. At each calibration frequency, synchronous measurement of input stimulus and steady-state output enables comprehensive amplitude–phase characterization. Subsequent curve-fitting of these experimental datasets can be used to derive the coil’s transfer function, establishing its frequency–domain response profile [17,18]. However, changes in the environmental medium or structural deformation will affect the calibration accuracy, and the calibration of the receiving system needs to be updated regularly. In addition, the frequency response method has high requirements for the uniformity of the calibration magnetic field, and the field source has poor versatility, making it difficult to meet the needs of on-site calibration [13]. Reference [16] proposed a receiving coil parameter calibration method based on time domain zero-state response, which transforms parameter estimation into a nonlinear programming problem, and combines the descent direction and step size strategy to perform parameter inversion through curve fitting. This approach eliminates reliance on the input signal for calibrating the coil parameters; however, the precision of the fitting has a direct impact on the final outcome. Reference [13] introduced a calibration method based on time domain feedback, utilizing the solution error from the induced electromotive force of the receiving coil as the feedback signal. It employs an exponential decay current as the calibration signal and integrates a τ value conversion algorithm to identify and correct errors. The calibration file is refreshed by reducing the non-steady-state interval, which effectively eliminates reliance on a uniform magnetic field. Additionally, the straightforward circuit design makes it ideal for on-site calibration of the receiving coil.

The above methods can determine the receiving coil parameters by analyzing the entire response signal of the receiving coil. However, the accuracy of the solution is affected by the error in the response signal throughout the entire process. This paper uses an active damped oscillation method to relate the coil parameters to the characteristic parameters of the response signal. Furthermore, through derivation, the solution of the coil parameters is transformed into the problem of solving nonlinear equations. This method is simpler and suitable for field calibration. Among the various existing numerical methods for solving nonlinear equations, Newton’s method is the most basic and important method for solving nonlinear equations [19,20]. In Reference [7], the Newton method is used to solve the nonlinear equation. However, due to the large difference between inductance and capacitance parameters in this paper, the inductance can be solved under a small error by Newton’s method, but the error of capacitance fluctuates greatly, and the overall stability of the solution needs to be improved. For this reason, it can be converted into a nonlinear least-squares problem. The Gauss–Newton method has fast convergence when the column is of full rank, but when the nonlinearity of F(x) is very large, the convergence speed is very slow or does not converge. The Levenberg–Marquardt method constructs a robust Newton-type iteration step by introducing a forced positive definite strategy, taking into account both numerical stability and second-order convergence, and can effectively deal with Jacobian singular or ill-conditioned problems [21].

To address the early errors in the response signal caused by coil parameter offset, the large order-of-magnitude differences between the inductance and distributed capacitance parameters to be solved during the coil parameter solution process, as well as their varying degrees of influence on the solution equation, and to prevent imbalanced optimization directions caused by excessively large parameter iteration steps, this paper analyzes the impact of coil parameter offset on the early signal accuracy based on the response characteristics of the coil to be calibrated. Subsequently, a physical mapping relationship between the underdamped response signal characteristics and the coil parameters to be solved is established using an active underdamped oscillation method. The equations are then solved using the LM method with a damping factor and an indentation factor. This method avoids large single-parameter errors in the solution of the inductance and distributed capacitance parameters, thereby accurately solving the receiving coil parameters. Finally, simulations and experiments verify that this method, compared with other nonlinear equation-solving methods, accurately solves the coil parameters to be calibrated, prevents early signal distortion caused by coil parameter offset, and facilitates coil sensor calibration.

2. Analysis of Dynamic Characteristics of Receiving Coil

2.1. Analysis of Receiving Coil Response Characteristics

Figure 1 shows the equivalent model of the excitation and receiving coils of the pulsed eddy current and the state switching circuit corresponding to the active damped oscillation. R₀ is the equivalent resistance, L is the inductance, C is the distributed capacitance, respectively, and R_b is the damping matching resistor. At this time, the transfer function, damping ratio ζ, and natural frequency ω_n of the receiving coil are:

$$\begin{array}{c}H(s)={\displaystyle \frac{1}{LC[s^2+s(\frac{R_0}{L}+\frac{1}{R_{b}C})+\frac{R_0+R_b}{R_{b}LC}]}}\\ \zeta ={\displaystyle \frac{R_b{R}_{0}C+L}{2\sqrt{R_{b}LC(R_b+R_0)}}}\\ {\omega }_n=\sqrt{{\displaystyle \frac{R_b+R_0}{R_{b}LC}}}\end{array}$$

(1)

From (1), we can see that when the receiving coil acquires the induced voltage v(t) caused by the change in the magnetic field, the actual signal u(t) obtained by the acquisition module is the signal after v(t) is acted upon by H(s). u(t) is the time domain convolution of v(t) and h(t). u(t) and v(t) can be converted to each other through the process shown in Formula (2) [22,23].

$$\begin{array}{c}u(t)=v(t)\ast h(t)={\mathcal{F}}^{-1}\left[V(\omega )\times H(\omega )\right]\\ v(t)={\mathcal{L}}^{-1}(V(s))={\mathcal{L}}^{-1}\left({\displaystyle \frac{\mathcal{L}u(t))}{H(s)}}\right)\end{array}$$

(2)

2.2. Analysis of the Accuracy of Receiving Coil Deconvolution

In order to discuss the relationship between the deconvolution result of the response signal and the true result under different receiving coil parameters, as well as the influence of parameter deviation on the deconvolution signal and the size of the signal error, the receiving coil parameters shown in

Table 1. Receiving coil parameters.

Receiving Coil Parameters	L (mH)	C (nF)	R (Ω)
Initial Value	10	0.25	2.6
True value	10.09	0.2516225	2.6

are defined, where the true value is obtained through actual measurement, and the initial value is offset based on the true value as the known initial value of the coil parameter. First, the step response of the receiving coil is calculated by (1), and then the step signal is deconvolved to obtain the corresponding input signal v(t) through the receiving coil transfer function under different parameter changes. In order to analyze the influence of parameter error on deconvolution under certain changes in different parameters, the early mean error MSE of the signal, the maximum error Max_error of the signal, and the steady-state error Steady-state_error of the signal shown in (3) are defined, respectively. The waveform and error of the receiving signal after deconvolution under different parameter changes are shown in Figure 2.

$$\begin{array}{c}\begin{array}{cc}MSE=mean(v(t)-\epsilon (t))& 0\le t\le 10us\end{array}\\ Max\_error=\max \left|v(t)-\epsilon (t)\right|\\ \begin{array}{cc}\mathrm{Steady}-\mathrm{state}\_\mathrm{error}=v(t)-\epsilon (t)& t\to \infty \end{array} \end{array}$$

(3)

As shown in Figure 2a–c, in terms of waveform, the parameter offset of inductance and capacitance will cause a large distortion in the early waveform of the signal, and the change of capacitance parameters will even cause oscillation in the early signal, compared with inductance. The offset of resistance parameters also has an impact on the early signal, but the impact is smaller. The impact of resistance parameters on the signal is mainly reflected in the steady-state value of the signal. As shown in Figure 2d–f, in terms of the early mean error, when the absolute error of the inductance parameter reaches 20%, the early mean error of the signal reaches 0.0731. Compared with the 0.016 of the capacitance parameter deviation under the same absolute error, the early signal error caused by inductance is larger; in terms of the maximum error point, the maximum error values caused by the offset of inductance and capacitance parameters are the same, but both are greater than the maximum error caused by resistance. Therefore, the inductance and capacitance parameters have a greater impact on the early error of the signal, and the resistance mainly affects the steady-state error. In order to ensure the accuracy of the collected signals and the precision of the early signals, it is necessary to accurately identify the various parameters of the receiving coil system, avoid signal distortion through the coil transition process, and ensure the accuracy of the signal and the accuracy of post-processing.

2.3. Active Underdamped Oscillation and Signal Characteristics Analysis

To suppress signal oscillations in the received waveform, impedance matching is typically employed to bring the system into a critically damped state [13,16]. The corresponding matching resistance under critical damping conditions can be calculated using the following expression:

$$R_b={\displaystyle \frac{L}{R_{0}C\pm 2\sqrt{LC}}}$$

(4)

In order to avoid underdamped oscillation of the signal because of uncertain factors, the system is generally kept in a slightly overdamped state. By actively connecting S₁ and S₂ in Figure 1, Figure 3 illustrates that the approach of enhancing the matching resistance and utilizing a parallel state switching capacitor can transition the system from a mildly overdamped condition to an underdamped condition. At this time, when the transmitting coil applies a ramp input, the receiving coil will sense the step signal ε(t). At this time, the overshoot σ% and the adjacent peak time ∆t_p corresponding to the response signal obtained by the acquisition module are expressed as:

$$\begin{array}{c}\sigma \%={\displaystyle \frac{y_m-y_s}{y_s}}=e^{\left(-{\displaystyle \frac{\zeta \pi }{\sqrt{1-{\zeta }^2}}}\right)}\times 100\%\\ \Delta t_p={\displaystyle \frac{\pi }{{\omega }_n\sqrt{1-{\zeta }^2}}}\end{array}$$

(5)

Equation (5) shows that Δt_p and σ% of the response signal depend solely on the system’s natural frequency and damping ratio. Therefore, the adjacent peak time and overshoot are obtained by extracting the characteristics of the response signal, and then the natural frequency and damping ratio of the system can be obtained by solving the equation. From (1), there is a nonlinear relationship between the natural frequency and damping ratio and the parameters to be identified, L and C. Therefore, the method of solving nonlinear equations can be used to solve for L and C, thereby completing the parameter identification of the receiving coil and ensuring the accuracy of the early signal.

3. Accept Coil Parameter Solution

3.1. Response Signal Feature Extraction and Analysis

According to the actual parameters corresponding to Table 1, the circuit simulation analysis was performed through Multisim, and the early signal of the receiving coil induced voltage under the parallel connection of different state switching capacitors and series state switching resistors was obtained as shown in Figure 4.

As illustrated in Figure 4, under ramp excitation of the transmitting coil, the receiving coil is theoretically expected to detect a step response. However, the presence of the coil’s inherent resonant frequency leads to an early-stage overshoot. It can be seen from the local amplification that the method of increasing the damping resistance has minimal effect on the improvement in the overshoot signal in the early stage, and the overshoot peak decreases minimally. By increasing parallel capacitance, the effect of the overshoot on the signal distortion can be significantly reduced. When the parallel state switching capacitance is greater than 5 nF, the early overshoot is very small. Therefore, this paper adopts the method of parallel state switching capacitors to make the system transition to the underdamped state.

It can be seen from the underdamped oscillation response waveform in Figure 5 that with an increase in the parallel state switching capacitor, the amplitude and rise time of the signal also increase, and the corresponding oscillation times also increase, which is consistent with the law in Figure 3. Moreover, there is almost no overshoot in the early stage of the signal. The adjacent peak time and overshoot of the response can be obtained by formula (5) to obtain the characteristic parameters of the response signal. In order to accurately extract the characteristic parameters, the error between the extracted value of the characteristic parameters of the curve under different state switching capacitors and the calculated value and the true value is shown in Figure 6.

From the characteristic parameter extraction and calculation error and capacitance curve in Figure 6, it follows that under the parameter design in Table 1 of this article, when the state switching capacitor value is 12.8 nF, the overshoot and peak time interval errors obtained by feature extraction of the underdamped oscillation response signal are the smallest, and the errors of the damping ratio and natural frequency calculated by the characteristic parameters are also the smallest. Therefore, the state switching capacitor selected in this article is 12.8 nF.

3.2. Parameter Solution Based on the Levenberg–Marquardt Algorithm

To determine the parameters of the receiving coil, the known characteristic features of the response signal are utilized, whereby the parameter identification problem is transformed into solving a system of two nonlinear equations.

$$\mathbf{F}=\left[\begin{array}{c}{f}_1\\ f_2\end{array}\right]=\left[\begin{array}{c}{\omega }_n-\sqrt{{\displaystyle \frac{R_b+R_0}{R_{b}LC}}}\\ \zeta -{\displaystyle \frac{R_b{R}_{0}C+L}{2\sqrt{R_{b}LC(R_b+R_0)}}}\end{array}\right]$$

(6)

When using a coil as a receiving system for signal acquisition, the parameters of the coil are often measured during initial use, and the measured values can be used as initial values for solving nonlinear equations, while the true values produce small deviations near the initial values. Under the premise of knowing the initial value, the classic Newton method can be used to solve the nonlinear equation. Among them, the Levenberg–Marquardt (LM) method is one of the most famous nonlinear methods proposed on the basis of Newton’s method [24]. The overall flow diagram is shown in Figure 7. The objective function is to minimize the residual sum of squares:

$${\min }_{\mathbf{x}}{\displaystyle \frac{1}{2}}\Vert \mathbf{F}(\mathbf{x}){\Vert }^2={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^2{f}_i^2}(\mathbf{x})$$

(7)

In each iteration, the Levenberg–Marquardt (LM) method is calculated with a step size of [25]:

$$d_k^{LM}=-{(J_k^T{J}_k+{\lambda }_{k}I)}^{-1}{J}_k^T{F}_k$$

(8)

Among them, F_k = F(x_k), J_k = F′(x_k) is the Jacobian matrix, λ_k = ‖F(x_k)‖^δ, δ∈[1, 2] is the LM parameter. λ_k is a non-negative constant. The LM method overcomes the problem of singular or near-singular J_k when applying the Newton method by introducing a non-negative parameter λ_k. In addition, it also prevents the problem of long step length ‖d_k‖. During the solution process, the step length can be adjusted by dynamically adjusting λ_k by evaluating the effect of the current step length.

As can be seen from Figure 8, the search directions corresponding to the inductance parameters and capacitance parameters when solving the natural frequency and damping ratio equations are consistent. However, when solving the natural frequency equation and the damping ratio equation, the two parameters have different sensitivity responses to the equations. The inductance parameter has a greater impact on the solution and the error of the objective function. Therefore, when solving nonlinear equations, it is easy to have a large inductance step, which affects the identification results of the capacitance parameters and causes an imbalance in the optimization direction. Therefore, it is necessary to adjust the solution step for the two parameters.

The step length indentation factor can solve the problem of updating the step size imbalance and enhance the capacitance sensitivity. Therefore, in order to avoid the excessive error of capacitance parameters caused by the over-fitting of inductance parameters in the process of parameter identification, the step size indent factor is introduced, and the step size in the iterative process is adjusted to:

$${\mathbf{x}}_{k+1}={\mathbf{x}}_k+S·d_k$$

(9)

$$S={[s_L,s_c]}^T$$

(10)

where S is the step size reduction factor, and S_L and S_C are the step size reduction parameters of L and C, respectively. The identifiability of the capacitance C is significantly improved by the current inductance parameter step size method.

Based on the parameter values shown in Table 1, the algorithm was implemented in the MATLAB environment using MTLAB R2022b, on a Windows 10 computer with an Intel Core i5 13400F CPU. The error convergence curves, solution parameters, and performance comparisons of the four solution methods during the parameter identification process are shown in Figure 9 and Figure 10, and

Table 2. Solution parameters of the three methods.

Parameters of the Solution	Iterations	L-Error Improvement	C-Error Improvement	L Relative Error	C Relative Error
BFGS	16	38.2765	8.1584	0.00231	7.84512 × 10−4
Gauss–Newton	12	8.4596 × 103	1.2764	1.05443 × 10−5	0.00502
LM+Shrinkage	20	59.4516	42.2612	0.0015	1.51457 × 10−4
LM	13	15.4721	2.1277	0.00577	0.00301

, respectively. The BFGS method, based on the Newton method, the LM algorithm, the Gauss–Newton method, and the LM algorithm with a step size reduction, was used.

As can be seen from Figure 9 and Figure 10 and Table 2, in terms of iteration speed, the Gauss–Newton method has the smallest number of iterations for solving nonlinear equations, but for the nonlinear solution problem in this paper, the three methods are less time-consuming; in terms of error improvement, the Gauss–Newton method has the best improvement in inductance error and solution accuracy, but the capacitance solution error is large, while the LM+Shrinkage method has the best improvement in capacitance error and solution accuracy, and also has a good effect in terms of inductance solution accuracy; among the three solution methods, the LM algorithm will increase the number of solution steps after adding the step size shrinkage factor, but the accuracy of the capacitance parameters has been improved. Since there are only two equations in this problem, the solution times for different methods are similar. Starting from the perspective of parameter solution accuracy, considering the accuracy of the capacitance solution and inductance solution comprehensively, the LM algorithm plus step size shrinkage factor method is selected to solve the receiving coil inductance and capacitance.

4. Experimental Testing

To confirm the correctness of the receiving coil parameter solution method based on the LM algorithm proposed in this paper, an experimental schematic diagram, as shown in Figure 11, was built based on the circuit model in Figure 1. The experimental schematic diagram is shown in Figure 11. The real parameters of the coil are shown in Table 1. By actively changing the dynamic response characteristics of the coil, the applied excitation, the obtained response signal waveform, and the corresponding characteristic parameters are shown in Figure 12 after the coil is in an underdamped oscillation state.

After the ramp current is applied to the excitation coil, the response voltage signal of the receiving coil is shown in Figure 12. The corresponding values of the response signal peak value, adjacent peak time, steady-state value, and overshoot obtained by extracting the characteristic parameters of the response signal are shown in the annotations of Figure 12.

After obtaining the characteristic parameters, the corresponding iterative process and the convergence effect of inductance and capacitance errors obtained after obtaining the nonlinear solution to Equation (6) based on the LM plus step size reduction factor are shown in Figure 13a.

Based on the characteristic parameters extracted from the signal, the receiving coil parameter model is solved using the Levenberg–Marquardt (LM) algorithm. The iterative process converges after 22 steps. By applying ramp excitation and performing feature extraction on the underdamped oscillation waveform, followed by parameter identification through the LM-based method proposed in this study, the relative error in capacitance estimation is reduced to 0.0159% (E_C), while the inductance estimation error is 0.158% (E_L). This demonstrates the method’s effectiveness in accurately identifying the parameters of the receiving coil. Compared with the capacitance solution error of 0.038% and inductance solution error of 0.17% corresponding to reference [7], the method proposed in this paper greatly improves the solution error of capacitance parameters while ensuring the solution error of inductance parameters. At the same time, compared with the parameter solution effect in Table 2, it can be seen that in actual applications, environmental noise will affect the feature extraction accuracy of the response signal and thus affect the final solution accuracy. Therefore, in actual application, it is necessary to avoid the influence of environmental noise on the response signal.

Table 3. Multiple test results.

Test Results	Iterations	E_C (%)	E_L (%)
Test 1	22	0.0161	0.16
Test 2	22	0.0154	0.154
Test 3	21	0.0173	0.165
Test 4	23	0.0148	0.151
Test 5	22	0.016	0.159
Test 6	22	0.0157	0.161
Mean	22	0.015883	0.158333
Standard deviation	0.632456	0.000838	0.005046

shows the results of multiple measurements and calculations of the coil under ambient noise. Table 3 shows that the mean values of the iteration number, inductance error, and capacitance error for the multiple measurements are 22, 0.1583%, and 0.01588%, respectively. The standard deviations of the inductance and capacitance solutions are 0.000838 and 0.005046, respectively. These test results demonstrate the stability of the measurement method.

In order to verify the influence of parameter error on the accuracy of signal deconvolution under this solution accuracy, the error analysis of the deconvolution signal machine under the solution accuracy is shown in Figure 13b. It can be seen from Figure 13b that under this solution accuracy, the inductance and capacitance parameters have a limited influence on the deconvolution solution. The maximum error caused by capacitance is only 1.6 × 10⁻⁵, and the maximum error caused by inductance is only 1.58 × 10⁻³. Under this level of solution accuracy, the pulse eddy current detection signal can be accurately restored after deconvolution, which avoids the early signal distortion caused by parameter offset.

5. Conclusions

In order to solve the problem that the deconvolution of the induced voltage signal is biased, and the early signal waveform is affected by the offset of the coil parameters during the application of the pulse eddy current receiving coil:

(1) This study formulates a nonlinear equation that describes the relationship between the characteristics of underdamped oscillations in the receiving coil and the parameters of the coil itself. Additionally, it introduces a method for actively damping these oscillations and correlates the solutions for the coil parameters with the solutions derived from the nonlinear equation through the process of feature extraction.

(2) Based on the background of coil parameter offset and known initial parameters, in order to avoid the influence of different magnitudes of inductance and capacitance parameters and different sensitivity responses to the equation on parameter identification, the LM algorithm plus step size retraction factor method based on the improved Newton method is used to iterate the nonlinear equation. The three methods of the Gauss–Newton method, the LM algorithm, and the LM algorithm plus step size retraction factor are compared and analyzed to solve the receiving coil parameter equation. The comparison shows that the LM algorithm plus step size retraction factor method can effectively improve the solution accuracy of capacitance parameters.

(3) Experimental verification shows that, following feature extraction and parameter equation solving, the identification errors for capacitance and inductance can be reduced to 0.0159% and 0.158%, respectively, enabling precise characterization of the receiving coil parameters.

(4) Based on the receiving coil parameters obtained in this paper, an accurate detection signal can be obtained by deconvolution, and the parameters of the device can be calibrated in real time because of the simple circuit and method. However, since the input signal needs to be applied by the excitation coil, it is mainly suitable for the central loop structure. In the future, the application range of this method can be further expanded, and the coil parameter calibration method under the excitation signal applied only to the single structure of the receiving coil can be studied.