Simulation of electromagnetic fields provides a means for direct observing the variation of eddy currents induced in the conductor which cannot be accomplished by analyzing the experimental signals. Through the graphical user interface (GUI) of the simulation software, the diffusion and attenuation of eddy currents can be intuitively presented in a graphical way. In this context, finite element simulation of the probe-generated electromagnetic field is conducted with the commercially available software ANSYS. The elements used in ANSYS for the simulation model, a PECT probe being lifted 1 mm above a carbon steel plate, are listed in Table 1. The applicable model for the probe coils is the ‘circuit-coupled stranded coil’ (ANSYS terminology). The carbon steel plate has a thickness of 8 mm and is assigned an electrical conductivity of 5 MS/m and a relative magnetic permeability of 300 [15].

## 3.1 Analysis of the conventional probe

A conventional probe consisting of two co-axial cylindrical coils is a structure with rotational symmetry around the axis of the coils. It is well-known that the problem of this type of probe being above a plate can be analytically solved, but herein in order to compare it with the presented probe we tackle the problem by means of numerical simulation. Since a 3D model is more difficult to generate than a 2D model and consumes much more computing time, a 2D half of the problem domain is used, as shown in Fig. 2. Details of the probe coils are given in Table 2. The left-hand two circuits modeled by ANSYS element CIRCU124 are coupled to the probe excitation coil and receiving coil, respectively. The upper circuit is an independent pulse current source of 3 A amplitude, 100 ms pulse width, 0 delay time, 4 ms rise/fall time (edge time) and 200 ms period connecting in series with a stranded coil, while the lower one is a 10 gigohm resistance connecting in series with a stranded coil. The high resistance is equivalent to an open-circuit terminal for measuring the voltage induced in the receiving coil. The right-hand part is the meshed finite element model. Mapped mesh with quadrilateral-shaped elements is applied to the plate and coils, while free mesh with triangle-shaped elements is used in the near air space; a single layer of INFIN110 elements is used in the remote air space to model the effect of far-field decay. In addition, refinements are introduced in regions where the field is changing fast. The finite element part and the circuits are connected by the current iK and electromagnetic force emfK.

Because of symmetry, the boundary condition on the axis is that the field is parallel to the axis; at the spherical outer boundary of the remote free space an exterior surface flag is imposed. The calculation interval for transient analysis is predefined by setting a time step of 0.5 ms and thus the number of steps for one calculation is 400.

Table 2

Parameters of the probe coils.

| Excitation coil | Receiving coil |

Inner diameter *φ*1 (mm) | 20 | 65 |

Outer diameter *φ*2 (mm) | 60 | 75 |

Height *h* (mm) | 40 | 5 |

No. of turns *n* | 800 | 900 |

Resistance *R* (Ω) | 2.2 | 340 |

Inductance *L* (mH) | 7.1 | 16 |

Figure 3 shows the contour plots of eddy currents in the cross section of the plate when the elapsed time, beginning from the front end of the pulse edge, is 1 ms, 2 ms, 3 ms, 4 ms, 5 ms, 8 ms, 10 ms, 12 ms, 15 ms, 30 ms and 36 ms, respectively. The varied distribution of eddy currents reveals the eddy-current diffusion evolved with time. It is observed that during the edge time (*t* = 1, 2, 3, 4 ms) the eddy current diffuses downwards and outwards until its maximum value (labeled with MX in the plots) enters the inside of the plate when *t* = 5 ms; then the outward diffusion almost stops, but the downward diffusion continues until the eddy current reaches the bottom when *t* = 12 ms; after that, the eddy current decays rapidly and diffuses outwards at a slow pace. As the radial range of the eddy-current distribution is related to the probe footprint, the footprint size can be calculated at a time after the edge time. It is worthy to mention that the time of 12 ms corresponds to the characteristic diffusion time of the transient field flowing out the conductor which can be calculated using \({\tau }_{\text{D}}=\mu \sigma {d}^{2}/{{\pi }}^{2}\), where \(\mu\), \(\sigma\) and \(d\) are the permeability, conductivity and thickness of the conductor, respectively [19]. Substitution of the parameters of the carbon steel plate into the formula yields \({\tau }_{\text{D}}\approx\) 12.2 ms, which proves the accuracy of finite element modeling.

Following ref. [15] and without loss of generality, the probe footprint is defined as the area within which the eddy current density is above 30% of the maximum eddy current density in the cross section of the carbon steel plate at the characteristic diffusion time (here, *t* = 12 ms). From Fig. 3, the footprint radius of this conventional probe is calculated as 63 mm, and the maximum eddy current density is read as 55248 A/m2.

## 3.2 Analysis of the presented probe

The presented probe is also rotationally symmetric about its axis and therefore a 2D finite element model is established as well, as shown in Fig. 4. The model is identical to the former one except that the numbers of the probe excitation coils and the circuits coupled to them are increased to four. The four excitation coils are of the same height and thickness (outer radius minus inner radius) and each one has a number of turns of 200. The independent pulse current sources Is,*i* which are respectively applied to the four coils are configured with the same amplitude of 3 A, pulse width of 100 ms, edge time of 4 ms and period of 200 ms but different delay time. Other settings including the meshing control, boundary condition and calculation interval are the same as those used in modeling the conventional probe.

The following simulations are aimed at analyzing the influence of the amount of delay time of the pulses and the applying sequence of the delayed pulses on the probe footprint. As the characteristic diffusion time of the eddy current field induced by the first pulse is calculated as 12 ms, two aspects of factors should be considered on the determination of the time delay of the succeeding pulse. On the one hand, a large delay time is beneficial for altering the distribution of the preceding eddy current which is decaying rapidly with time; on the other hand, the increase of the delay time reduces the acquisition time for the receiving coil to measure the variation of the resultant eddy current. In this context, the initial delay time of the second pulse is set as 10 ms. The join of the second pulse will introduce a new round of eddy current diffusion and the updated characteristic time is selected as the delay time of the third pulse, and the same determination method goes for the fourth pulse.

Based on the described procedure, the delay time of the pulses are initially set as 0 ms, 10 ms, 20 ms and 29 ms. Figure 5 shows the diffusion of eddy currents induced by the presented probe consisting of four excitation coils which are, respectively, from outermost to innermost, driven by the four pulses. It is evident that every time the new delayed pulse is introduced, the pattern of the eddy current distribution is altered and the center of the eddy current with the maximum density is moved inwards. After the successive regulating steps at the moments of 10 ms, 20 ms and 29 ms, the final eddy current field presents a smaller distribution on the cross section of the plate; in the radial direction, the MX labeled zone in the plot at *t* = 36 ms is nearly moved to half of the coordinate of that at *t* = 5 ms. The footprint radius calculated at the characteristic diffusion time of the final eddy current field is 54 mm, which is considerably reduced compared with that of the conventional probe. In addition, due to the sequential excitation, the density of the final eddy current field is greatly improved; the eddy current at *t* = 36 ms has the maximum density 28776 A/m2 which is much stronger than the conventional probe-induced eddy current at the same time which has the maximum density only 7262 A/m2 (see Fig. 3).

A. Effect of the sequence of excitation

Since there are four probe sub-coils and accordingly four pulses of different delay time *t*d,*i* (*i* = 1, 2, 3, 4), varying the sequence of applying the pulses to the sub-coils might cause different patterns of eddy current distribution, thereby affecting the probe footprint size. In total, there are 24 possible sequences of excitation, as listed in Table 3. Simulations for all the cases are conducted to obtain the probe footprint sizes. Figure 6 plots variation of the footprint radius against the excitation sequence number. It is revealed that the probe footprint radius is reduced as the delay time of the excitation pulse in the outermost sub-coil, namely *t*d1, decreases, and vice versa. The last sequence which respectively applies pulses having delay time of 0, 10, 20 and 29 ms to the four sub-coils yields a minimum footprint radius. The result corroborates the excitation scheme described in the above section. To conclude, to maximize the reduction in probe footprint, it is a necessity to drive the outermost sub-coil with a zero-delay pulse and the inner ones with pulses having gradually increasing delay time.

Table 3

Possible sequences of excitation pulses applied to the probe sub-coils.

| Excitation sequence number |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

Delay time *t*d1 (ms) | 29 | 29 | 29 | 29 | 29 | 29 | 20 | 20 | 20 | 20 | 20 | 20 |

Delay time *t*d2 (ms) | 20 | 20 | 10 | 10 | 0 | 0 | 29 | 29 | 10 | 10 | 0 | 0 |

Delay time *t*d3 (ms) | 10 | 0 | 20 | 0 | 20 | 10 | 10 | 0 | 29 | 0 | 29 | 10 |

Delay time *t*d4 (ms) | 0 | 10 | 0 | 20 | 10 | 20 | 0 | 10 | 0 | 29 | 10 | 29 |

| Excitation sequence number |

13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

Delay time *t*d1 (ms) | 10 | 10 | 10 | 10 | 10 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |

Delay time *t*d2 (ms) | 29 | 29 | 20 | 20 | 0 | 0 | 29 | 29 | 20 | 20 | 10 | 10 |

Delay time *t*d3 (ms) | 20 | 0 | 29 | 0 | 29 | 20 | 20 | 10 | 29 | 10 | 29 | 20 |

Delay time *t*d4 (ms) | 0 | 20 | 0 | 29 | 20 | 29 | 10 | 20 | 10 | 29 | 20 | 29 |

B. Effect of the delay amount

Since the sequentially induced eddy current fields are superposed at last, the delay amount between adjacent excitation pulses can affect the final eddy current field and thus the probe footprint as well. The behavior of eddy currents can be characterized by the voltage signals induced in the probe receiving coil, namely the PECT signals. For a specimen made of ferromagnetic material, the time-dependent PECT signals are usually plotted in a semi-logarithmic graph to reveal the small differences in amplitude of the late-phase signals [15]. Figure 7 shows the simulated PECT signals when the delay amount of excitation pulses applied to the adjacent sub-coils is changed. The outermost sub-coil is applied by a zero-delayed pulse. The delay amount of other applied pulses is represented as ∆*t*d,*i* which equals to *t*d,*i*+1 - *t*d,*i* (*i* = 1, 2, 3). For comparison, the conventional PECT probe signal is also presented. Compared with the reference signal, the presented probe’s signals exhibit three extra bumps at the early phase which correspond to the subsequent excitation pulses after the initial one, while the late-phase signals are straight lines, just like the reference signal. Fortunately, for ferromagnetic material PECT signal, usually only the late-phase signal is used to obtained the signal features such as the decay rate for defect quantification. Due to the subsequent pulses, the early-phase signal which is supposed to experience a continuous decay is increased, making the signal have a larger amplitude at the late phase than the reference signal. This is of great practical significance because the experiment signal will have a better signal-to-noise ratio (SNR) which is beneficial for accurate extraction of the decay rate (i.e., the slope of the straight part). Apart from that, as the delay amount ∆*t*d,*i* increases, the amplitude of the late-phase signal increases but the beginning of the signal is delayed, which means that the interval for feature extraction is shortened. Therefore, in practical applications, a compromise should be made to determine the delay amount. On the premise that the interval of the late-phase signal is of enough length and SNR for accurate extraction of the signal feature, the delay amount of the excitation pulses can be designed as long as possible.

Figure 8 plots the variation of the probe footprint with the delay amount of the excitation pulses. The footprint radius shows a linear decrease as the delay amount increases. The trend is attributed to the fact that the larger the delay amount is, the smaller is the influence of the already-diffused eddy current field on the resultant field. On the other hand, the long time of delay will result in a relatively weak field and thus a smaller signal amplitude. The delay amount is supposed to have a limit at which the eddy current induced by the preceding excitation pulse is vanished. To demonstrate that, a group of simulations are carried out by gradually increasing the delay amount and for comparison, a reference group of simulations using the same model but with the sub-coil 1 at null excitation are also conducted. Figure 9 shows the result. It is in evidence that when the delay amount is small, the former signal (solid) has a visibly larger amplitude than the reference signal (dashed), but as the delay amount increases the difference becomes smaller. Specifically, when the delay time *t*d2, *t*d3 and *t*d4 respectively reaches 21, 40 and 58 ms, the two signals almost coincide. At this time, the contribution of the first pulse applied to the sub-coil 1 is negligible, which indicates the limit of the delay amount. Therefore, to make all the sub-coils contribute to the PECT signal, the delay amount should be within the limit.