#### 3.1. Mathematical Basics

The sensors developed in this work use the eddy current principles. Their electromagnetic parts are, basically, RLC series circuits. An RLC series circuit is an association in series of a resistor, an inductor, and a capacitor. In this case,

L is the inductance of a sensor coil,

L_{c},

R is the sum of the coil resistance

R_{c} and any other additional resistances

R_{a}, and

C is the capacitance of an external capacitive array,

C_{a}, placed in series with the coil.

Figure 2 shows an electrical equivalent circuit for the sensor.

The following simplifying assumptions limit the use of this equivalent circuit: (1) the capacitors do not present electrical resistances; in other words, only displacement currents are considered within the capacitors; (2) if external resistors are added to the sensor, they do not present both inductive and capacitive effects and only conduction current will be present in these resistors; (3) the operational frequency of the sensors will be in the range of 7.5–15 kHz, so no capacitive effects will be considered in the sensor coil; (4) the model does not consider parasitic capacitances; (5) if contact resistances are known, they can be added to the model.

**Figure 2.**
The electrical equivalent circuit of the ECT sensor for reinforcement inspections.

**Figure 2.**
The electrical equivalent circuit of the ECT sensor for reinforcement inspections.

In

Figure 2, an input

V_{source}, with angular frequency

ω = 2πf, is the voltage applied to the terminals of the RLC circuit,

V_{cap} is the voltage at the capacitive array terminals, and

i_{sensor} is the loop current in the sensor. Initially, the analysis of the electrical circuit of

Figure 2 will be done for the no-load condition (no ferromagnetic material placed under the sensor). The second law of Kirchoff allows to express, for the voltage at the source terminals:

and for the current:

Before proceeding with the analysis of the equivalent circuit it is necessary to carry out an analysis of the behavior of the electromagnetic field in the region of interest, with the presence of a steel bar. As the sensor is fed by a time-varying voltage at its terminals, the resultant time-varying electromagnetic field will induce eddy current loops in the conducting body. The magnitude and behavior of the eddy current will depend on the magnetic flux density distribution, the metal conductivity, metal permeability, and the electrical frequency. The induced eddy currents will create a counter time-varying magnetic field that will disturb the original field. As an illustration,

Figure 3 shows a field mapping for a 900 turn coil, fed by a voltage source of 5.0 V

_{rms}, with a frequency equal to 8.05 kHz. The non-commercial FEMM software [

25] was used to perform the 2D simulations.

Figure 3a shows the flux distribution in the region under the coil, without the presence of the steel bar.

Figure 3b shows the flux distribution in the region under the coil with the presence of the steel bar, and

Figure 3c shows details of the flux within the bar and in the region surrounding it.

Inspecting the maps of

Figure 3, it is possible to see some expected phenomena from the electromagnetic theory: (1) the high permeability of the steel distorts the lines of flux around the bar; as is known, in a magnetic/non-magnetic interface, the lines of flux are perpendicular to the interface in the non-magnetic medium; (2) the high conductivity of steel does not permit the penetration of the magnetic flux within the bar; the flux density (and consequently the eddy currents) is confined to a tiny region around the interface between the bar and the surrounding non-conductor medium; in fact, the skin depth for the steel bar (calculated using the formula

$\delta =1/\sqrt{\mu f\sigma}$), considering a relative permeability,

${\mu}_{r},$ equal to 1000, electric conductivity,

$\sigma ,$ equal to 5.9 MS/m [

16], and frequency

$f$ equal to 8.05 kHz is about 0.13 mm, is very consistent with the figures in

Figure 3. Finally, magnitude and phase of the flux density are disturbed point by point.

Table 1 shows the components of the magnetic induction vector at the center of the red line in

Figure 3b.

**Figure 3.**
Simulation of the flux distribution under an ECT sensor. (**a**) Without the steel bar; (**b**) With the steel bar; (**c**) Details of the flux in the steel bar and region around it.

**Figure 3.**
Simulation of the flux distribution under an ECT sensor. (**a**) Without the steel bar; (**b**) With the steel bar; (**c**) Details of the flux in the steel bar and region around it.

**Table 1.**
2D components of the magnetic induction vector, with and without the steel bar.

**Table 1.**
2D components of the magnetic induction vector, with and without the steel bar.
| Bx (Wb/m^{2}) | By (Wb/m^{2}) |
---|

Without the bar | −3.665 × 10^{−6} − j6.346 × 10^{−7} | 2.526 × 10^{−4} − j1.224 × 10^{−6} |

With the bar | −4.240 × 10^{−8} − j5.172 × 10^{−11} | 2.522 × 10^{−4} + j1.815 × 10^{−7} |

As a result of the above discussion, the parameters of the electrical equivalent circuit will be affected in the following way: (1) ohmic losses will occur in the steel bar; this fact can be taken into account by adding a resistance ∆R

_{e} in the equivalent circuit; (2) The coil inductance is no longer the original value,

${L}_{c}$. A new effective inductance

${L}_{ef}$ will be defined as:

where

$\Delta {L}_{e}$ is a little change of the coil inductance, caused by the changes in the original magnetic field, by the presence of eddy currents in the steel bar.

The voltage at the sensor terminals will be now be expressed as:

or:

The sensor is designed to operate at its resonant frequency at no load. In other words, the inductive and capacitive reactances within the parentheses in Equation (5) will have the same value. or:

and:

At the resonant frequency, the current in the sensor is:

which after some algebraic operations are expressed as the sum of a real and an imaginary parts:

The voltage at the capacitor (after some algebraic operations) is:

The phase angle for the sensor current, ${\varphi}_{c},$ is ${\mathrm{tg}}^{-1}(-\omega \Delta {L}_{e}/\left({R}_{c}+{R}_{a}+\Delta {R}_{e}\right))$, and phase angle for the voltage at the capacitor, ${\varphi}_{v},$ is $t{g}^{-1}(\left({R}_{c}+{R}_{a}+\Delta {R}_{e}\right)/\omega \Delta {L}_{e})$. The phase angle for the relation ${V}_{cap}/{i}_{sensor}$will be always $\varphi =-\pi /2$.

At no-load condition,

$\Delta {L}_{e}=\Delta {R}_{e}=$ 0, and Equations (9) and (10) become:

and:

Equations (11) and (12) express the maximum values of the current at the sensor, and of the voltage at the probe capacitor, respectively. Connecting a potentiometer in series with the sensor coil, the no-load condition (values of ${i}_{sensor}$and ${V}_{cap}$ without the presence of a steel bar under the sensor) can be periodically calibrated.

#### 3.3. Finite Element Simulations and Experimental Comparisons for an ECT Sensor

Finite element analysis is a very interesting way to investigate the behavior of electromagnetic devices. Through it, a good understanding of the phenomena involved can be obtained, in addition to the mathematical modeling of the problem. Moreover, prototypes are built with more confidence, if the expected results for their operation can be accurately predicted.

This subsection will present the construction details of an ECT sensor built from the mathematical model presented in the previous section.

Figure 4 shows the electromagnetic component of this sensor. It is composed of a multi-turn coil with 900 turns of 24 AWG wire, connected in series with a capacitive array with capacitance equal to 5 nF and an additional resistance of 50 Ω (not shown in the figure for clarity). The dimensions of the coil are also provided in this figure.

The authors performed 3D finite element frequency-domain simulations for this sensor using the commercial software COMSOL Multiphysics [

26].

Figure 5 shows the magnetic flux density at the coil surface and the steel bar surface. The gauge of the bar is 20 mm, and the distance between the top of the bar and the sensor is 25 mm.

Figure 6 shows the mapping of the eddy current induced in the steel bar. As can be seen from these pictures, the magnetic induction is very low elsewhere (magnetic saturation is not present), and the eddy currents are concentrated in the region of the steel bar right below of the sensor.

**Figure 4.**
The electromagnetic components of an ECT sensor to inspect the reinforcement of concrete structures. (**a**) Perspective view; (**b**) Coil dimensions.

**Figure 4.**
The electromagnetic components of an ECT sensor to inspect the reinforcement of concrete structures. (**a**) Perspective view; (**b**) Coil dimensions.

**Figure 5.**
Magnetic flux density at the surface of the coil and the steel bar.

**Figure 5.**
Magnetic flux density at the surface of the coil and the steel bar.

**Figure 6.**
Induced current in the steel bar represented by the red arrows. (**a**) Perspective view; (**b**) Top view; (**c**) side view.

**Figure 6.**
Induced current in the steel bar represented by the red arrows. (**a**) Perspective view; (**b**) Top view; (**c**) side view.

These figures are very interesting to understand the behavior of the field quantities involved. However, to evaluate if this sensor will produce the expected results other variables should be analyzed, such as the effective resistance and inductance of the winding, changes in the phase angle of the current in the sensor, and the voltage drop at the capacitive array. COMSOL Multiphysics was prepared to automatically perform simulations for two bar gauges, as well as for different positions of the steel bar in relation to the sensor.

Simulations were done for a steel bar with gauge of 20.0 mm, placed at 25.0 and 45.0 mm under the sensor.

Figure 7 show the results for the effective coil resistance, effective coil inductance, the voltage at the capacitive array and phase angle of the current in the sensor. The graphics also present the experimental values obtained for this sensor, but the methodology used for the experimental tests will be present in the subsequent sections. As can be seen, the simulated and experimental results agree very well each other.

**Figure 7.**
Simulated (color marks) and experimental results (hollow black marks) for the effective resistance (**a**); effective inductance (**b**); voltage (**c**); and phase angle (**d**) for a 20 mm steel bar. Red marks: Steel bar placed 25 mm under the sensor. Blue marks: Steel bar placed 45 mm under the sensor.

**Figure 7.**
Simulated (color marks) and experimental results (hollow black marks) for the effective resistance (**a**); effective inductance (**b**); voltage (**c**); and phase angle (**d**) for a 20 mm steel bar. Red marks: Steel bar placed 25 mm under the sensor. Blue marks: Steel bar placed 45 mm under the sensor.