## 1 Introduction  
Carbon steels are common in a wide variety of applications owing to their low cost and excellent mechanical properties. These range from marine applications, buildings and bridges, domestic appliances, the automotive industry, the oil and gas industry, and especially to power plants, among them nuclear power plants, which have many piping systems carrying fluids and gases [1; 2; 3; 4; 5]. Welding and joining processes such as arc welding, laser welding, and friction stir welding are used to assemble and join the main parts of such structures [6; 7; 8]. Welding has the advantage of having relatively few restrictions on the thickness and shape of the target material, reducing the weight of the equipment, and solving the problems of size limits, as parts produced in small sizes can be combined with the welding process to become larger parts [9; 10; 11]. However, despite the fact that welding is considered one of the most important manufacturing processes, during this process the weld area undergoes compression and expansion caused by rapid temperature changes and mechanical stresses, residual stresses that lead to a reduction in structural integrity and which affect the mechanics of the joint. Among the most important types of failures, which occur frequently, are fatigue and stress corrosion cracking (SCC), which may lead to cracking in service, with consequences such as leakage and even structural failure. In particular, the possibility of such failures is greatly increased during long-term operation in high-temperature, high-pressure, and corrosive environments [12; 13; 14; 15; 16; 17]. Therefore, to secure the safety of the installations (plants), the integrity of the pipes and weld joints should be assured. To achieve this, non-destructive testing (NDT) is performed on the welds to be inspected during the planned preventive maintenance period to confirm the integrity evaluation of pipe weld inspections according to the ASME code requirements. During this period, the correct welding parameters/procedures should be selected to prevent the formation of defects of various types [18; 19; 20; 21]. Numerous non-destructive testing (NDT) methods, namely eddy current testing (ECT), radiography testing, ultrasonic testing, magnetic particle testing, penetrant testing, and so on, can be applied as industrial methods to investigate the integrity of welds in assembled pipes [22; 23; 24; 25]. The ECT technique has many advantages such as high detection sensitivity, no coupling agent, no contact between the test specimen and the sensor, and the ability to detect parts in-line. In addition, the technique is appropriate for a variety of conductive materials, either ferromagnetic or non-ferromagnetic [26; 27; 28]. Certain factors affect the response of an eddy current probe, namely the frequency, magnetic permeability, and conductivity of the specimen. The electrical conductivity and magnetic permeability of the specimen depend on the microstructure of the specimen. They are related to the skin depth, given by the expression (1) [29; 30]:  
\[\delta_{s}=\sqrt{\frac{2}{\omega\sigma\mu}} \tag{1}\]  
where \(\omega\), \(\sigma\), and \(\mu\) are the angular frequency, the conductivity, and the permeability, respectively. In addition, the ECT is strongly affected by the lift-off, which can be defined as the separating distance between the surface of the excitation coil and the surface of the conducting material. A number of research studies have been performed to enhance defect characterization with regard to the reduction of the lift-off effect, new design and optimization of probes, and feature extraction [31; 32]. Our focus is on research studies on crack detection in carbon steel specimens using the finite element method (FEM). [PERSON] et al. [33] proposed a new rotating field EC probe for defect detection on carbon steel using finite element analysis and found that the excitation strategy gives the designed probe advantages for arbitrary orientation defect detection. In a similar way, [PERSON] et al.