## 3 Results and Discussion  
In this section, the MMM signals of three different rectangular defects of an ASTM A-36 steel pipe are presented. This pipe has the following dimensions: 247.0 mm length, 48.06 mm outer diameter and 3.65 mm thickness. Figure 3 shows the measurement system of MMM signals around three different rectangular defects of an ASTM A-36 steel pipe. This system includes a rotatory mechanism, a magnetoresistive sensor (MAG3110), an Arduino nano (ATmega328) and a virtual instrumentation developed in Delphi Borland code [31]. The rotatory mechanism uses a motor (Buhler GmbH, Braunschweig, Germany) that is supplied with 1.5 V dc to generate a rotational motion of 2 rpm. The measurement system is fabricated of non-magnetic materials such as nilamide and aluminium, which do not affect the self-magnetic leakage flux of the ferromagnetic pipe. The pipe sample is collocated in the rotatory supports, keeping a constant distance of 2 mm between the external surface of the pipe and the magnetoresistive sensor. The pipe has three rectangular defects with different depth and width along its length. These defects (see Figure 4) have the following dimensions: S1 (4.36 mm width and 3.65 mm depth), S2 (3.80 mm width and 2.0 mm depth) and S3 (2.80 mm width and 0.50 mm depth). The tangential and normal MMM signals around the three defects are measured using the magnetoresistive sensor. The sensor has a resolution of 30 nT, a sensibility of 100 nT, and it uses an I\({}^{2}\)C interface to communicate with Arduino nano (ATmega328). The measured MMM signals of the three defects are processed using a virtual instrumentation, which is developed in Delphi Borland code [31]. For the analytical model, values of \(H_{0}\) for each defect are determined to estimate the best fit of Equations (1)-(3) for the corresponding experimental data. For these cases, we assumed a relative permeability of 180. Figure 5 depicts the tangential MMM signal of the first rectangular defect (S1), which is obtained using the analytical model and proposed measurement system. The measu  
Figure 4: Dimensions of three rectangular defects of the ASTM A-36 steel pipe. Figure 3: Measurement system of the tangential and normal MMM signals around three rectangular defects of the ASTM A-36 steel pipe. exhibits a peak (173.2 \(\upmu\)T) at the defect center (\(y\) = 0 mm). This peak corresponds with the maximum depth (3.65 mm) of the defect S1. This peak width is related to the defect width (4.36 mm). The width of the measured tangential MMM signal is less than the signal of the analytical model. This difference can occur because the 2D magnetic dipole method considers a constant magnetic permeability of the ferromagnetic sample. In the positions from \(-\)5.0 mm to 5.0 mm, the analytical tangential MMM signal has a high absolute difference (18.52 \(\upmu\)T and 47.18 \(\upmu\)T) with respect to the measured MMM signal. On the other hand, the normal MMM signal of defect S1 has two peaks (112.2 \(\upmu\)T and \(-\)143.6 \(\upmu\)T) close to both edges of the defect S1 (see Figure 6). These two peaks represent a change in the polarity of the normal MMM signal due to different polarities of the magnetic charges on the defect edges, in which the normal MMM signal reached a value zero at the defect center (\(y\) = 0 mm). The distance along \(y\)-axis between the two peaks values is related to the defect width (4.36 mm). The maximum and minimum magnitudes of the normal MMM signal are achieved along the positions close to \(\pm\)2.0 mm. The normal MMM signal registers a magnetic flux offset about \(\pm\)80.0 \(\upmu\)T. For this case, the normal MMM signal obtained through analytical model has a good approximation in comparison with the measured MMM signal. In the positions from \(-\)5.0 mm to 5.0 mm, these signals have an absolute difference of 8.22 \(\upmu\)T and 24.52 \(\upmu\)T, respectively. Figure 7 shows the measured tangential MMM signal of the second rectangular defect (S2). This MMM signal has a maximum value (137.1 \(\upmu\)T) at the defect center.