## 2 Principles of the Sensor Probes  
This section summarizes our previous works on the development of the three ET methods, including MFLT, ECT and PSECT, for inspection of corrosion in the MSR tubes. The sensor probes were constructed by multiple Hall sensor elements (15 InSb Hall sensors) for efficient detection of the corrosion [11; 12; 13]. The use of multiple Hall sensors has two main benefits over the sensing coils using in the conventional methods (i.e., ECT and PSECT method): (1) Hall sensor directly measures the magnetic field distributed around corrosion rather than the indirect measurement via total magnetic flux rate using the sensing coils; thus, the sensitivity could be improved and not strongly depend on the exciting frequency; (2) spatial resolution of the sensor array could be improved by decrease the size of magnetic sensors while the sensing size has a limited size (e.g., diameter over 1.0 mm). The sketch of the three sensor probes is shown in Figure 1. The MFLT probe includes a permanent magnet to produces a DC magnetic field into the tube and a Hall sensor array measures the magnetic flux leakage (MFL) around the corrosion for the detection of the corrosion. The bobbin coil in the ECT probe is supplied an AC to produce an eddy current in the tube according to the [PERSON]'s law. According to [PERSON]'s law, the eddy current flow is distorted by the presence of corrosion and produces the secondary magnetic field. The Hall sensor array could measure the secondary magnetic field, and the corrosion could be detected. On the other hand, the eddy current penetrates into the thickness of the tube according to the skin effect, in which the intensity of eddy current at a depth \(d\) exponentially decreases with the depth, permeability (\(\mu\)) and conductivity (\(\sigma\)) of the tube and exciting frequency (\(f\)), as described in Equation (1). The skin depth (\(\delta\)) is expressed as Equation (2), which is used to estimate the exciting frequency. For instance, the estimated exciting frequency should be used for ferritin stainless steel (SS439) MSR tube with a thickness of 1.2 mm ranges from 300 Hz to 2000 Hz [9; 12; 14]. \[J_{t}=J_{0}e^{-d\sqrt{\pi f\mu\alpha}} \tag{1}\]  
\[\delta=\frac{1}{\sqrt{\pi f\mu\alpha}} \tag{2}\]  
Because of the high permeability of the SS439 material, the skin depth will be low; thus, the eddy current is concentrated near to the inner surface of the MSR tube that limits the detection of the corrosion on the outer surface. Therefore, a magnet is additionally used in the PSECT method to saturate the MSR tube partially (i.e., decrease the permeability). Then, the skin depth could deeper penetrate up to the outer surface and allows better detection of the corrosion. It is noted that the MSR tube has fins that are used to increase the contact surface, and the fins are considered as multiple corrosion on the outer surface that produces noise to the real corrosion signal. On the other hand, the interaction between the magnet and the fins could produce [PERSON]'s force on the MFLT/PSECT probes that changes according to the profile of the fins and produces vibration of the MFLT/PSECT probes during the scan. Moreover, the degradation of MSR tube material after the long period operation could result in the changes in the homogeneous magnetic permeability of the tube. Thus, it could produce another noise to the corrosion signal. The total magnetic field at a Hall sensor (\(B(t)\)) is the superposition of the magnetic field from the magnet (\(B_{M}\)), probe variation (\(B_{V}\)), material degradation (\(B_{D}\)), exciting bobbin coil (\(B_{B}\)), eddy current at corrosion (\(B_{C}\)) and eddy current at fins (\(B_{F}\)), as expressed in Equation (3); where, \(B_{B0}\), \(B_{C0i}\) and \(B_{F0j}\) are amplitudes and zero, \(\Delta\alpha_{i}\) and \(\Delta\beta_{j}\) are phase-shifts of the time-varying magnetic fields.