## 2 Materials and Methods
### Measurement of Magnetic Flux Density
On the other hand, when using the Hall sensor, the magnetic flux density in the vertical direction can be measured by the Hall sensor output voltage \(V_{H}\), as expressed in Equation (7) (relative permeability of the air is assumed to 1):  
\[H_{y}=\alpha_{1}(d-\alpha_{2})^{2}+\alpha_{3} \tag{6}\]  
\[V_{H}=kIB\cos\theta=kIH_{y} \tag{7}\]  
where \(V_{H}\), \(k\), \(B\), \(I\), \(\theta\) denote the Hall voltage, the Hall constant, the magnetic flux density incident on the Hall sensor, the Hall sensor input current, and the direction angle of the magnetic flux density perpendicularly incident on the Hall sensor. The Hall voltage \(V_{H}\) by Equation (7) is linearly proportional to \(Hy\), the magnetic flux density in the vertical direction. On the other hand, if the Hall constant (\(k\)) and the Hall sensor input current (\(I\)) are constant, and Equation (6) is substituted into Equation (7), it is expressed as Equations (8) and (9). That is, by measuring the magnetic flux density in the vertical direction, the depth of the defect can be quantitatively evaluated, where \(c_{1}\) and \(c_{2}\) are constants:  
\[H_{\text{LU}}=\frac{+m}{4\pi\mu}\int_{-\frac{D}{2}-W}^{\frac{D}{2}}\frac{y_{p} }{\left\{\left(x_{p}-u\right)^{2}+\left(y_{p}\right)^{2}\right\}^{\frac{3}{2} }}du \tag{2}\]0  
\[H_{\text{LU}}=\frac{+m}{4\pi\mu}\int_{-\frac{D}{2}-W}^{\frac{D}{2}}\frac{y_{p} }{\left\{\left(x_{p}-u\right)^{2}+\left(y_{p}\right)^{2}\right\}^{\frac{3}{2} }}du \tag{2}\]1  
Figure 4 shows the block diagram of the signal processing for a single Hall sensor element. The output voltage of the Hall sensor \(V_{H}\) was low-pass filtered (LPF) to remove the high-frequency noise signal. The first stage amplifier was used to gain the signal before transferring to the main signal processing circuits. The LPFs and first stage amplifier were attached near to the Hall sensor in the sensor probe. The second stage amplifier has a controllable gain, which was adjusted by the software in the PC. The signal was then converted to digital via A/D converter and real-time display/process in the PC. The proposed inspection system uses multiple Hall sensors; thus, the number of LPFs, first amplifiers and second amplifiers are the same as the number of Hall sensors for simultaneous signal processing. \[H_{\text{LU}}=\frac{+m}{4\pi\mu}\int_{-\frac{D}{2}-W}^{\frac{D}{2}}\frac{y_{p} }{\left\{\left(x_{p}-u\right)^{2}+\left(y_{p}\right)^{2}\right\}^{\frac{3}{2} }}du \tag{2}\]2 (**a**) Relationship between the depth of damage \(d\) and the vertical component of magnetic flux intensity \(H_{y}\) and (**b**) the normalized \(H_{y}\) at different positions from the yoke’s center in the \(x\)-axis. Figure 5 shows the proposed magnetic leakage testing system to inspect the corrosion in the water-cooled tube wall. A magnetic sensor array and magnetizer were manufactured to fit with the water-cooled tube's surface, as shown in the left and middle drawing. The magnetic sensor array was placed at the middle of magnetizer poles for measuring the distribution of vertical magnetic flux, as discussed in the previous paragraphs. There were three wheels (a front and two rears wheels) used to maintain the lift-off between the sensor and the tube and help scan the tube easily.