# A Method for Detecting Surface Defects in Railhead by Magnetic Flux Leakage
## 3 Defect Detection Method  
When a ferromagnetic object such as a rail is magnetized, a magnetic field will be generated. If a defect is in the surface, the magnetic refraction occurs at the interface of the rail and the air inside the defect, and part of the magnetic flux refracts out of the rail, forming a leakage magnetic field [24]. As shown in Figure 1a, the object under test, such as a rail, is magnetized by the applied excitation magnetic field. Near a defect, some magnetic lines leak out of the rail, forming a magnetic leakage field as shown in Figure 1b. Such a phenomenon does not appear in the non-defect place, so a sensor is usually installed under the yoke to detect the change of the magnetic field near the surface of the rail to find the defects. Figure 2 shows the section of the \(x\)-\(z\) plane near a surface defect. The defect width is \(2a\) and the depth is \(b\). The detection direction is also shown in Figure 2.  
Figure 1: MFL detection. The MFL field intensity of point \(P(x,z)\) above the defect is \(H(x,z)\). The \(x\)-component is \(H_{x}(x,z)\) and the \(z\)-component is \(H_{z}(x,z)\). They can be obtained by Equations (1) and (2), respectively. \[H_{x}(x,z)=\frac{\sigma_{ms}}{2\pi}\left[tan^{-1}\frac{b+z}{a-x}+tan^{-1}\frac {b+z}{a+x}-tan^{-1}\frac{z}{a-x}-tan^{-1}\frac{z}{a+x}\right] \tag{1}\]  
\[H_{z}(x,z)=\frac{\sigma_{ms}}{4\pi}ln\frac{\left[\left(x+z\right)^{2}+\left(z +b\right)^{2}\right]\left[\left(x-z\right)^{2}+z^{2}\right]}{\left[\left(x-a \right)^{2}+\left(z+b\right)^{2}\right]\left[\left(x+a\right)^{2}+z^{2}\right]} \tag{2}\]  
where \(\sigma_{ms}\) is the magnetic charge density of the defect side, and it is calculated by Equation (3). \[\sigma_{ms}=5.3\bigg{(}\frac{b/a+1}{b/(a\mu)}\bigg{)}H \tag{3}\]  
In Equation (3), \(\mu\) is the magnetic permeability of the material and \(H\) is the applied magnetic field strength. According to Equations (1) and (2), the sensor signal amplitude of the same defect is different with various lift-offs. In patrol detection, when the sensor signal changes, it is difficult to distinguish whether the change is caused by a defect or the lift-off change. Due to the random interference during patrol detection, it further increases the difficulty. If \(z=1\) mm, \(a=1\) mm, \(b=1\) mm, \(\sigma_{ms}/2\pi=1\), the magnetic field distribution in the \(x\) and \(z\) directions of the defect is shown in Figure 3. It can be seen from Figure 3, as the sensor approaching the defect from far to near, the \(x\)-component of its signal increases and the \(z\)-component increases first, reaches the maximum and then decreases. When the sensor is directly above the defect, the \(x\)-component is the maximum, and the \(z\)-component is 0. As the sensor moving away from the defect, the \(x\)-component of its signal is decreases and the \(z\)-component decreases first, reaches the minimum, and then increases. If multiple sensors are arranged in the \(x\)-direction, the relative values of their signals are regular when they pass a defect. When a sensor is directly above the defect, the \(x\)-component of its signal is more significant than that of the sensor set at other place, the \(z\)-component of its signal is more significant than those of the sensors which have passed the defect and smaller than that of the sensor which has not passed the defect yet. The effect of lift-off on sensor signal is different from that of a defect. The smaller the lift-off, the more significant the signal, whether \(x\) or \(z\) component. If the lift-off of sensors is similar, the effect of lift-off change on the signal of each sensor is similar. Therefore, the defect signal can be distinguished from lift-off interference by the relationship among the \(x\) and \(z\) components of the sensors. Figure 2: The \(x\)-\(z\) plane near a defect. As shown in Figure 2, one main sensor and four auxiliary sensors are arranged longitudinally along the \(x\)-direction.