# A Direction-Sensitive Microwave Sensor for Metal Crack Detection
## 2 Sensor Design
### Equivalent Circuit Model  
To demonstrate the working principle of the proposed sensor, its equivalent circuit is described in this section. Figure 2a presents a physical multilayer model of the rectangular patch resonator when there is no crack on the metal ground plane.  
In Figure 2a, \(J_{D}\) is the displacement current which is created by the electric field between the rectangular patch and ground plane, and this can be regarded as a capacitance. Additionally, the conduction current \(J_{C}\), which is excited on both rectangular patch resonator and metal plane, can be considered as an inductance in the equivalent circuit. According to the physical model, the equivalent circuit of the proposed sensor can be simplified as Figure 2b shows, where \(L\) and \(C\) are the series inductance and capacitance of rectangular patch resonator, respectively. Therefore, the input admittance \(Y_{in}\) can be expressed as  
\[Y_{in}=jwC+\frac{1}{jwL}. \tag{5}\]  
Additionally, the reference resonant frequency \(f_{0}\) of the sensor can be expressed as  
\[f_{0}=\frac{1}{2\pi\sqrt{LC}}. \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Value (mm)** & **Variables** & **Value (mm)** \\ \hline W & 40 & WS & 0.5 \\ W1 & 5.5 & W50 & 3 \\ W2 & 6 & L & 50 \\ W3 & 37 & L1 & 27 \\ W4 & 4.1 & L2 & 9.3 \\ W5 & 6 & L3 & 22.4 \\ W6 & 46 & L4 & 8.8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimensional parameters of the proposed sensor.  
Figure 2: Equivalent model of the sensor without crack on the ground plane. (**a**) Physical model. (**b**) Equivalent Circuit.  
When there is a crack on the metal ground plane, the equivalent circuit of the proposed sensor can be modeled as shown in Figure 3.  
The model presented in Figure 3a is similar to the model shown in Figure 2, in which \(J_{D}\) and \(J_{C}\) still exist.  
Compared with the model in Figure 2a, it can be found that the extra conduction currents \(J_{C1}\), \(J_{C2}\) and \(J_{C3}\) are exited along the crack boundaries except \(J_{C}\) due to the discontinuity on the metal ground plane. Additionally, the displacement current \(J_{D1}\) is created by the electric field between two crack boundaries where \(J_{C2}\) and \(J_{C3}\) located. So, when there is a metal crack on the metal ground plane, the additional displacement and conduction currents will introduce additional capacitances and inductances into the equivalent circuit, which can be regarded as an additional capacitance \(C^{\prime}\). The input admittance \({Y^{\prime}}_{in}\) of circuit can be expressed as  
\[{Y^{\prime}}_{in}=j\omega L+j\omega(C+C^{\prime}), \tag{7}\]  
where \(C^{\prime}\) is the capacitance introduced by a crack, and its expression can written as  
\[C^{\prime}=\left(\frac{1}{\frac{1}{j\omega_{C1}+1/j\omega L_{2}}+j\omega L_{1 }+j\omega L_{3}}\right)/j\omega. \tag{8}\]  
Then, the resonant frequency of the sensor for crack detection can be expressed as  
\[f=\frac{1}{2\pi\sqrt{L(C+C^{\prime})}}. \tag{9}\]  
In summary, an additional capacitance introduced by a metal crack will lead to a decreased resonant frequency of the sensor. So, when there is a crack on the metal ground surface, the resonant frequency will deviate from the reference value. Additionally, as the capacitance \(C^{\prime}\) is determined by the width, depth and direction of the metal crack, detailed characteristics of the crack can be inferred from the resonant frequency response of sensor.