## 2 Theory
### Enhancement with a Soft Magnetic Patch  
In wave generation, the dynamic magnetic field in the magnetostrictive patch is supplied by the coils loaded with sinusoidal pulses. It can be described with the magnetic [PERSON]'s law qualitatively. As shown in Figure 4, the coils in the middle were the magnetic source, which supplied a constant magnetomotive force \(F\) when the input power of the equipment stayed unchanged. The magnetic flux circulated around the coils constructing a space named the magnetic circuit. According to the magnetic [PERSON]'s law, the relation between the magnetomotive force \(F\) and the magnetic flux \(\varnothing\) is as follows:  
Figure 3: Magnetostrictive patch transducer for shear wave inspection.  
Figure 2: Magnetic domain movement in the magnetostrictive effect.  
\[\varnothing=\frac{F}{\sum_{i=1}^{4}R_{i}},\ (i=1,2,3,4) \tag{7}\]  
where  
\[R_{i}=\frac{L_{i}}{\mu_{i}\mu_{0}A_{i}},\ (i=1,2,3,4) \tag{8}\]  
\(R_{i}\) is the magnetic resistance of part \(i\) in the magnetic circuit as marked in Figure 4; \(L_{i}\) and \(A_{i}\) is the length and sectional area, respectively; and \(\mu_{i}\) is the relative permeability. Furthermore, \(\mu_{0}=4\pi\times 10^{-7}\ N\cdot\Lambda^{-2}\) is the permeability of the vacuum. Part 1 denotes the magnetostrictive patch. Parts 2, 3 and 4 were full of air. The magnetostrictive patch was not magnetized along the direction of the dynamic magnetic field and thus, the magnetic intensity \(\mathbf{H_{d}}\) in part 1 could be calculated as:  
\[\mathbf{H_{d}}=\frac{\varnothing}{\mu_{1}\mu_{0}A_{1}} \tag{9}\]  
The magnetic resistance of parts 2 and 4 were the same (i.e., \(R_{2}=R_{4}\)). Combined with Equations (7) and (8), Equation (9) can be written as follows:  
\[\mathbf{H_{d}}=\frac{F}{L_{1}+2\frac{\mu_{1}A_{1}}{\mu_{2}A_{2}}L_{2}+\frac{\mu_{ 1}A_{1}}{\mu_{3}A_{3}}L_{3}} \tag{10}\]  
It could be found that without changing the material property (\(\mu_{1}\)) and sizes (\(L_{1}\) and \(A_{1}\)) of the magnetostrictive patch (part 1), the magnetic intensity \(\mathbf{H_{d}}\) could be improved with an increase in the relative permeability of part 3 (\(\mu_{3}\)). It could be achieved by covering a soft magnetic patch (SMP) over the coils. \(\mathbf{H_{d}}\) could be improved further by increasing the thickness of the SMP (related to \(A_{3}\)) and decreasing the gap (\(L_{2}\)) between the SMP and magnetostrictive patch. It should be noted that Equation (10) qualitatively explains the improvement by the SMP in wave generation. The magnetic flux was distributed unevenly in the magnetic circuit, which also does not hold a regular geometric structure.  
In wave detection, the magnetostrictive patch becomes a magnetic source because of the inverse magnetostrictive effect. The magnetic field induced by the magnetostrictive patch was proportional to the wave strength without additional magnetic fields. With the SMP above the coils, the induced magnetic field may have been restricted within the area around the coils and the induced current in the coils would be larger with an amplitude increase in the induced dynamic magnetic field. The improvement of the SMP can affect both processes when a MPT is used as an actuator and sensor at the same time. Furthermore, the electromagnetic interference, which are ubiquitous in the testing field, can be blocked by the SMP, improving the signal-to-noise ratio (SNR) of the detection signals.  
Figure 4: Diagram of dynamic magnetic fields in the MPTs (magnetostrictive patch transducers).