## 3 Simulation Setups and Results
### Results
Figure 6a-c show the gain profile for an enclosure with the conductivity of \(10^{6}\), \(10^{7}\), and \(10^{8}\) S/m, respectively; the hidden object is once again assumed to be a PEC cube which, has a length of 0.2 m with the center of the PEC. The thickness and size of the shell are the same as shown in Figure 4. The G values are measured on \(xy\) plane at \(z=-0.005\) m. The difference between the maximum and minimum of the G values is 15 dB. When the conductivity increases from \(10^{6}\) to \(10^{7}\) S/m, the G values are almost the same. The location and area of the PEC inside box are easily discernable. However, at \(10^{8}\) S/m, the image of the inside box is slightly distorted from that of the lower conductivity shield cases. This is an interesting feature as  
Figure 5: The G values for (**a**) different sizes of PEC inside box as a function of width (\(x\)), length (\(y\)), and height (\(z\)). (**b**) 0.2, 0.4, and 0.8 height with 0.2 width and 0.2 length on \(xy\) plane at \(z=-0.005\) m and \(xyz\) geometry. The relative permeability of the inside box is fixed at 1.  
it indicates a threshold conductivity requirement before the fields are appreciably distorted on the observation plane. Figure 6d shows the gain when the conductivities of the hidden box are changed but the outer shell has a fixed \(10^{6}\) S/m conductivity. The G value is measured at the center of the \(xy\) plane along the negative z-direction. In the PEC case, the G value is the highest (29.6 dB) and sharply decreases along the negative z-direction. Far from \(z=-0.02\) m, the cases of PEC and \(10^{8}\) S/m conductivity have the same G value. When the conductivity of the inside box decreases from \(10^{8}\) to \(10^{7}\) S/m, the G value is reduced from 19.2 to 0.9 dB. For a conductivity of \(10^{6}\) S/m, the G value is 0. This suggests that the object is not easily detectable when the conductivity of both inside and outside boxes is the same. This suggests that ELF/VLF magnetic fields have difficulty discerning hidden objects when the conductivity contrast is not appreciably high and is an important limitation of the method. Figure 7 shows that the \(H\) fields are measured on the \(xy\) plane at \(z=-0.005\) m for conductive (\(\mu_{r}=1\)) and ferromagnetic (\(\mu_{r}=1000\)) materials through an outer shell at 50 Hz case. The inside objects in cube geometry have a length of 0.6 m with \(\sigma_{in}=10^{6-8}\) S/m conductivity. The outer shells have 2.7 mm thickness with \(\sigma_{out}=10^{6-8}\) S/m conductivity. The size of the outer shells and the location of magnetic dipoles are the same as shown in Figure 4. Compared to only inside materials in Figures 2 and 3, there is a conductive outer shell. When an outer shell has lower conductivity than an inside object, the size and location of the inside object can be imaged as shown in Figure 7e,f,i,j. From the boundary of the inside box, the magnetic fields decrease when the measurement point goes to the center (\(x=y=0.6\) m). The reason is that a negative-induced dipole induces an enhanced magnetic field with gathered magnetic flux as shown in Figures 2 and 3. However, at \(\sigma_{in}=10^{7}\) S/m, the magnetic field in the case of the ferromagnetic material increases again as the area of measurement is less than a half in the radius of the object \(\left(\left(x-0.6\right)^{2}+\left(y-0.6\right)^{2}<0.3^{2}\right)\), as shown in Figure 7f. At \(\sigma_{in}=10^{8}\) S/m, the \(H\) field continually decreases toward the center, but the amplitude is enhanced compared to the conductivity material case, as shown in Figure 7i,j. When the conductivities of the inside box and outer shell are the same, both boxes can be considered as an unseparated one box, as shown in Figure 7a,b. At \(\sigma_{out}=10^{8}\), the location and area of the hidden PEC box are easily discernable in Figure 6c, but the inside materials with \(\sigma_{in}=10^{6-8}\) S/m cannot be imaged because of less distortion field.