## 2 Materials and Methods
### Underground Probe Design  
The goal of this research was to create a precise probe model to forecast the dielectric characteristics of soil, as shown in Figure 14. The model uses Frequency Domain Reflectometry (FDR) techniques to determine how these properties affect signal attenuation. We offer a solid technical solution that takes into account the limitations and specifications of electrical soil models, in order to create a probe suitable for overcoming the problem of low signal amplitudes. This solution seeks to significantly amplify signal amplitudes, ensuring  
Figure 11: Equivalent electric circuit of the soil.  
Figure 12: Simulated conductance for different circular plate separations \(d\).  
that electromagnetic signals propagate effectively through the soil, which enables precise extraction of complex dielectric permittivity.  
The proposed probe design primarily consists of a pair of circular parallel plates configured as a capacitor, along with a coil, a varactor, and an oscillator, with the soil serving as the dielectric medium. The varactor is employed to tune the resonant frequency of the overall circuit to 433 MHz. As demonstrated later, this configuration aids in extracting soil parameters from the measured data [19; 20].  
The dielectric permittivity of soil can be measured using a variety of techniques, such as the coaxial probe, the transmission line, and the parallel plate capacitance; however, as Table 3 illustrates, each technique has limitations of its own. The ideal approach depends on a number of factors, including price, size, accuracy, and ease. Temperature, material type, and frequency range are some factors to take into account. We employed the parallel plate capacitance method, to save on production costs, because it is easy to build a parallel plate capacitor, which is just two parallel metal plates, and because the uniform electric field between the plates facilitates computation and analysis. The primary benefits of employing parallel plate capacitance are its suitability for high-loss materials and its comparatively simpler measurements; additionally, it functions well at low frequencies compared to transmission lines, while the coaxial probe method is better suited for high frequencies and the measurement of dielectric permittivity for both solids and liquids. For high-loss materials, the coaxial probe delivers great accuracy. The coaxial probe method's primary disadvantages, however, are the need for repeated calibrations each time a permittivity measurement is desired and the potential for mistakes due to air gaps forming between the probe and the liquid being tested [21; 22].  
The Table 4 provides a detailed comparison between the TDR (Time Domain Reflectometry) and FDR (Frequency Domain Reflectometry) methods. It highlights their operating principles, analysis domains, main applications, advantages, disadvantages, and the contexts in which each method is best suited. The aim is to offer a clearer understanding of the key differences between these two techniques used for analyzing and detecting faults in electrical systems and transmission lines.  
Figure 14: Schematic of the proposed buried probe.  
Figure 13: The simulated imaginary part of the admittance for a different circular plate separation \(d\).