## 2 Materials and Methods
### Underground Probe Design
#### 2.4.4 Transmission Line  
When coupling an RF signal source to a buried probe for measuring the reflection coefficient, the selection of the transmission line is crucial. Its primary function is to transport signals from the source to the buried probe. The environmental conditions of the soil significantly influence the choice of transmission line, taking into account factors such as the allowable range in the medium, rating, wave attenuation sensitivity, and cost.  
Various types of transmission lines can be considered, including parallel lines such as coaxial cables, planar lines, and twisted lines. However, due to its low attenuation for UHF frequencies, coaxial cable is often the preferred choice. The established connection utilizes a coaxial cable, where the signal propagates as a voltage difference between the signal and the ground (shield) conductors, allowing for the observation of power reflection during underground communication.  
The impedance, denoted by \(Z_{in}\), is expressed as follows:  
\[Z_{in}=\frac{Z_{Load}+Z_{C}j\tan(\beta l)}{Z_{c}+Z_{Load}j\tan(\beta l)}Z_{c} \tag{25}\]  
where \(\beta\) is defined as  
\[\beta=\frac{2\pi}{\lambda} \tag{26}\]  
The electrical length \(l\) of the transmission line is given by  
Figure 21: Results of the dielectric permittivity estimation.  
Figure 20: Resonance curve for dielectric permittivity \(\varepsilon_{r}=10\).  
\[l=\frac{\lambda}{4} \tag{27}\]  
And the wavelength \(\lambda\) can be expressed as  
\[\lambda=\frac{c}{\sqrt{\epsilon_{r}}} \tag{28}\]  
In the above equations, the terms are defined as follows:  
* \(Z_{Load}\): load impedance;
* \(l\): electrical length of the transmission line;
* \(\beta\): propagation constant of the transmission line;
* \(Z_{c}\): characteristic impedance of the transmission line.  
Figure 22 illustrates the equivalent circuit of the transmission line, while Figure 23 presents the probe configuration with the transmission line:  
The results presented in Figure 24 indicate that, for an inductance of \(1\,\mathrm{nH}\), resonance could be observed for various dielectric permittivities, achieving a reflection coefficient of \(-26\,\mathrm{dB}\) at \(\epsilon_{r}=12\).  
Figures 25 and 26 illustrate the results of the estimated dielectric permittivities for a transmission line with an electrical length of \(E=\frac{\pi}{2}\):  
Figure 27 illustrates the quantified error between the theoretical dielectric permittivity and the estimated values for the proposed probe. The error ranged from 270% to 4% for dielectric permittivity values between 2 and 15, using transmission lines of lengths equal to \(\frac{\pi}{2}\) and \(\pi\). Notably, the error for low dielectric permittivity values was minimized to 93% with a transmission line of electrical length equal to \(\frac{\pi}{2}\). This limitation arose from the varicap diode; thus, a new varicap model could provide greater variation in capacitance. The model without a transmission line could successfully determine high values of dielectric permittivity, reaching up to 6, which can be attributed to the high reflection coefficient at the resonance frequency. Conversely, the poor estimation of dielectric permittivity for low values was linked to the limited reflection coefficient at the resonance frequency, influenced by inductance values that fail to achieve optimal resonance. The model using a transmission line of length \(\frac{\pi}{2}\) yielded better estimates of permittivity values.  
Figure 23: Probe diagram. The red arrow represents a variable capacity.  
Figure 22: Schematic representation of a transmission line.  
Figure 24: Reflection coefficient for different dielectric soil permittivities \(\epsilon_{r}\).  
Figure 25: Results of the dielectric permittivity estimation for an electrical length \(E=\frac{\pi}{2}\).