# Passive Harmonic RFID System for Buried Assets Localization
## 3 Read Range  
### Soil Attenuation Model  
In order to estimate the maximum underground range for tag detection, a study of the soil's attenuation loss is required. The attenuation can be obtained from the complex dielectric constant of the soil. It can be divided into two parts. The first one, called the reflection attenuation, is the result of the reflection produced when the wave changes medium (air-soil), and the second one is the attenuation due to the soil's loss. These components can be estimated from the dielectric constant of the soil. The reflection attenuation (\(L_{r}\)) is independent of the distance, and can be calculated for normal incidence using the Fresnel coefficient.  
\[L_{r}(dB)=-10\log\left(1-\left|\frac{\sqrt{\varepsilon_{s}}-1}{\sqrt{ \varepsilon_{s}}+1}\right|^{2}\right) \tag{1}\]  
The attenuation loss due to the propagation (\(L_{p}\)) can be calculated from the complex propagation constant \(\gamma\) and the propagation distance \(d\).  
\[L_{p}(dB)=8.68\epsilon\alpha(Np/m)\times d \tag{2}\]  
\[\alpha(Np/m)=-\mathrm{Im}(\gamma)=\frac{2\pi f}{c}\sqrt{\frac{\varepsilon_{s} f}{2}\left(\sqrt{1+\left(\frac{\varepsilon_{s}^{\prime\prime}}{\varepsilon_{s}f} \right)^{2}}-1\right)} \tag{3}\]  
where \(\varepsilon_{s}=\varepsilon_{s}f-j\varepsilon_{s}^{\prime\prime}\) is the complex dielectric relative constant of the soil and \(c\) is the speed of light in the vacuum.  
The dielectric constant depends on soil physical properties including bulk density, soil texture, and the level of water content. The dielectric soil properties can be measured following the procedure described in the literature [43] from the measurement of the S parameters with an especial coaxial fixture. After de-embedding the S parameter of the coaxial line with the soil as a dielectric, the dielectric constant can be found using the Nicholson and Ross algorithm. In this work, we use the model of [PERSON] et al. [46; 47], which is validated for the frequency range of 0.3-1.3 GHz. This model provides expressions for soil dielectric constant as a function of the soil volumetric water content \(\theta_{\mathrm{V}}\), the frequency \(f\), the fraction of sand particles \(S\), the fraction of clay particles \(C\), the density of the soil particles \(\rho_{\mathrm{S}}\) (a typical value is 2.66 g/cm\({}^{3}\)), and the bulk density of the soil \(\rho_{\mathrm{B}}\) (an average value is 1.6 g/cm\({}^{3}\)).  
Figure 10: Measured spectrum at \(2f_{0}=1732\) MHz (\(\Delta f=0\)) using three different modulation frequencies.  
Using Equations (1)-(3) and the dielectric constant given by [PERSON]'s model, Figures 11 and 12 depict the simulations of the attenuation coefficient \(\alpha\) in (dB/m) and the reflection loss at \(f_{0}\) (865 MHz) and \(2f_{0}\) (1730 MHz) for two soil cases; namely, case 1 (sand fraction S = 15%, clay fraction C = 20%) and case 2 (sand fraction S = 60%, clay fraction C = 20%). The attenuation coefficient increases with the frequency especially when the water content increases, and it is noticeably higher at the second harmonic. These simulations show that the attenuation can vary considerably depending on the composition of the soil. On the other hand, the loss due to signal reflection at the boundaries of the two mediums (air and soil) is considerably lower than the attenuation due to propagation.