## 4 Parameter Derivation
### Circuit Based FRVF Model  
The basic circuit model as shown in Figure 6 is used to introduce the grounding grid high-frequency behaviors for many years. The passive parameters of resistance, inductance, and capacitance (RLC branch) are found in (32)-(35) [9; 10; 11; 12; 13; 14; 15].  
\[\mathrm{R_{c}}=\frac{\mathrm{\rho}}{2\pi\mathrm{I}}\left(\ln\left(\frac{4l}{a }\right)-1\right) \tag{32}\]  
\[\mathrm{Z_{c}}=\sqrt{\frac{L}{C}}=\frac{1}{2\pi}\left(\ln\left(\frac{4l}{a} \right)-1\right)\sqrt{\frac{\mathrm{\mu_{0}}}{\mathrm{\varepsilon_{0}}}} \tag{33}\]  
\[\mathrm{L_{c}}=2l(\mathrm{Z_{c}}-\mathrm{R_{c}}) \tag{34}\]  
Figure 4: The main soil components.  
Figure 5: Complete model of soil ionization behavior.  
\[R_{\rm s}=Z_{\rm c}-R_{\rm c} \tag{35}\]  
where \(a\) and \(l\) represent the electrode diameter and length, respectively.  
Currently, in order to evaluate the high-frequency behavior of the grounding system, along with the effect of soil ionization, Figure 7 shows the circuit-based diagram of the proposed strategy. As is observed, there are some series impedance branches with their real or complex conjugate poles; those are enforced to be stable. It means that the real part of all fitted poles will have a negative value to be located at the left side of the imaginary axis in the complex plane. In (24), however, all the \(c_{n}\) and \(a_{n}\) coefficients could be real or complex, but \(d\) and \(h\) are certainly real if they exist. The passive parameters of Figure 7 will be calculated with the following algorithm:  
* The \(d\) and \(h\) coefficients represent the resistance and the inductance of grounding grid in the rest mode, which are determined by \(R_{0}\) and \(L_{0}\) respectively. \[R_{0}=d\] (36) \[L_{0}=h\]
* If the partial transfer function is defined as (37), then the poles are real and stable, thus we have: \[Z_{i}(s)=\frac{c_{i}}{s+a_{i}}=\frac{\frac{1}{C_{1}}}{s+\frac{1}{R_{1}C_{1}}}\] (37) \[C_{1}=\frac{1}{c_{i}}\;\&R_{1}=\frac{a_{i}}{c_{i}}\] (38)  
Other criteria must be applied to the parameter determination algorithm which prevent the resistance to be negative.  
Figure 6: Conventional grounding system impedance model.  
Figure 7: Proposed high frequency grounding system behavior considering soil ionization.  
reachability to 90\({}^{\circ}\) mostly. As the soil resistivity increases, impedance amplitude growth begins at lower frequencies. As a result, in more fertile soils, the frequency response angle becomes faster to achieve to 90\({}^{\circ}\).  
In the investigation of the optimal length of the horizontal electrodes of the grounding system, the impedance frequency response is calculated for a length of 2 m to 100 m at the same condition. When the same current amplitude is injected into one of the two ends of the electrode, the impedance changes as the soil resistivity changes, which is shown in Figure 11. This plot diagram, which is drawn at a frequency of 50 Hz, reflects the electrode impedance, decreasing since the length is increased. However, as the frequency of the injected current increases, the impedance reduction will stop after a while, and its magnitude will be fixed at a definite constant value. The length of the electrode, when the impedance amplitude begins to be constant, is called the optimal length, and should be used in the grounding grid designs. These results indicate that the lower the frequency, the effective length will be less for a specific reduction. For example, at a frequency of 1 kHz, the effective length for a specific resistance of 10,000 ohms-m is about 1000 m long. However, this effective length can be reduced to 40 m in the soil resistivity of 10 ohms. The point that was previously expected is that, as the frequency increases, the effective length of the horizontal electrodes decreases.  
Figure 8: A typical horizontal electrode as a grounding grid.  
Figure 10: Phase angle of a typical horizontal electrode; (a) \(\rho=100\)\(\Omega\cdot\)m, (b) \(\rho=300\)\(\Omega\cdot\)m, (c) \(\rho=1000\)\(\Omega\cdot\)m.