For a Single-Line-to-Ground (SLG) fault, the positive ($Z_1$), negative ($Z_2$), and zero ($Z_0$) sequence networks are connected in series. This is because the sequence currents are equal: $I_1 = I_2 = I_0 = I_f / 3$. The neutral grounding impedance ($Z_N$) is included in the zero-sequence loop as $3Z_N$.

The neutral grounding resistor $R_N$ carries the total neutral current $I_N$, which is the sum of the three zero-sequence phase currents ($I_N = I_{0a} + I_{0b} + I_{0c} = 3I_0$). The voltage drop across the resistor is $V_N = I_N \times R_N = 3I_0 \times R_N$. When solving calculations on a single-phase per-phase basis, this voltage drop is represented as $I_0 \times (3R_N)$. Hence, the resistor is scaled by 3 in zero-sequence circuit calculations.

Zero-sequence currents can only flow if a neutral return path is grounded. For a Delta-Star (Dyn11) transformer, the primary delta winding isolates zero sequence current, while the secondary star winding neutral provides a path for zero-sequence current to flow into a ground fault. For a Star-Star (YNyn0) transformer without a tertiary delta winding, the zero-sequence path is highly restricted (impedance is 10 to 15 times positive sequence impedance) because the zero-sequence magnetic flux must return through the air or steel tank wall.

In isolated/unearthed systems, an arcing line-to-ground fault can continuously charge and discharge the network's phase-to-ground capacitances. If this happens at system resonance, voltage peaks can escalate to 5 or 6 times the nominal line voltage, causing system insulation failure. HRG inserts a resistor $R_N$ sized such that the resistive fault current is equal to or greater than the network's total charging current ($I_R \ge I_{co}$). This damping resistor absorbs the capacitive energy, limiting transient overvoltages to a safe 2.5 times nominal voltage.

A Peterson Coil is a variable-reactance tuning reactor ($L_p$) placed between the transformer neutral and earth. It is tuned to resonate with the total system phase-to-earth capacitances ($3C_{ph}$). By establishing resonance, the inductive current from the Peterson coil cancels out the capacitive earth fault current at the fault point, reducing the residual fault current to near zero, which extinguishes the ground fault arc without tripping the circuit breaker.

IEEE Std 80 defines allowable step and touch voltage thresholds to prevent dangerous levels of current from passing through a human body during a fault. Permissible voltages are computed using the formulas:

\(V_{touch} = \frac{116 + 0.17 C_s \rho_s}{\sqrt{t_f}}\)      \(V_{step} = \frac{116 + 0.7 C_s \rho_s}{\sqrt{t_f}}\)

Where $\rho_s$ is the soil resistivity, $C_s$ is the surface layer derating factor, and $t_f$ is the fault duration. These boundaries indicate that faster relay trip times ($t_f$) yield higher safe voltage thresholds.

Ground Potential Rise (GPR) is the maximum potential relative to remote earth that a substation grounding grid reaches when a ground fault current flows into it: $GPR = I_f \times R_{grid}$, where $R_{grid}$ is the grid grounding resistance. If the GPR is lower than the allowable touch voltage, the installation is safe. If the GPR exceeds the allowable touch voltage, a detailed safety analysis of the mesh and step voltages is required to verify that people will not be exposed to touch potentials that exceed safe thresholds.

Feeder lines add series resistance ($R_{cable}$) and reactance ($X_{cable}$) to the positive, negative, and zero-sequence fault loops. Over long distances, this added cable impedance increases the total impedance of the fault loop, which reduces the ground fault current at the fault point. This reduction must be analyzed to ensure that the fault current remains high enough to trip overcurrent or ground fault protection relays.

In isolated systems, there is no direct connection between neutral and earth. During a single-line-to-ground fault, the fault loop is completed only through the system's phase-to-ground capacitance, resulting in a low fault current. However, the voltage of the neutral shifts to phase voltage relative to ground, causing the healthy phases' voltages to rise to the line-to-line voltage ($\sqrt{3} \times V_{ph}$), which increases the voltage stress on the system insulation by 73%.

Low Resistance Grounding (LRG) limits the fault current to a moderate value (typically 100A to 400A) to provide enough current for standard overcurrent protective relays to operate and clear the fault quickly. High Resistance Grounding (HRG) limits the fault current to a very low value (typically 5A to 10A). This low current allows the system to continue operating with a single line-to-ground fault, while suppressing transient overvoltages until the fault can be located and cleared during a scheduled shutdown.