Earth Grid Design

Overview

The Earth Grid Design calculator performs substation earthing grid analysis per IEEE Std 80 — Guide for Safety in AC Substation Grounding. It sizes a rectangular buried conductor grid with optional ground rods and verifies that touch and step voltages remain within safe limits during ground faults.

The calculator provides:

Grid resistance — IEEE 80-2013 Eq. 57 (Sverak)
Ground Potential Rise (GPR) — I_f × R_g
Mesh (touch) voltage — E_m within the grid mesh
Step voltage — E_s at the grid perimeter
Tolerable voltage limits — for 50 kg or 70 kg body weight with surface layer
Safety assessment — PASS if E_m ≤ E_touch AND E_s ≤ E_step
Grid conductor sizing — minimum CSA by IEEE 80-2013 Eq. 37 with the Table 1 constants of the conductor material

For complex non-rectangular grids or 3D visualization, use the Earth Grid 3D Tool in the Tools menu.

Input Parameters

Grid Geometry

Parameter	Default	Range	Description
`grid_length`	60 m	1 – 500	Grid length (X direction)
`grid_width`	40 m	1 – 500	Grid width (Y direction)
`spacing`	6 m	1 – 50	Conductor spacing in both directions
`burial_depth`	0.5 m	0.3 – 2.0	Burial depth below ground surface

Conductor

Parameter	Default	Description
`conductor_diameter`	12.78 mm	Conductor diameter (≈120 mm² Cu)
`conductor_material`	Cu_annealed	IEEE 80 Table 1 material: `Cu_annealed`, `Cu_hard_drawn`, `Cu_clad_steel` (40 %), `Cu_clad_steel_30`, `Al_clad_steel`, `Steel_1020`, `Stainless_clad_steel`, `Steel_galvanized` (zinc-coated steel rod), `Stainless_304`
`ambient_temperature`	40 °C	T_a in Eq. 37 (−40 – 60)
`joint_type`	bolted	Grid connection type setting T_m in Eq. 37 = min(joint limit, material fusing temperature): `exothermic_weld` (fusing temperature), `brazed` (450 °C), `bolted` (250 °C; bolted, clamp and compression / pressure-type connectors). See the joint table under Formulas.
`max_conductor_temperature`	none (from joint type)	Explicit T_m override (100 – 1510 °C, above the ambient and not above the material's fusing temperature), e.g. a connector qualified to IEEE Std 837 for a higher temperature. When given it replaces the joint limit.

Ground Rods

Parameter	Default	Description
`num_rods`	20	Number of vertical ground rods
`rod_length`	3 m	Length of each rod
`rod_diameter`	16 mm	Rod diameter (documentation only — not used by Eq. 57 or the mesh/step equations)

Fault & Soil Parameters

Parameter	Default	Description
`soil_resistivity`	100 Ω·m	Soil resistivity (1 – 10,000)
`fault_current`	10 kA	Maximum grid fault current
`clearing_time`	0.5 s	Fault clearing time (0.01 – 3)

Safety Criteria

Parameter	Default	Description
`body_weight`	70 kg	Body weight for tolerable voltage (50 or 70 kg)
`surface_resistivity`	3,000 Ω·m	Surface layer resistivity (e.g. crushed rock)
`surface_thickness`	0.1 m	Surface layer thickness

Output Results

Output	Symbol	Unit	Description
Grid Resistance	R_g	Ω	Total grid-to-earth resistance
GPR	GPR	V	Ground Potential Rise = I_f × R_g
Mesh Voltage	E_m	V	Maximum touch voltage within grid mesh
Step Voltage	E_s	V	Maximum step voltage at grid perimeter
Tolerable Touch	E_touch	V	Maximum safe touch voltage for body weight
Tolerable Step	E_step	V	Maximum safe step voltage for body weight
Touch Safe	—	PASS / FAIL	E_m ≤ E_touch
Step Safe	—	PASS / FAIL	E_s ≤ E_step
Overall	—	PASS / FAIL	Both touch and step must pass
Joint type and T_m used	T_m	°C	Connection type and the maximum conductor temperature used in Eq. 37, with its basis (joint limit, fusing temperature or override)
Grid conductor k	k	A·√s/mm²	Equivalent constant from Eq. 37 and Table 1
Grid conductor minimum / selected	A	mm²	Eq. 37 minimum and next standard size

Formulas

Grid Resistance — IEEE 80-2013 Eq. 57 (Sverak)

R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))]

Where A = L × W (grid area), L_T = total buried conductor length (grid + rods), h = burial depth (valid for 0.25 – 2.5 m). The depth correction multiplies the area term only; the 1/L_T term is not depth-corrected.

Grid Conductor Size — IEEE 80-2013 Eq. 37

A_mm² = I_kA / √[ (TCAP × 10^-4 / (t_c α_r ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a)) ]
         = I √t_c / k,   k = 1000 √[ (TCAP × 10^-4 / (α_r ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a)) ]

Material constants (IEEE 80-2013 Table 1). The last column, the Eq. 41 constant K_f for T_m = fusing temperature and T_a = 40 °C, is reproduced by the other constants and is shown as a check; k is the resulting A·√s/mm² value at those temperatures.

Material	Conductivity (%)	α_r at 20 °C (1/°C)	K₀ (°C)	Fusing T_m (°C)	ρ_r at 20 °C (µΩ·cm)	TCAP (J/(cm³·°C))	K_f	k (40 °C, fusing)
Copper, annealed soft-drawn	100.0	0.00393	234	1083	1.72	3.42	7.00	281.8
Copper, commercial hard-drawn	97.0	0.00381	242	1084	1.78	3.42	7.06	279.4
Copper-clad steel wire	40.0	0.00378	245	1084	4.40	3.85	10.45	188.8
Copper-clad steel wire	30.0	0.00378	245	1084	5.86	3.85	12.06	163.6
Aluminium-clad steel wire	20.3	0.00360	258	657	8.48	3.58	17.20	114.7
Steel, 1020	10.8	0.00160	605	1510	15.90	3.28	15.95	123.7
Stainless-clad steel rod	9.8	0.00160	605	1400	17.50	4.44	14.72	134.1
Zinc-coated steel rod	8.6	0.00320	293	419	20.10	3.93	28.96	68.1
Stainless steel, 304	2.4	0.00130	749	1400	72.00	4.03	30.05	65.7

Maximum Conductor Temperature T_m — joint type

The conductor is sized for the weakest element of the conductor and its connections, so T_m is the lower of the joint temperature limit and the material fusing temperature:

T_m = min(T_joint, T_fusing)     (or the explicit max_conductor_temperature when given)

`joint_type`	Connection	T_joint (°C)	Annealed Cu, 10 kA, 0.5 s, T_a 40 °C
`exothermic_weld`	Exothermic (thermit) weld — fuses with the conductor	fusing temperature (Table 1)	T_m 1083 °C, k 281.8, 25.09 mm² → 35 mm²
`brazed`	Brazed joint	450	k 215.1, 32.87 mm² → 35 mm²
`bolted` (default)	Bolted, clamp and compression (pressure-type) connectors	250	k 169.7, 41.68 mm² → 50 mm²

Limits as tabulated by IEEE Std 80-1986 for the connecting means (pressure-type connectors 250 – 350 °C, the lower bound used here; brazed 450 °C; exothermic weld = conductor fusing temperature) and carried into the conductor/connection selection discussion of IEEE Std 80-2013 §11.3, which also recommends staying at or below 250 °C where hard-drawn copper must not anneal. BS 7430:2011 gives the same 450 °C brazed and 250 °C bolted values. bolted is the default because it is the common field connection and the conservative choice. A connector qualified to IEEE Std 837 for a higher temperature is entered with max_conductor_temperature. Where the joint limit is above the fusing temperature (zinc-coated steel, 419 °C, with a brazed joint) the fusing temperature governs.

Surface Layer Derating (C_s)

C_s = 1 - 0.09 × (1 - ρ_soil/ρ_surface) / (2h_s + 0.09)

Tolerable Voltages

E_touch = (1000 + 1.5 × C_s × ρ_s) × C_b / √t
E_step  = (1000 + 6 × C_s × ρ_s) × C_b / √t

C_b = 0.116 (50 kg) or 0.157 (70 kg).

Mesh & Step Voltages

E_m = (ρ × I_f × K_m × K_i) / (L_M + L_R)
E_s = (ρ × I_f × K_s × K_i) / (0.75L_s + 0.85L_R)

Mesh Spacing Factor (K_m) — IEEE 80 eq. 81

K_m = (1/2π) × [ln(D²/(16hd) + (D+2h)²/(8Dd) - h/(4d)) + (K_ii/K_h) × ln(8/(π(2n-1)))]

Where D = spacing, h = burial depth, d = conductor diameter, n = effective conductors. K_h = √(1+h), K_ii = 1 (with rods) or 1/(2n)^2/n (without rods).

Step Spacing Factor (K_s) — IEEE 80 eq. 83

K_s = (1/π) × [1/(2h) + 1/(D+h) + (1/D)(1 - 0.5^n-2)]

Irregularity Factor (K_i) — IEEE 80 eq. 89

K_i = 0.644 + 0.148n

Worked Examples

The 18 examples below are the validation cases for this calculation. Each one was produced by the calculation engine from the inputs shown, every figure was checked against the independent reference quoted at the end of the example, and the same cases are run against every engine build (see validation scorecard, tool fe_calc_earthing; worked examples for every calculation).

#	Case	R_g (Ω)	GPR (V)	E_m / E_touch (V)	E_s / E_step (V)	T_m (°C)	Conductor (mm²)	Result
1	Small 20 × 20 m grid in 50 Ω·m soil, four corner rods, 1 kA	1.260	1,260	200 / 239	187 / 289	250	6	PASS
2	60 × 40 m grid, 20 rods, crushed rock, 10 kA	1.002	10,016	1,511 / 921	1,164 / 3,020	250	50	FAIL
3	Large 100 × 80 m grid, 10 m spacing, 40 rods, 8 kA	0.547	4,372	758 / 870	447 / 2,815	250	35	PASS
4	Fine 2.5 m mesh buried 1 m, hard-drawn 8 mm copper, 70 kg	1.671	6,683	1,082 / 1,186	794 / 3,882	250	16	PASS
5	The same fine mesh assessed for a 50 kg body	1.671	6,683	1,082 / 876	794 / 2,868	250	16	FAIL
6	Copper-clad steel grid, 2 kA for 1 s	1.833	3,665	525 / 655	461 / 2,150	1,084	16	PASS
7	Galvanised steel grid, 0.2 s clearing, 5000 Ω·m rock	3.782	15,127	2,121 / 2,216	1,840 / 7,811	419	35	PASS
8	30 × 30 m grid, 200 Ω·m, no rods, no surface layer	3.354	6,708	1,360 / 289	934 / 488	250	10	FAIL
9	Coarse 10 m mesh: touch voltage fails, step passes	1.856	1,856	549 / 255	256 / 355	250	6	FAIL
10	Rocky 1000 Ω·m site, 1 s clearing: both limits fail	12.869	38,606	7,805 / 392	4,749 / 1,099	250	25	FAIL
11	30 % copper-clad steel with bolted joints limited to 250 °C	1.219	6,093	854 / 921	747 / 3,020	250	50	PASS
12	1020 steel grid, 4 kA for 1 s	1.422	5,689	604 / 655	697 / 2,150	1,510	35	PASS
13	304 stainless steel at 20 °C ambient	1.576	7,879	924 / 1,039	1,158 / 3,295	1,400	50	PASS
14	40 × 40 m grid buried 2 m: depth term of Eq. 57	1.155	3,464	605 / 921	207 / 3,020	250	16	PASS
15	Joint type: exothermic-welded annealed copper, 10 kA for 0.5 s	0.782	7,822	793 / 921	940 / 3,020	1,083	35	PASS
16	Joint type: brazed annealed copper (450 °C), 10 kA for 0.5 s	0.782	7,822	793 / 921	940 / 3,020	450	35	PASS
17	Joint type: bolted annealed copper (250 °C), 10 kA for 0.5 s	0.782	7,822	793 / 921	940 / 3,020	250	50	PASS
18	Tm override: IEEE 837-qualified compression connectors at 350 °C	0.782	7,822	793 / 921	940 / 3,020	350	50	PASS

Example 1 — Small 20 × 20 m grid in 50 Ω·m soil, four corner rods, 1 kA

A compact distribution-substation grid in good soil without a surface layer: four rods set Kii = 1 and the 1 kA grid current keeps both voltages under the limits.

Given: Soil ρ = 50 Ω·m; grid current I_G = 1 kA for t_s = 0.5 s (no Sf/Df applied). Grid 20 × 20 m, conductors at D = 4 m spacing buried h = 0.5 m, 12.78 mm annealed copper with bolted / pressure-type joints; 4 rods × 3 m (16 mm); no surface layer; body weight 70 kg.

Geometry	n_x = ⌊20/4⌋ + 1 = 6; n_y = ⌊20/4⌋ + 1 = 6; L_C = 6 × 20 + 6 × 20 = 240 m; L_R = 4 × 3 = 12 m; L_T = 252 m; A = 20 × 20 = 400 m²; L_p = 2 × (20 + 20) = 80 m
Surface layer (Eq. 27)	No surface layer: ρ_s = ρ = 50 Ω·m, so C_s = 1.0000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 1.0000 × 50) × 0.157/√0.5 = 238.7 V (Eq. 33); E_step = (1000 + 6 × 1.0000 × 50) × 0.157/√0.5 = 288.6 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 50 × [1/252 + (1/√(20 × 400)) × (1 + 1/(1 + 0.5 × √(20/400)))] = 50 × [0.003968 + 0.011180 × 1.8994] = 1.2602 Ω; GPR = I_G × R_g = 1,000 × 1.2602 = 1,260.2 V vs E_touch 238.7 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 240/80 = 6.0000; n_b = √(L_p/(4√A)) = √(80/(4 × √400)) = 1.0000; n_c = n_d = 1 (rectangle); n = 6.0000 × 1.0000 = 6.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(4²/(16 × 0.5 × 0.01278) + (4 + 2 × 0.5)²/(8 × 4 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 6.000 − 1)))] = (1/2π) × [ln(207.84) + 0.8165 × ln(0.2315)] = 0.6592
Irregularity factor Ki	K_i = 0.644 + 0.148 × 6.000 = 1.5320
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 50 × 1,000 × 0.6592 × 1.5320 / (240 + 12) = 200.4 V → 200.4 vs limit 238.7 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(4 + 0.5) + (1/4) × (1 − 0.5^{6.000 − 2})] = (1/π) × [1.0000 + 0.2222 + 0.2344] = 0.4636
Step voltage Es	L_S = 0.75 × 240 + 0.85 × 12 = 190.2 m; E_s = ρ × I_G × K_s × K_i / L_S = 50 × 1,000 × 0.4636 × 1.5320 / 190.2 = 186.7 V → 186.7 vs limit 288.6 V PASS
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,083)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(484/274)] = 169.66 A·√s/mm²; A = I_G × √t / k = 1,000 × √0.5 / 169.66 = 4.17 mm² → next standard size 6 mm²
Result	2/2 checks pass → PASS; touch 84 % and step 65 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: no surface layer so ρ_s = ρ and Cs = 1; E_touch70 = (1000 + 1.5 × 1.0000 × 50) × 0.157/√0.5 = 238.7 V (Eq. 33), E_step70 = (1000 + 6 × 1.0000 × 50) × 0.157/√0.5 = 288.6 V (Eq. 30); grid 6 × 6 conductors, L_C = 6 × 20 + 6 × 20 = 240 m, L_R = 4 × 3 = 12 m, L_T = 252 m, A = 400 m²; n = n_a·n_b = 6.0000 × 1.0000 = 6.000 (§16.5.1), Ki = 0.644 + 0.148 × 6.000 = 1.5320, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.6592 (Eq. 86), Em = 50 × 1000 × 0.6592 × 1.5320/252 = 200.4 V (Eq. 85); Ks = 0.4636, L_S = 0.75 × 240 + 0.85 × 12 = 190.2 m, Es = 50 × 1000 × 0.4636 × 1.5320/190.2 = 186.7 V (§16.5.2); R_g = 50 × [1/252 + (1/√(20 × 400)) × (1 + 1/(1 + 0.5√(20/400)))] = 50 × [0.003968 + 0.011180 × 1.8994] = 1.2602 Ω (Eq. 57, Sverak), GPR = 1000 × 1.2602 = 1260.2 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 250 °C = min(250 bolted/pressure-type (default) joint limit, 1083 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 1/√[(3.42 × 1e-4/(0.5 × 0.00393 × 1.72)) × ln(484/274)] = 4.17 mm² (Eq. 37) (validation case `c0036-20x20-small-grid-4-rods`)

Example 2 — 60 × 40 m grid, 20 rods, crushed rock, 10 kA

A zone-substation grid with 20 rods and a 3000 Ω·m crushed-rock layer (Cs = 0.70): with the full 10 kA taken as grid current (no split factor) the mesh voltage still exceeds the touch limit; the step voltage passes.

Given: Soil ρ = 100 Ω·m; grid current I_G = 10 kA for t_s = 0.5 s (no Sf/Df applied). Grid 60 × 40 m, conductors at D = 6 m spacing buried h = 0.5 m, 12.78 mm annealed copper with bolted / pressure-type joints; 20 rods × 3 m (16 mm); surface layer 3,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊60/6⌋ + 1 = 11; n_y = ⌊40/6⌋ + 1 = 7; L_C = 7 × 60 + 11 × 40 = 860 m; L_R = 20 × 3 = 60 m; L_T = 920 m; A = 60 × 40 = 2,400 m²; L_p = 2 × (60 + 40) = 200 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 100/3,000) / (2 × 0.1 + 0.09) = 0.7000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7000 × 3,000) × 0.157/√0.5 = 921.4 V (Eq. 33); E_step = (1000 + 6 × 0.7000 × 3,000) × 0.157/√0.5 = 3,019.6 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 100 × [1/920 + (1/√(20 × 2,400)) × (1 + 1/(1 + 0.5 × √(20/2,400)))] = 100 × [0.001087 + 0.004564 × 1.9563] = 1.0016 Ω; GPR = I_G × R_g = 10,000 × 1.0016 = 10,016.4 V vs E_touch 921.4 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 860/200 = 8.6000; n_b = √(L_p/(4√A)) = √(200/(4 × √2,400)) = 1.0103; n_c = n_d = 1 (rectangle); n = 8.6000 × 1.0103 = 8.688
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(6²/(16 × 0.5 × 0.01278) + (6 + 2 × 0.5)²/(8 × 6 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 8.688 − 1)))] = (1/2π) × [ln(422.21) + 0.8165 × ln(0.1555)] = 0.7203
Irregularity factor Ki	K_i = 0.644 + 0.148 × 8.688 = 1.9299
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 100 × 10,000 × 0.7203 × 1.9299 / (860 + 60) = 1,511.0 V → 1,511.0 vs limit 921.4 V FAIL
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(6 + 0.5) + (1/6) × (1 − 0.5^{8.688 − 2})] = (1/π) × [1.0000 + 0.1538 + 0.1651] = 0.4198
Step voltage Es	L_S = 0.75 × 860 + 0.85 × 60 = 696.0 m; E_s = ρ × I_G × K_s × K_i / L_S = 100 × 10,000 × 0.4198 × 1.9299 / 696.0 = 1,164.1 V → 1,164.1 vs limit 3,019.6 V PASS
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,083)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(484/274)] = 169.66 A·√s/mm²; A = I_G × √t / k = 10,000 × √0.5 / 169.66 = 41.68 mm² → next standard size 50 mm²
Result	1/2 checks pass → FAIL; touch 164 % and step 39 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 100/3000)/(2 × 0.1 + 0.09) = 0.7000 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7000 × 3000) × 0.157/√0.5 = 921.4 V (Eq. 33), E_step70 = (1000 + 6 × 0.7000 × 3000) × 0.157/√0.5 = 3019.6 V (Eq. 30); grid 11 × 7 conductors, L_C = 7 × 60 + 11 × 40 = 860 m, L_R = 20 × 3 = 60 m, L_T = 920 m, A = 2400 m²; n = n_a·n_b = 8.6000 × 1.0103 = 8.688 (§16.5.1), Ki = 0.644 + 0.148 × 8.688 = 1.9299, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.7203 (Eq. 86), Em = 100 × 10000 × 0.7203 × 1.9299/920 = 1511.0 V (Eq. 85); Ks = 0.4198, L_S = 0.75 × 860 + 0.85 × 60 = 696 m, Es = 100 × 10000 × 0.4198 × 1.9299/696 = 1164.1 V (§16.5.2); R_g = 100 × [1/920 + (1/√(20 × 2400)) × (1 + 1/(1 + 0.5√(20/2400)))] = 100 × [0.001087 + 0.004564 × 1.9563] = 1.0016 Ω (Eq. 57, Sverak), GPR = 10000 × 1.0016 = 10016.4 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 250 °C = min(250 bolted/pressure-type (default) joint limit, 1083 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 10/√[(3.42 × 1e-4/(0.5 × 0.00393 × 1.72)) × ln(484/274)] = 41.68 mm² (Eq. 37) (validation case `c0036-60x40-grid-rods-crushed-rock`)

Example 3 — Large 100 × 80 m grid, 10 m spacing, 40 rods, 8 kA

A transmission-substation grid: 1900 m of buried conductor and a 0.15 m crushed-rock layer keep an 8 kA fault within the touch limit at 87 % utilisation.

Given: Soil ρ = 100 Ω·m; grid current I_G = 8 kA for t_s = 0.5 s (no Sf/Df applied). Grid 100 × 80 m, conductors at D = 10 m spacing buried h = 0.5 m, 12.78 mm annealed copper with bolted / pressure-type joints; 40 rods × 3 m (16 mm); surface layer 2,500 Ω·m, 0.15 m thick; body weight 70 kg.

Geometry	n_x = ⌊100/10⌋ + 1 = 11; n_y = ⌊80/10⌋ + 1 = 9; L_C = 9 × 100 + 11 × 80 = 1,780 m; L_R = 40 × 3 = 120 m; L_T = 1,900 m; A = 100 × 80 = 8,000 m²; L_p = 2 × (100 + 80) = 360 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 100/2,500) / (2 × 0.15 + 0.09) = 0.7785
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7785 × 2,500) × 0.157/√0.5 = 870.2 V (Eq. 33); E_step = (1000 + 6 × 0.7785 × 2,500) × 0.157/√0.5 = 2,814.7 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 100 × [1/1,900 + (1/√(20 × 8,000)) × (1 + 1/(1 + 0.5 × √(20/8,000)))] = 100 × [0.000526 + 0.002500 × 1.9756] = 0.5465 Ω; GPR = I_G × R_g = 8,000 × 0.5465 = 4,372.3 V vs E_touch 870.2 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 1,780/360 = 9.8889; n_b = √(L_p/(4√A)) = √(360/(4 × √8,000)) = 1.0031; n_c = n_d = 1 (rectangle); n = 9.8889 × 1.0031 = 9.920
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(10²/(16 × 0.5 × 0.01278) + (10 + 2 × 0.5)²/(8 × 10 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 9.920 − 1)))] = (1/2π) × [ln(1,086.66) + 0.8165 × ln(0.1352)] = 0.8526
Irregularity factor Ki	K_i = 0.644 + 0.148 × 9.920 = 2.1121
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 100 × 8,000 × 0.8526 × 2.1121 / (1,780 + 120) = 758.2 V → 758.2 vs limit 870.2 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(10 + 0.5) + (1/10) × (1 − 0.5^{9.920 − 2})] = (1/π) × [1.0000 + 0.0952 + 0.0996] = 0.3803
Step voltage Es	L_S = 0.75 × 1,780 + 0.85 × 120 = 1,437.0 m; E_s = ρ × I_G × K_s × K_i / L_S = 100 × 8,000 × 0.3803 × 2.1121 / 1,437.0 = 447.2 V → 447.2 vs limit 2,814.7 V PASS
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,083)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(484/274)] = 169.66 A·√s/mm²; A = I_G × √t / k = 8,000 × √0.5 / 169.66 = 33.34 mm² → next standard size 35 mm²
Result	2/2 checks pass → PASS; touch 87 % and step 16 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 100/2500)/(2 × 0.15 + 0.09) = 0.7785 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7785 × 2500) × 0.157/√0.5 = 870.2 V (Eq. 33), E_step70 = (1000 + 6 × 0.7785 × 2500) × 0.157/√0.5 = 2814.7 V (Eq. 30); grid 11 × 9 conductors, L_C = 9 × 100 + 11 × 80 = 1780 m, L_R = 40 × 3 = 120 m, L_T = 1900 m, A = 8000 m²; n = n_a·n_b = 9.8889 × 1.0031 = 9.920 (§16.5.1), Ki = 0.644 + 0.148 × 9.920 = 2.1121, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.8526 (Eq. 86), Em = 100 × 8000 × 0.8526 × 2.1121/1900 = 758.2 V (Eq. 85); Ks = 0.3803, L_S = 0.75 × 1780 + 0.85 × 120 = 1437 m, Es = 100 × 8000 × 0.3803 × 2.1121/1437 = 447.2 V (§16.5.2); R_g = 100 × [1/1900 + (1/√(20 × 8000)) × (1 + 1/(1 + 0.5√(20/8000)))] = 100 × [0.000526 + 0.002500 × 1.9756] = 0.5465 Ω (Eq. 57, Sverak), GPR = 8000 × 0.5465 = 4372.3 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 250 °C = min(250 bolted/pressure-type (default) joint limit, 1083 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 8/√[(3.42 × 1e-4/(0.5 × 0.00393 × 1.72)) × ln(484/274)] = 33.34 mm² (Eq. 37) (validation case `c0036-100x80-large-grid-40-rods`)

Example 4 — Fine 2.5 m mesh buried 1 m, hard-drawn 8 mm copper, 70 kg

A dense mesh at 1 m depth with no rods: Kh = √2 and the no-rod Kii apply, and the 0.3 s clearing time keeps the limits high enough to pass at 91 % touch utilisation.

Given: Soil ρ = 70 Ω·m; grid current I_G = 4 kA for t_s = 0.3 s (no Sf/Df applied). Grid 25 × 15 m, conductors at D = 2.5 m spacing buried h = 1 m, 8 mm hard-drawn copper with bolted / pressure-type joints; no rods; surface layer 3,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊25/2.5⌋ + 1 = 11; n_y = ⌊15/2.5⌋ + 1 = 7; L_C = 7 × 25 + 11 × 15 = 340 m; no rods; L_T = 340 m; A = 25 × 15 = 375 m²; L_p = 2 × (25 + 15) = 80 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 70/3,000) / (2 × 0.1 + 0.09) = 0.6969
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.6969 × 3,000) × 0.157/√0.3 = 1,185.6 V (Eq. 33); E_step = (1000 + 6 × 0.6969 × 3,000) × 0.157/√0.3 = 3,882.3 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 70 × [1/340 + (1/√(20 × 375)) × (1 + 1/(1 + 1 × √(20/375)))] = 70 × [0.002941 + 0.011547 × 1.8124] = 1.6708 Ω; GPR = I_G × R_g = 4,000 × 1.6708 = 6,683.3 V vs E_touch 1,185.6 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 340/80 = 8.5000; n_b = √(L_p/(4√A)) = √(80/(4 × √375)) = 1.0163; n_c = n_d = 1 (rectangle); n = 8.5000 × 1.0163 = 8.638
Mesh factor Km (Eq. 86)	K_h = √(1 + 1) = 1.4142; K_ii = 1/(2 × 8.638)^2/8.638 = 0.5170; K_m = (1/2π) × [ln(2.5²/(16 × 1 × 0.008) + (2.5 + 2 × 1)²/(8 × 2.5 × 0.008) − 1/(4 × 0.008)) + (0.5170/1.4142) × ln(8/(π × (2 × 8.638 − 1)))] = (1/2π) × [ln(144.14) + 0.3656 × ln(0.1565)] = 0.6832
Irregularity factor Ki	K_i = 0.644 + 0.148 × 8.638 = 1.9225
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 70 × 4,000 × 0.6832 × 1.9225 / (340) = 1,081.6 V → 1,081.6 vs limit 1,185.6 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 1) + 1/(2.5 + 1) + (1/2.5) × (1 − 0.5^{8.638 − 2})] = (1/π) × [0.5000 + 0.2857 + 0.3960] = 0.3761
Step voltage Es	L_S = 0.75 × 340 + 0.85 × 0 = 255.0 m; E_s = ρ × I_G × K_s × K_i / L_S = 70 × 4,000 × 0.3761 × 1.9225 / 255.0 = 794.0 V → 794.0 vs limit 3,882.3 V PASS
Grid conductor (Eq. 37, Table 1)	hard-drawn copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00381/°C, ρ_r = 1.78 µΩ·cm, K₀ = 242 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,084)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00381 × 1.78)) × ln(492/282)] = 167.53 A·√s/mm²; A = I_G × √t / k = 4,000 × √0.3 / 167.53 = 13.08 mm² → next standard size 16 mm²
Result	2/2 checks pass → PASS; touch 91 % and step 20 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 70/3000)/(2 × 0.1 + 0.09) = 0.6969 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.6969 × 3000) × 0.157/√0.3 = 1185.6 V (Eq. 33), E_step70 = (1000 + 6 × 0.6969 × 3000) × 0.157/√0.3 = 3882.3 V (Eq. 30); grid 11 × 7 conductors, L_C = 7 × 25 + 11 × 15 = 340 m, no rods, L_T = 340 m, A = 375 m²; n = n_a·n_b = 8.5000 × 1.0163 = 8.638 (§16.5.1), Ki = 0.644 + 0.148 × 8.638 = 1.9225, Kii = 0.5170, Kh = √(1 + 1) = 1.4142, Km = 0.6832 (Eq. 86), Em = 70 × 4000 × 0.6832 × 1.9225/340 = 1081.6 V (Eq. 85); Ks = 0.3761, L_S = 0.75 × 340 + 0.85 × 0 = 255 m, Es = 70 × 4000 × 0.3761 × 1.9225/255 = 794.0 V (§16.5.2); R_g = 70 × [1/340 + (1/√(20 × 375)) × (1 + 1/(1 + 1√(20/375)))] = 70 × [0.002941 + 0.011547 × 1.8124] = 1.6708 Ω (Eq. 57, Sverak), GPR = 4000 × 1.6708 = 6683.3 V; grid conductor (commercial hard-drawn copper, Table 1: TCAP 3.42, αr 0.00381, ρr 1.78, K0 242, Tm 250 °C = min(250 bolted/pressure-type (default) joint limit, 1084 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 4/√[(3.42 × 1e-4/(0.3 × 0.00381 × 1.78)) × ln(492/282)] = 13.08 mm² (Eq. 37) (validation case `c0036-25x15-fine-mesh-deep-burial`)

Example 5 — The same fine mesh assessed for a 50 kg body

The fine-mesh grid that passes at 70 kg fails at 50 kg: Cb falls from 0.157 to 0.116, cutting the tolerable voltages by 26 %, and the mesh voltage is unchanged.

Geometry	n_x = ⌊25/2.5⌋ + 1 = 11; n_y = ⌊15/2.5⌋ + 1 = 7; L_C = 7 × 25 + 11 × 15 = 340 m; no rods; L_T = 340 m; A = 25 × 15 = 375 m²; L_p = 2 × (25 + 15) = 80 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 70/3,000) / (2 × 0.1 + 0.09) = 0.6969
Tolerable voltages (50 kg)	C_b = 0.116; E_touch = (1000 + 1.5 × 0.6969 × 3,000) × 0.116/√0.3 = 876.0 V (Eq. 32); E_step = (1000 + 6 × 0.6969 × 3,000) × 0.116/√0.3 = 2,868.5 V (Eq. 29)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 70 × [1/340 + (1/√(20 × 375)) × (1 + 1/(1 + 1 × √(20/375)))] = 70 × [0.002941 + 0.011547 × 1.8124] = 1.6708 Ω; GPR = I_G × R_g = 4,000 × 1.6708 = 6,683.3 V vs E_touch 876.0 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 340/80 = 8.5000; n_b = √(L_p/(4√A)) = √(80/(4 × √375)) = 1.0163; n_c = n_d = 1 (rectangle); n = 8.5000 × 1.0163 = 8.638
Mesh factor Km (Eq. 86)	K_h = √(1 + 1) = 1.4142; K_ii = 1/(2 × 8.638)^2/8.638 = 0.5170; K_m = (1/2π) × [ln(2.5²/(16 × 1 × 0.008) + (2.5 + 2 × 1)²/(8 × 2.5 × 0.008) − 1/(4 × 0.008)) + (0.5170/1.4142) × ln(8/(π × (2 × 8.638 − 1)))] = (1/2π) × [ln(144.14) + 0.3656 × ln(0.1565)] = 0.6832
Irregularity factor Ki	K_i = 0.644 + 0.148 × 8.638 = 1.9225
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 70 × 4,000 × 0.6832 × 1.9225 / (340) = 1,081.6 V → 1,081.6 vs limit 876.0 V FAIL
Step factor Ks	K_s = (1/π) × [1/(2 × 1) + 1/(2.5 + 1) + (1/2.5) × (1 − 0.5^{8.638 − 2})] = (1/π) × [0.5000 + 0.2857 + 0.3960] = 0.3761
Step voltage Es	L_S = 0.75 × 340 + 0.85 × 0 = 255.0 m; E_s = ρ × I_G × K_s × K_i / L_S = 70 × 4,000 × 0.3761 × 1.9225 / 255.0 = 794.0 V → 794.0 vs limit 2,868.5 V PASS
Grid conductor (Eq. 37, Table 1)	hard-drawn copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00381/°C, ρ_r = 1.78 µΩ·cm, K₀ = 242 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,084)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00381 × 1.78)) × ln(492/282)] = 167.53 A·√s/mm²; A = I_G × √t / k = 4,000 × √0.3 / 167.53 = 13.08 mm² → next standard size 16 mm²
Result	1/2 checks pass → FAIL; touch 124 % and step 28 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 70/3000)/(2 × 0.1 + 0.09) = 0.6969 (Eq. 27); E_touch50 = (1000 + 1.5 × 0.6969 × 3000) × 0.116/√0.3 = 876.0 V (Eq. 32), E_step50 = (1000 + 6 × 0.6969 × 3000) × 0.116/√0.3 = 2868.5 V (Eq. 29); grid 11 × 7 conductors, L_C = 7 × 25 + 11 × 15 = 340 m, no rods, L_T = 340 m, A = 375 m²; n = n_a·n_b = 8.5000 × 1.0163 = 8.638 (§16.5.1), Ki = 0.644 + 0.148 × 8.638 = 1.9225, Kii = 0.5170, Kh = √(1 + 1) = 1.4142, Km = 0.6832 (Eq. 86), Em = 70 × 4000 × 0.6832 × 1.9225/340 = 1081.6 V (Eq. 85); Ks = 0.3761, L_S = 0.75 × 340 + 0.85 × 0 = 255 m, Es = 70 × 4000 × 0.3761 × 1.9225/255 = 794.0 V (§16.5.2); R_g = 70 × [1/340 + (1/√(20 × 375)) × (1 + 1/(1 + 1√(20/375)))] = 70 × [0.002941 + 0.011547 × 1.8124] = 1.6708 Ω (Eq. 57, Sverak), GPR = 4000 × 1.6708 = 6683.3 V; grid conductor (commercial hard-drawn copper, Table 1: TCAP 3.42, αr 0.00381, ρr 1.78, K0 242, Tm 250 °C = min(250 bolted/pressure-type (default) joint limit, 1084 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 4/√[(3.42 × 1e-4/(0.3 × 0.00381 × 1.78)) × ln(492/282)] = 13.08 mm² (Eq. 37) (validation case `c0036-25x15-fine-mesh-50kg-body`)

Example 6 — Copper-clad steel grid, 2 kA for 1 s

A slow 1 s backup clearing time lowers the tolerable voltages by √2 relative to 0.5 s; the 40 % copper-clad steel conductor, exothermic-welded so Tm is its 1084 °C fusing temperature, is sized from its own IEEE 80 Table 1 constants (k = 188.8 against 281.8 for welded annealed copper), so it needs half as much again as copper would.

Given: Soil ρ = 150 Ω·m; grid current I_G = 2 kA for t_s = 1 s (no Sf/Df applied). Grid 40 × 40 m, conductors at D = 5 m spacing buried h = 0.5 m, 12.78 mm 40 % copper-clad steel with exothermic-welded joints; 10 rods × 3 m (16 mm); surface layer 3,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊40/5⌋ + 1 = 9; n_y = ⌊40/5⌋ + 1 = 9; L_C = 9 × 40 + 9 × 40 = 720 m; L_R = 10 × 3 = 30 m; L_T = 750 m; A = 40 × 40 = 1,600 m²; L_p = 2 × (40 + 40) = 160 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 150/3,000) / (2 × 0.1 + 0.09) = 0.7052
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7052 × 3,000) × 0.157/√1 = 655.2 V (Eq. 33); E_step = (1000 + 6 × 0.7052 × 3,000) × 0.157/√1 = 2,149.8 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 150 × [1/750 + (1/√(20 × 1,600)) × (1 + 1/(1 + 0.5 × √(20/1,600)))] = 150 × [0.001333 + 0.005590 × 1.9471] = 1.8327 Ω; GPR = I_G × R_g = 2,000 × 1.8327 = 3,665.3 V vs E_touch 655.2 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 720/160 = 9.0000; n_b = √(L_p/(4√A)) = √(160/(4 × √1,600)) = 1.0000; n_c = n_d = 1 (rectangle); n = 9.0000 × 1.0000 = 9.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(5²/(16 × 0.5 × 0.01278) + (5 + 2 × 0.5)²/(8 × 5 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 9.000 − 1)))] = (1/2π) × [ln(305.16) + 0.8165 × ln(0.1498)] = 0.6638
Irregularity factor Ki	K_i = 0.644 + 0.148 × 9.000 = 1.9760
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 150 × 2,000 × 0.6638 × 1.9760 / (720 + 30) = 524.7 V → 524.7 vs limit 655.2 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(5 + 0.5) + (1/5) × (1 − 0.5^{9.000 − 2})] = (1/π) × [1.0000 + 0.1818 + 0.1984] = 0.4393
Step voltage Es	L_S = 0.75 × 720 + 0.85 × 30 = 565.5 m; E_s = ρ × I_G × K_s × K_i / L_S = 150 × 2,000 × 0.4393 × 1.9760 / 565.5 = 460.6 V → 460.6 vs limit 2,149.8 V PASS
Grid conductor (Eq. 37, Table 1)	40 % copper-clad steel: TCAP = 3.85 J/(cm³·°C), α_r = 0.00378/°C, ρ_r = 4.4 µΩ·cm, K₀ = 245 °C, T_m = 1,084 °C (exothermic weld: fusing temperature), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.85 × 10⁻⁴/(0.00378 × 4.4)) × ln(1,329/285)] = 188.79 A·√s/mm²; A = I_G × √t / k = 2,000 × √1 / 188.79 = 10.59 mm² → next standard size 16 mm²
Result	2/2 checks pass → PASS; touch 80 % and step 21 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 150/3000)/(2 × 0.1 + 0.09) = 0.7052 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7052 × 3000) × 0.157/√1 = 655.2 V (Eq. 33), E_step70 = (1000 + 6 × 0.7052 × 3000) × 0.157/√1 = 2149.8 V (Eq. 30); grid 9 × 9 conductors, L_C = 9 × 40 + 9 × 40 = 720 m, L_R = 10 × 3 = 30 m, L_T = 750 m, A = 1600 m²; n = n_a·n_b = 9.0000 × 1.0000 = 9.000 (§16.5.1), Ki = 0.644 + 0.148 × 9.000 = 1.9760, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.6638 (Eq. 86), Em = 150 × 2000 × 0.6638 × 1.9760/750 = 524.7 V (Eq. 85); Ks = 0.4393, L_S = 0.75 × 720 + 0.85 × 30 = 565.5 m, Es = 150 × 2000 × 0.4393 × 1.9760/565.5 = 460.6 V (§16.5.2); R_g = 150 × [1/750 + (1/√(20 × 1600)) × (1 + 1/(1 + 0.5√(20/1600)))] = 150 × [0.001333 + 0.005590 × 1.9471] = 1.8327 Ω (Eq. 57, Sverak), GPR = 2000 × 1.8327 = 3665.3 V; grid conductor (40 % copper-clad steel wire, Table 1: TCAP 3.85, αr 0.00378, ρr 4.4, K0 245, Tm 1084 °C = fusing temperature, exothermic weld joint, Ta 40 °C) A = 2/√[(3.85 × 1e-4/(1 × 0.00378 × 4.4)) × ln(1329/285)] = 10.59 mm² (Eq. 37), agreeing with Eq. 41: 2 × Kf 10.45 × √1 kcmil × 0.5067 = 10.59 mm² (validation case `c0036-40x40-copper-clad-steel-1s`)

Example 7 — Galvanised steel grid, 0.2 s clearing, 5000 Ω·m rock

Fast 0.2 s clearing lifts the tolerable voltages so a 4 kA fault in 300 Ω·m soil passes; the zinc-coated steel conductor has brazed joints, but the 450 °C brazed limit is above the 419 °C zinc fusing temperature, so Tm = min(450, 419) = 419 °C (k = 68.1) and it needs about four times the welded copper cross-section.

Given: Soil ρ = 300 Ω·m; grid current I_G = 4 kA for t_s = 0.2 s (no Sf/Df applied). Grid 50 × 30 m, conductors at D = 5 m spacing buried h = 0.5 m, 12.78 mm zinc-coated (galvanised) steel with brazed joints; 16 rods × 3 m (16 mm); surface layer 5,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊50/5⌋ + 1 = 11; n_y = ⌊30/5⌋ + 1 = 7; L_C = 7 × 50 + 11 × 30 = 680 m; L_R = 16 × 3 = 48 m; L_T = 728 m; A = 50 × 30 = 1,500 m²; L_p = 2 × (50 + 30) = 160 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 300/5,000) / (2 × 0.1 + 0.09) = 0.7083
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7083 × 5,000) × 0.157/√0.2 = 2,215.9 V (Eq. 33); E_step = (1000 + 6 × 0.7083 × 5,000) × 0.157/√0.2 = 7,810.5 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 300 × [1/728 + (1/√(20 × 1,500)) × (1 + 1/(1 + 0.5 × √(20/1,500)))] = 300 × [0.001374 + 0.005774 × 1.9454] = 3.7816 Ω; GPR = I_G × R_g = 4,000 × 3.7816 = 15,126.6 V vs E_touch 2,215.9 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 680/160 = 8.5000; n_b = √(L_p/(4√A)) = √(160/(4 × √1,500)) = 1.0163; n_c = n_d = 1 (rectangle); n = 8.5000 × 1.0163 = 8.638
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(5²/(16 × 0.5 × 0.01278) + (5 + 2 × 0.5)²/(8 × 5 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 8.638 − 1)))] = (1/2π) × [ln(305.16) + 0.8165 × ln(0.1565)] = 0.6694
Irregularity factor Ki	K_i = 0.644 + 0.148 × 8.638 = 1.9225
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 300 × 4,000 × 0.6694 × 1.9225 / (680 + 48) = 2,121.4 V → 2,121.4 vs limit 2,215.9 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(5 + 0.5) + (1/5) × (1 − 0.5^{8.638 − 2})] = (1/π) × [1.0000 + 0.1818 + 0.1980] = 0.4392
Step voltage Es	L_S = 0.75 × 680 + 0.85 × 48 = 550.8 m; E_s = ρ × I_G × K_s × K_i / L_S = 300 × 4,000 × 0.4392 × 1.9225 / 550.8 = 1,839.6 V → 1,839.6 vs limit 7,810.5 V PASS
Grid conductor (Eq. 37, Table 1)	zinc-coated (galvanised) steel: TCAP = 3.93 J/(cm³·°C), α_r = 0.0032/°C, ρ_r = 20.1 µΩ·cm, K₀ = 293 °C, T_m = 419 °C (brazed joint: min(450, fusing 419) = fusing), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.93 × 10⁻⁴/(0.0032 × 20.1)) × ln(712/333)] = 68.14 A·√s/mm²; A = I_G × √t / k = 4,000 × √0.2 / 68.14 = 26.25 mm² → next standard size 35 mm²
Result	2/2 checks pass → PASS; touch 96 % and step 24 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 300/5000)/(2 × 0.1 + 0.09) = 0.7083 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7083 × 5000) × 0.157/√0.2 = 2215.9 V (Eq. 33), E_step70 = (1000 + 6 × 0.7083 × 5000) × 0.157/√0.2 = 7810.5 V (Eq. 30); grid 11 × 7 conductors, L_C = 7 × 50 + 11 × 30 = 680 m, L_R = 16 × 3 = 48 m, L_T = 728 m, A = 1500 m²; n = n_a·n_b = 8.5000 × 1.0163 = 8.638 (§16.5.1), Ki = 0.644 + 0.148 × 8.638 = 1.9225, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.6694 (Eq. 86), Em = 300 × 4000 × 0.6694 × 1.9225/728 = 2121.4 V (Eq. 85); Ks = 0.4392, L_S = 0.75 × 680 + 0.85 × 48 = 550.8 m, Es = 300 × 4000 × 0.4392 × 1.9225/550.8 = 1839.6 V (§16.5.2); R_g = 300 × [1/728 + (1/√(20 × 1500)) × (1 + 1/(1 + 0.5√(20/1500)))] = 300 × [0.001374 + 0.005774 × 1.9454] = 3.7816 Ω (Eq. 57, Sverak), GPR = 4000 × 3.7816 = 15126.6 V; grid conductor (zinc-coated steel rod, Table 1: TCAP 3.93, αr 0.0032, ρr 20.1, K0 293, Tm 419 °C = min(450 brazed joint limit, 419 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 4/√[(3.93 × 1e-4/(0.2 × 0.0032 × 20.1)) × ln(712/333)] = 26.25 mm² (Eq. 37), agreeing with Eq. 41: 4 × Kf 28.96 × √0.2 kcmil × 0.5067 = 26.25 mm² (validation case `c0036-50x30-galvanised-steel-0.2s`)

Example 8 — 30 × 30 m grid, 200 Ω·m, no rods, no surface layer

A bare mesh in moderate soil with nothing to help it: both mesh and step voltages exceed the 70 kg limits.

Given: Soil ρ = 200 Ω·m; grid current I_G = 2 kA for t_s = 0.5 s (no Sf/Df applied). Grid 30 × 30 m, conductors at D = 5 m spacing buried h = 0.5 m, 10 mm annealed copper with bolted / pressure-type joints; no rods; no surface layer; body weight 70 kg.

Geometry	n_x = ⌊30/5⌋ + 1 = 7; n_y = ⌊30/5⌋ + 1 = 7; L_C = 7 × 30 + 7 × 30 = 420 m; no rods; L_T = 420 m; A = 30 × 30 = 900 m²; L_p = 2 × (30 + 30) = 120 m
Surface layer (Eq. 27)	No surface layer: ρ_s = ρ = 200 Ω·m, so C_s = 1.0000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 1.0000 × 200) × 0.157/√0.5 = 288.6 V (Eq. 33); E_step = (1000 + 6 × 1.0000 × 200) × 0.157/√0.5 = 488.5 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 200 × [1/420 + (1/√(20 × 900)) × (1 + 1/(1 + 0.5 × √(20/900)))] = 200 × [0.002381 + 0.007454 × 1.9306] = 3.3542 Ω; GPR = I_G × R_g = 2,000 × 3.3542 = 6,708.4 V vs E_touch 288.6 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 420/120 = 7.0000; n_b = √(L_p/(4√A)) = √(120/(4 × √900)) = 1.0000; n_c = n_d = 1 (rectangle); n = 7.0000 × 1.0000 = 7.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1/(2 × 7.000)^2/7.000 = 0.4705; K_m = (1/2π) × [ln(5²/(16 × 0.5 × 0.01) + (5 + 2 × 0.5)²/(8 × 5 × 0.01) − 0.5/(4 × 0.01)) + (0.4705/1.2247) × ln(8/(π × (2 × 7.000 − 1)))] = (1/2π) × [ln(390.00) + 0.3841 × ln(0.1959)] = 0.8499
Irregularity factor Ki	K_i = 0.644 + 0.148 × 7.000 = 1.6800
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 200 × 2,000 × 0.8499 × 1.6800 / (420) = 1,359.8 V → 1,359.8 vs limit 288.6 V FAIL
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(5 + 0.5) + (1/5) × (1 − 0.5^{7.000 − 2})] = (1/π) × [1.0000 + 0.1818 + 0.1938] = 0.4379
Step voltage Es	L_S = 0.75 × 420 + 0.85 × 0 = 315.0 m; E_s = ρ × I_G × K_s × K_i / L_S = 200 × 2,000 × 0.4379 × 1.6800 / 315.0 = 934.1 V → 934.1 vs limit 488.5 V FAIL
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,083)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(484/274)] = 169.66 A·√s/mm²; A = I_G × √t / k = 2,000 × √0.5 / 169.66 = 8.34 mm² → next standard size 10 mm²
Result	0/2 checks pass → FAIL; touch 471 % and step 191 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: no surface layer so ρ_s = ρ and Cs = 1; E_touch70 = (1000 + 1.5 × 1.0000 × 200) × 0.157/√0.5 = 288.6 V (Eq. 33), E_step70 = (1000 + 6 × 1.0000 × 200) × 0.157/√0.5 = 488.5 V (Eq. 30); grid 7 × 7 conductors, L_C = 7 × 30 + 7 × 30 = 420 m, no rods, L_T = 420 m, A = 900 m²; n = n_a·n_b = 7.0000 × 1.0000 = 7.000 (§16.5.1), Ki = 0.644 + 0.148 × 7.000 = 1.6800, Kii = 0.4705, Kh = √(1 + 0.5) = 1.2247, Km = 0.8499 (Eq. 86), Em = 200 × 2000 × 0.8499 × 1.6800/420 = 1359.8 V (Eq. 85); Ks = 0.4379, L_S = 0.75 × 420 + 0.85 × 0 = 315 m, Es = 200 × 2000 × 0.4379 × 1.6800/315 = 934.1 V (§16.5.2); R_g = 200 × [1/420 + (1/√(20 × 900)) × (1 + 1/(1 + 0.5√(20/900)))] = 200 × [0.002381 + 0.007454 × 1.9306] = 3.3542 Ω (Eq. 57, Sverak), GPR = 2000 × 3.3542 = 6708.4 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 250 °C = min(250 bolted/pressure-type (default) joint limit, 1083 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 2/√[(3.42 × 1e-4/(0.5 × 0.00393 × 1.72)) × ln(484/274)] = 8.34 mm² (Eq. 37) (validation case `c0036-30x30-grid-200ohm-m-no-rods`)

Example 9 — Coarse 10 m mesh: touch voltage fails, step passes

Wide conductor spacing (only 4 × 4 conductors) drives the mesh voltage to twice the touch limit while the step voltage, with its 6×ρ allowance, still passes.

Given: Soil ρ = 100 Ω·m; grid current I_G = 1 kA for t_s = 0.5 s (no Sf/Df applied). Grid 30 × 30 m, conductors at D = 10 m spacing buried h = 0.5 m, 12.78 mm annealed copper with bolted / pressure-type joints; no rods; no surface layer; body weight 70 kg.

Geometry	n_x = ⌊30/10⌋ + 1 = 4; n_y = ⌊30/10⌋ + 1 = 4; L_C = 4 × 30 + 4 × 30 = 240 m; no rods; L_T = 240 m; A = 30 × 30 = 900 m²; L_p = 2 × (30 + 30) = 120 m
Surface layer (Eq. 27)	No surface layer: ρ_s = ρ = 100 Ω·m, so C_s = 1.0000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 1.0000 × 100) × 0.157/√0.5 = 255.3 V (Eq. 33); E_step = (1000 + 6 × 1.0000 × 100) × 0.157/√0.5 = 355.3 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 100 × [1/240 + (1/√(20 × 900)) × (1 + 1/(1 + 0.5 × √(20/900)))] = 100 × [0.004167 + 0.007454 × 1.9306] = 1.8557 Ω; GPR = I_G × R_g = 1,000 × 1.8557 = 1,855.7 V vs E_touch 255.3 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 240/120 = 4.0000; n_b = √(L_p/(4√A)) = √(120/(4 × √900)) = 1.0000; n_c = n_d = 1 (rectangle); n = 4.0000 × 1.0000 = 4.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1/(2 × 4.000)^2/4.000 = 0.3536; K_m = (1/2π) × [ln(10²/(16 × 0.5 × 0.01278) + (10 + 2 × 0.5)²/(8 × 10 × 0.01278) − 0.5/(4 × 0.01278)) + (0.3536/1.2247) × ln(8/(π × (2 × 4.000 − 1)))] = (1/2π) × [ln(1,086.66) + 0.2887 × ln(0.3638)] = 1.0662
Irregularity factor Ki	K_i = 0.644 + 0.148 × 4.000 = 1.2360
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 100 × 1,000 × 1.0662 × 1.2360 / (240) = 549.1 V → 549.1 vs limit 255.3 V FAIL
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(10 + 0.5) + (1/10) × (1 − 0.5^{4.000 − 2})] = (1/π) × [1.0000 + 0.0952 + 0.0750] = 0.3725
Step voltage Es	L_S = 0.75 × 240 + 0.85 × 0 = 180.0 m; E_s = ρ × I_G × K_s × K_i / L_S = 100 × 1,000 × 0.3725 × 1.2360 / 180.0 = 255.8 V → 255.8 vs limit 355.3 V PASS
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,083)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(484/274)] = 169.66 A·√s/mm²; A = I_G × √t / k = 1,000 × √0.5 / 169.66 = 4.17 mm² → next standard size 6 mm²
Result	1/2 checks pass → FAIL; touch 215 % and step 72 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: no surface layer so ρ_s = ρ and Cs = 1; E_touch70 = (1000 + 1.5 × 1.0000 × 100) × 0.157/√0.5 = 255.3 V (Eq. 33), E_step70 = (1000 + 6 × 1.0000 × 100) × 0.157/√0.5 = 355.3 V (Eq. 30); grid 4 × 4 conductors, L_C = 4 × 30 + 4 × 30 = 240 m, no rods, L_T = 240 m, A = 900 m²; n = n_a·n_b = 4.0000 × 1.0000 = 4.000 (§16.5.1), Ki = 0.644 + 0.148 × 4.000 = 1.2360, Kii = 0.3536, Kh = √(1 + 0.5) = 1.2247, Km = 1.0662 (Eq. 86), Em = 100 × 1000 × 1.0662 × 1.2360/240 = 549.1 V (Eq. 85); Ks = 0.3725, L_S = 0.75 × 240 + 0.85 × 0 = 180 m, Es = 100 × 1000 × 0.3725 × 1.2360/180 = 255.8 V (§16.5.2); R_g = 100 × [1/240 + (1/√(20 × 900)) × (1 + 1/(1 + 0.5√(20/900)))] = 100 × [0.004167 + 0.007454 × 1.9306] = 1.8557 Ω (Eq. 57, Sverak), GPR = 1000 × 1.8557 = 1855.7 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 250 °C = min(250 bolted/pressure-type (default) joint limit, 1083 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 1/√[(3.42 × 1e-4/(0.5 × 0.00393 × 1.72)) × ln(484/274)] = 4.17 mm² (Eq. 37) (validation case `c0036-30x30-coarse-mesh-touch-fail`)

Example 10 — Rocky 1000 Ω·m site, 1 s clearing: both limits fail

High soil resistivity with no surface layer: even 3 kA produces mesh and step voltages far above the limits, so a surface layer, more conductor or faster clearing is needed.

Given: Soil ρ = 1,000 Ω·m; grid current I_G = 3 kA for t_s = 1 s (no Sf/Df applied). Grid 40 × 40 m, conductors at D = 8 m spacing buried h = 0.5 m, 12.78 mm annealed copper with bolted / pressure-type joints; 8 rods × 3 m (16 mm); no surface layer; body weight 70 kg.

Geometry	n_x = ⌊40/8⌋ + 1 = 6; n_y = ⌊40/8⌋ + 1 = 6; L_C = 6 × 40 + 6 × 40 = 480 m; L_R = 8 × 3 = 24 m; L_T = 504 m; A = 40 × 40 = 1,600 m²; L_p = 2 × (40 + 40) = 160 m
Surface layer (Eq. 27)	No surface layer: ρ_s = ρ = 1,000 Ω·m, so C_s = 1.0000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 1.0000 × 1,000) × 0.157/√1 = 392.5 V (Eq. 33); E_step = (1000 + 6 × 1.0000 × 1,000) × 0.157/√1 = 1,099.0 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 1,000 × [1/504 + (1/√(20 × 1,600)) × (1 + 1/(1 + 0.5 × √(20/1,600)))] = 1,000 × [0.001984 + 0.005590 × 1.9471] = 12.8685 Ω; GPR = I_G × R_g = 3,000 × 12.8685 = 38,605.5 V vs E_touch 392.5 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 480/160 = 6.0000; n_b = √(L_p/(4√A)) = √(160/(4 × √1,600)) = 1.0000; n_c = n_d = 1 (rectangle); n = 6.0000 × 1.0000 = 6.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(8²/(16 × 0.5 × 0.01278) + (8 + 2 × 0.5)²/(8 × 8 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 6.000 − 1)))] = (1/2π) × [ln(715.23) + 0.8165 × ln(0.2315)] = 0.8559
Irregularity factor Ki	K_i = 0.644 + 0.148 × 6.000 = 1.5320
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 1,000 × 3,000 × 0.8559 × 1.5320 / (480 + 24) = 7,805.2 V → 7,805.2 vs limit 392.5 V FAIL
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(8 + 0.5) + (1/8) × (1 − 0.5^{6.000 − 2})] = (1/π) × [1.0000 + 0.1176 + 0.1172] = 0.3931
Step voltage Es	L_S = 0.75 × 480 + 0.85 × 24 = 380.4 m; E_s = ρ × I_G × K_s × K_i / L_S = 1,000 × 3,000 × 0.3931 × 1.5320 / 380.4 = 4,749.0 V → 4,749.0 vs limit 1,099.0 V FAIL
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,083)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(484/274)] = 169.66 A·√s/mm²; A = I_G × √t / k = 3,000 × √1 / 169.66 = 17.68 mm² → next standard size 25 mm²
Result	0/2 checks pass → FAIL; touch 1,989 % and step 432 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: no surface layer so ρ_s = ρ and Cs = 1; E_touch70 = (1000 + 1.5 × 1.0000 × 1000) × 0.157/√1 = 392.5 V (Eq. 33), E_step70 = (1000 + 6 × 1.0000 × 1000) × 0.157/√1 = 1099.0 V (Eq. 30); grid 6 × 6 conductors, L_C = 6 × 40 + 6 × 40 = 480 m, L_R = 8 × 3 = 24 m, L_T = 504 m, A = 1600 m²; n = n_a·n_b = 6.0000 × 1.0000 = 6.000 (§16.5.1), Ki = 0.644 + 0.148 × 6.000 = 1.5320, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.8559 (Eq. 86), Em = 1000 × 3000 × 0.8559 × 1.5320/504 = 7805.2 V (Eq. 85); Ks = 0.3931, L_S = 0.75 × 480 + 0.85 × 24 = 380.4 m, Es = 1000 × 3000 × 0.3931 × 1.5320/380.4 = 4749.0 V (§16.5.2); R_g = 1000 × [1/504 + (1/√(20 × 1600)) × (1 + 1/(1 + 0.5√(20/1600)))] = 1000 × [0.001984 + 0.005590 × 1.9471] = 12.8685 Ω (Eq. 57, Sverak), GPR = 3000 × 12.8685 = 38605.5 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 250 °C = min(250 bolted/pressure-type (default) joint limit, 1083 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 3/√[(3.42 × 1e-4/(1 × 0.00393 × 1.72)) × ln(484/274)] = 17.68 mm² (Eq. 37) (validation case `c0036-40x40-rocky-1000ohm-m-both-fail`)

Example 11 — 30 % copper-clad steel with bolted joints limited to 250 °C

Table 1 constants for 30 % copper-clad steel with bolted (pressure-type) joints: Tm = min(250, 1084) = 250 °C instead of the fusing temperature, so k falls from 163.6 to 98.0 and the conductor grows from 35 to 50 mm².

Given: Soil ρ = 100 Ω·m; grid current I_G = 5 kA for t_s = 0.5 s (no Sf/Df applied). Grid 40 × 40 m, conductors at D = 5 m spacing buried h = 0.5 m, 12.78 mm 30 % copper-clad steel with bolted / pressure-type joints; 16 rods × 3 m (16 mm); surface layer 3,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊40/5⌋ + 1 = 9; n_y = ⌊40/5⌋ + 1 = 9; L_C = 9 × 40 + 9 × 40 = 720 m; L_R = 16 × 3 = 48 m; L_T = 768 m; A = 40 × 40 = 1,600 m²; L_p = 2 × (40 + 40) = 160 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 100/3,000) / (2 × 0.1 + 0.09) = 0.7000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7000 × 3,000) × 0.157/√0.5 = 921.4 V (Eq. 33); E_step = (1000 + 6 × 0.7000 × 3,000) × 0.157/√0.5 = 3,019.6 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 100 × [1/768 + (1/√(20 × 1,600)) × (1 + 1/(1 + 0.5 × √(20/1,600)))] = 100 × [0.001302 + 0.005590 × 1.9471] = 1.2186 Ω; GPR = I_G × R_g = 5,000 × 1.2186 = 6,093.2 V vs E_touch 921.4 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 720/160 = 9.0000; n_b = √(L_p/(4√A)) = √(160/(4 × √1,600)) = 1.0000; n_c = n_d = 1 (rectangle); n = 9.0000 × 1.0000 = 9.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(5²/(16 × 0.5 × 0.01278) + (5 + 2 × 0.5)²/(8 × 5 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 9.000 − 1)))] = (1/2π) × [ln(305.16) + 0.8165 × ln(0.1498)] = 0.6638
Irregularity factor Ki	K_i = 0.644 + 0.148 × 9.000 = 1.9760
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 100 × 5,000 × 0.6638 × 1.9760 / (720 + 48) = 853.9 V → 853.9 vs limit 921.4 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(5 + 0.5) + (1/5) × (1 − 0.5^{9.000 − 2})] = (1/π) × [1.0000 + 0.1818 + 0.1984] = 0.4393
Step voltage Es	L_S = 0.75 × 720 + 0.85 × 48 = 580.8 m; E_s = ρ × I_G × K_s × K_i / L_S = 100 × 5,000 × 0.4393 × 1.9760 / 580.8 = 747.4 V → 747.4 vs limit 3,019.6 V PASS
Grid conductor (Eq. 37, Table 1)	30 % copper-clad steel: TCAP = 3.85 J/(cm³·°C), α_r = 0.00378/°C, ρ_r = 5.86 µΩ·cm, K₀ = 245 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,084)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.85 × 10⁻⁴/(0.00378 × 5.86)) × ln(495/285)] = 97.96 A·√s/mm²; A = I_G × √t / k = 5,000 × √0.5 / 97.96 = 36.09 mm² → next standard size 50 mm²
Result	2/2 checks pass → PASS; touch 93 % and step 25 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 100/3000)/(2 × 0.1 + 0.09) = 0.7000 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7000 × 3000) × 0.157/√0.5 = 921.4 V (Eq. 33), E_step70 = (1000 + 6 × 0.7000 × 3000) × 0.157/√0.5 = 3019.6 V (Eq. 30); grid 9 × 9 conductors, L_C = 9 × 40 + 9 × 40 = 720 m, L_R = 16 × 3 = 48 m, L_T = 768 m, A = 1600 m²; n = n_a·n_b = 9.0000 × 1.0000 = 9.000 (§16.5.1), Ki = 0.644 + 0.148 × 9.000 = 1.9760, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.6638 (Eq. 86), Em = 100 × 5000 × 0.6638 × 1.9760/768 = 853.9 V (Eq. 85); Ks = 0.4393, L_S = 0.75 × 720 + 0.85 × 48 = 580.8 m, Es = 100 × 5000 × 0.4393 × 1.9760/580.8 = 747.4 V (§16.5.2); R_g = 100 × [1/768 + (1/√(20 × 1600)) × (1 + 1/(1 + 0.5√(20/1600)))] = 100 × [0.001302 + 0.005590 × 1.9471] = 1.2186 Ω (Eq. 57, Sverak), GPR = 5000 × 1.2186 = 6093.2 V; grid conductor (30 % copper-clad steel wire, Table 1: TCAP 3.85, αr 0.00378, ρr 5.86, K0 245, Tm 250 °C = min(250 bolted/pressure-type joint limit, 1084 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 5/√[(3.85 × 1e-4/(0.5 × 0.00378 × 5.86)) × ln(495/285)] = 36.09 mm² (Eq. 37) (validation case `c0036-40x40-ccs30-bolted-250c`)

Example 12 — 1020 steel grid, 4 kA for 1 s

A plain exothermic-welded 1020 steel grid conductor (10.8 % conductivity, ρr = 15.9 µΩ·cm, Tm = 1510 °C fusing): k = 123.7, so 32.3 mm² where welded annealed copper would need 14.2 mm²; the 4 m mesh with 24 rods passes touch at 92 %.

Given: Soil ρ = 150 Ω·m; grid current I_G = 4 kA for t_s = 1 s (no Sf/Df applied). Grid 50 × 50 m, conductors at D = 4 m spacing buried h = 0.5 m, 14 mm 1020 steel with exothermic-welded joints; 24 rods × 3 m (16 mm); surface layer 3,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊50/4⌋ + 1 = 13; n_y = ⌊50/4⌋ + 1 = 13; L_C = 13 × 50 + 13 × 50 = 1,300 m; L_R = 24 × 3 = 72 m; L_T = 1,372 m; A = 50 × 50 = 2,500 m²; L_p = 2 × (50 + 50) = 200 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 150/3,000) / (2 × 0.1 + 0.09) = 0.7052
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7052 × 3,000) × 0.157/√1 = 655.2 V (Eq. 33); E_step = (1000 + 6 × 0.7052 × 3,000) × 0.157/√1 = 2,149.8 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 150 × [1/1,372 + (1/√(20 × 2,500)) × (1 + 1/(1 + 0.5 × √(20/2,500)))] = 150 × [0.000729 + 0.004472 × 1.9572] = 1.4223 Ω; GPR = I_G × R_g = 4,000 × 1.4223 = 5,689.0 V vs E_touch 655.2 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 1,300/200 = 13.0000; n_b = √(L_p/(4√A)) = √(200/(4 × √2,500)) = 1.0000; n_c = n_d = 1 (rectangle); n = 13.0000 × 1.0000 = 13.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(4²/(16 × 0.5 × 0.014) + (4 + 2 × 0.5)²/(8 × 4 × 0.014) − 0.5/(4 × 0.014)) + (1.0000/1.2247) × ln(8/(π × (2 × 13.000 − 1)))] = (1/2π) × [ln(189.73) + 0.8165 × ln(0.1019)] = 0.5380
Irregularity factor Ki	K_i = 0.644 + 0.148 × 13.000 = 2.5680
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 150 × 4,000 × 0.5380 × 2.5680 / (1,300 + 72) = 604.2 V → 604.2 vs limit 655.2 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(4 + 0.5) + (1/4) × (1 − 0.5^{13.000 − 2})] = (1/π) × [1.0000 + 0.2222 + 0.2499] = 0.4686
Step voltage Es	L_S = 0.75 × 1,300 + 0.85 × 72 = 1,036.2 m; E_s = ρ × I_G × K_s × K_i / L_S = 150 × 4,000 × 0.4686 × 2.5680 / 1,036.2 = 696.8 V → 696.8 vs limit 2,149.8 V PASS
Grid conductor (Eq. 37, Table 1)	1020 steel: TCAP = 3.28 J/(cm³·°C), α_r = 0.0016/°C, ρ_r = 15.9 µΩ·cm, K₀ = 605 °C, T_m = 1,510 °C (exothermic weld: fusing temperature), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.28 × 10⁻⁴/(0.0016 × 15.9)) × ln(2,115/645)] = 123.74 A·√s/mm²; A = I_G × √t / k = 4,000 × √1 / 123.74 = 32.33 mm² → next standard size 35 mm²
Result	2/2 checks pass → PASS; touch 92 % and step 32 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 150/3000)/(2 × 0.1 + 0.09) = 0.7052 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7052 × 3000) × 0.157/√1 = 655.2 V (Eq. 33), E_step70 = (1000 + 6 × 0.7052 × 3000) × 0.157/√1 = 2149.8 V (Eq. 30); grid 13 × 13 conductors, L_C = 13 × 50 + 13 × 50 = 1300 m, L_R = 24 × 3 = 72 m, L_T = 1372 m, A = 2500 m²; n = n_a·n_b = 13.0000 × 1.0000 = 13.000 (§16.5.1), Ki = 0.644 + 0.148 × 13.000 = 2.5680, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.5380 (Eq. 86), Em = 150 × 4000 × 0.5380 × 2.5680/1372 = 604.2 V (Eq. 85); Ks = 0.4686, L_S = 0.75 × 1300 + 0.85 × 72 = 1036.2 m, Es = 150 × 4000 × 0.4686 × 2.5680/1036.2 = 696.8 V (§16.5.2); R_g = 150 × [1/1372 + (1/√(20 × 2500)) × (1 + 1/(1 + 0.5√(20/2500)))] = 150 × [0.000729 + 0.004472 × 1.9572] = 1.4223 Ω (Eq. 57, Sverak), GPR = 4000 × 1.4223 = 5689.0 V; grid conductor (1020 steel, Table 1: TCAP 3.28, αr 0.0016, ρr 15.9, K0 605, Tm 1510 °C = fusing temperature, exothermic weld joint, Ta 40 °C) A = 4/√[(3.28 × 1e-4/(1 × 0.0016 × 15.9)) × ln(2115/645)] = 32.33 mm² (Eq. 37), agreeing with Eq. 41: 4 × Kf 15.95 × √1 kcmil × 0.5067 = 32.33 mm² (validation case `c0036-50x50-steel-1020-1s`)

Example 13 — 304 stainless steel at 20 °C ambient

The highest-resistivity Table 1 material (ρr = 72 µΩ·cm), exothermic-welded (Tm = 1400 °C fusing), with a 20 °C ambient instead of the 40 °C behind the tabulated Kf: k = 66.5 and 41.2 mm² for 5 kA in 0.3 s.

Given: Soil ρ = 80 Ω·m; grid current I_G = 5 kA for t_s = 0.3 s (no Sf/Df applied). Grid 30 × 20 m, conductors at D = 3 m spacing buried h = 0.5 m, 12 mm 304 stainless steel with exothermic-welded joints; 8 rods × 3 m (16 mm); surface layer 2,500 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊30/3⌋ + 1 = 11; n_y = ⌊20/3⌋ + 1 = 7; L_C = 7 × 30 + 11 × 20 = 430 m; L_R = 8 × 3 = 24 m; L_T = 454 m; A = 30 × 20 = 600 m²; L_p = 2 × (30 + 20) = 100 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 80/2,500) / (2 × 0.1 + 0.09) = 0.6996
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.6996 × 2,500) × 0.157/√0.3 = 1,038.6 V (Eq. 33); E_step = (1000 + 6 × 0.6996 × 2,500) × 0.157/√0.3 = 3,294.6 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 80 × [1/454 + (1/√(20 × 600)) × (1 + 1/(1 + 0.5 × √(20/600)))] = 80 × [0.002203 + 0.009129 × 1.9163] = 1.5757 Ω; GPR = I_G × R_g = 5,000 × 1.5757 = 7,878.6 V vs E_touch 1,038.6 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 430/100 = 8.6000; n_b = √(L_p/(4√A)) = √(100/(4 × √600)) = 1.0103; n_c = n_d = 1 (rectangle); n = 8.6000 × 1.0103 = 8.688
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(3²/(16 × 0.5 × 0.012) + (3 + 2 × 0.5)²/(8 × 3 × 0.012) − 0.5/(4 × 0.012)) + (1.0000/1.2247) × ln(8/(π × (2 × 8.688 − 1)))] = (1/2π) × [ln(138.89) + 0.8165 × ln(0.1555)] = 0.5434
Irregularity factor Ki	K_i = 0.644 + 0.148 × 8.688 = 1.9299
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 80 × 5,000 × 0.5434 × 1.9299 / (430 + 24) = 923.9 V → 923.9 vs limit 1,038.6 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(3 + 0.5) + (1/3) × (1 − 0.5^{8.688 − 2})] = (1/π) × [1.0000 + 0.2857 + 0.3301] = 0.5143
Step voltage Es	L_S = 0.75 × 430 + 0.85 × 24 = 342.9 m; E_s = ρ × I_G × K_s × K_i / L_S = 80 × 5,000 × 0.5143 × 1.9299 / 342.9 = 1,157.9 V → 1,157.9 vs limit 3,294.6 V PASS
Grid conductor (Eq. 37, Table 1)	304 stainless steel: TCAP = 4.03 J/(cm³·°C), α_r = 0.0013/°C, ρ_r = 72 µΩ·cm, K₀ = 749 °C, T_m = 1,400 °C (exothermic weld: fusing temperature), T_a = 20 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(4.03 × 10⁻⁴/(0.0013 × 72)) × ln(2,149/769)] = 66.52 A·√s/mm²; A = I_G × √t / k = 5,000 × √0.3 / 66.52 = 41.17 mm² → next standard size 50 mm²
Result	2/2 checks pass → PASS; touch 89 % and step 35 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 80/2500)/(2 × 0.1 + 0.09) = 0.6996 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.6996 × 2500) × 0.157/√0.3 = 1038.6 V (Eq. 33), E_step70 = (1000 + 6 × 0.6996 × 2500) × 0.157/√0.3 = 3294.6 V (Eq. 30); grid 11 × 7 conductors, L_C = 7 × 30 + 11 × 20 = 430 m, L_R = 8 × 3 = 24 m, L_T = 454 m, A = 600 m²; n = n_a·n_b = 8.6000 × 1.0103 = 8.688 (§16.5.1), Ki = 0.644 + 0.148 × 8.688 = 1.9299, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.5434 (Eq. 86), Em = 80 × 5000 × 0.5434 × 1.9299/454 = 923.9 V (Eq. 85); Ks = 0.5143, L_S = 0.75 × 430 + 0.85 × 24 = 342.9 m, Es = 80 × 5000 × 0.5143 × 1.9299/342.9 = 1157.9 V (§16.5.2); R_g = 80 × [1/454 + (1/√(20 × 600)) × (1 + 1/(1 + 0.5√(20/600)))] = 80 × [0.002203 + 0.009129 × 1.9163] = 1.5757 Ω (Eq. 57, Sverak), GPR = 5000 × 1.5757 = 7878.6 V; grid conductor (304 stainless steel, Table 1: TCAP 4.03, αr 0.0013, ρr 72.0, K0 749, Tm 1400 °C = fusing temperature, exothermic weld joint, Ta 20 °C) A = 5/√[(4.03 × 1e-4/(0.3 × 0.0013 × 72.0)) × ln(2149/769)] = 41.17 mm² (Eq. 37) (validation case `c0036-30x20-stainless-304-ambient-20c`)

Example 14 — 40 × 40 m grid buried 2 m: depth term of Eq. 57

Burying the grid at 2 m instead of 0.5 m cuts the Eq. 57 depth factor from 1.947 to 1.817 on the area term only, so Rg falls from 1.227 to 1.155 Ω; the step voltage falls far more because Ks depends on h.

Given: Soil ρ = 100 Ω·m; grid current I_G = 3 kA for t_s = 0.5 s (no Sf/Df applied). Grid 40 × 40 m, conductors at D = 5 m spacing buried h = 2 m, 12.78 mm annealed copper with bolted / pressure-type joints; no rods; surface layer 3,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊40/5⌋ + 1 = 9; n_y = ⌊40/5⌋ + 1 = 9; L_C = 9 × 40 + 9 × 40 = 720 m; no rods; L_T = 720 m; A = 40 × 40 = 1,600 m²; L_p = 2 × (40 + 40) = 160 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 100/3,000) / (2 × 0.1 + 0.09) = 0.7000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7000 × 3,000) × 0.157/√0.5 = 921.4 V (Eq. 33); E_step = (1000 + 6 × 0.7000 × 3,000) × 0.157/√0.5 = 3,019.6 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 100 × [1/720 + (1/√(20 × 1,600)) × (1 + 1/(1 + 2 × √(20/1,600)))] = 100 × [0.001389 + 0.005590 × 1.8173] = 1.1548 Ω; GPR = I_G × R_g = 3,000 × 1.1548 = 3,464.3 V vs E_touch 921.4 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 720/160 = 9.0000; n_b = √(L_p/(4√A)) = √(160/(4 × √1,600)) = 1.0000; n_c = n_d = 1 (rectangle); n = 9.0000 × 1.0000 = 9.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 2) = 1.7321; K_ii = 1/(2 × 9.000)^2/9.000 = 0.5261; K_m = (1/2π) × [ln(5²/(16 × 2 × 0.01278) + (5 + 2 × 2)²/(8 × 5 × 0.01278) − 2/(4 × 0.01278)) + (0.5261/1.7321) × ln(8/(π × (2 × 9.000 − 1)))] = (1/2π) × [ln(180.46) + 0.3037 × ln(0.1498)] = 0.7351
Irregularity factor Ki	K_i = 0.644 + 0.148 × 9.000 = 1.9760
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 100 × 3,000 × 0.7351 × 1.9760 / (720) = 605.2 V → 605.2 vs limit 921.4 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 2) + 1/(5 + 2) + (1/5) × (1 − 0.5^{9.000 − 2})] = (1/π) × [0.2500 + 0.1429 + 0.1984] = 0.1882
Step voltage Es	L_S = 0.75 × 720 + 0.85 × 0 = 540.0 m; E_s = ρ × I_G × K_s × K_i / L_S = 100 × 3,000 × 0.1882 × 1.9760 / 540.0 = 206.6 V → 206.6 vs limit 3,019.6 V PASS
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,083)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(484/274)] = 169.66 A·√s/mm²; A = I_G × √t / k = 3,000 × √0.5 / 169.66 = 12.50 mm² → next standard size 16 mm²
Result	2/2 checks pass → PASS; touch 66 % and step 7 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 100/3000)/(2 × 0.1 + 0.09) = 0.7000 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7000 × 3000) × 0.157/√0.5 = 921.4 V (Eq. 33), E_step70 = (1000 + 6 × 0.7000 × 3000) × 0.157/√0.5 = 3019.6 V (Eq. 30); grid 9 × 9 conductors, L_C = 9 × 40 + 9 × 40 = 720 m, no rods, L_T = 720 m, A = 1600 m²; n = n_a·n_b = 9.0000 × 1.0000 = 9.000 (§16.5.1), Ki = 0.644 + 0.148 × 9.000 = 1.9760, Kii = 0.5261, Kh = √(1 + 2) = 1.7321, Km = 0.7351 (Eq. 86), Em = 100 × 3000 × 0.7351 × 1.9760/720 = 605.2 V (Eq. 85); Ks = 0.1882, L_S = 0.75 × 720 + 0.85 × 0 = 540 m, Es = 100 × 3000 × 0.1882 × 1.9760/540 = 206.6 V (§16.5.2); R_g = 100 × [1/720 + (1/√(20 × 1600)) × (1 + 1/(1 + 2√(20/1600)))] = 100 × [0.001389 + 0.005590 × 1.8173] = 1.1548 Ω (Eq. 57, Sverak), GPR = 3000 × 1.1548 = 3464.3 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 250 °C = min(250 bolted/pressure-type (default) joint limit, 1083 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 3/√[(3.42 × 1e-4/(0.5 × 0.00393 × 1.72)) × ln(484/274)] = 12.50 mm² (Eq. 37) (validation case `c0036-40x40-burial-2m-grid-resistance`)

Example 15 — Joint type: exothermic-welded annealed copper, 10 kA for 0.5 s

Exothermic welds fuse with the conductor, so Tm is annealed copper's 1083 °C fusing temperature (IEEE 80 Table 1): k = 281.8 and 25.09 mm² → 35 mm². The grid itself passes touch at 86 %.

Given: Soil ρ = 100 Ω·m; grid current I_G = 10 kA for t_s = 0.5 s (no Sf/Df applied). Grid 60 × 60 m, conductors at D = 4 m spacing buried h = 0.5 m, 12.78 mm annealed copper with exothermic-welded joints; 24 rods × 3 m (16 mm); surface layer 3,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊60/4⌋ + 1 = 16; n_y = ⌊60/4⌋ + 1 = 16; L_C = 16 × 60 + 16 × 60 = 1,920 m; L_R = 24 × 3 = 72 m; L_T = 1,992 m; A = 60 × 60 = 3,600 m²; L_p = 2 × (60 + 60) = 240 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 100/3,000) / (2 × 0.1 + 0.09) = 0.7000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7000 × 3,000) × 0.157/√0.5 = 921.4 V (Eq. 33); E_step = (1000 + 6 × 0.7000 × 3,000) × 0.157/√0.5 = 3,019.6 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 100 × [1/1,992 + (1/√(20 × 3,600)) × (1 + 1/(1 + 0.5 × √(20/3,600)))] = 100 × [0.000502 + 0.003727 × 1.9641] = 0.7822 Ω; GPR = I_G × R_g = 10,000 × 0.7822 = 7,821.7 V vs E_touch 921.4 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 1,920/240 = 16.0000; n_b = √(L_p/(4√A)) = √(240/(4 × √3,600)) = 1.0000; n_c = n_d = 1 (rectangle); n = 16.0000 × 1.0000 = 16.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(4²/(16 × 0.5 × 0.01278) + (4 + 2 × 0.5)²/(8 × 4 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 16.000 − 1)))] = (1/2π) × [ln(207.84) + 0.8165 × ln(0.0821)] = 0.5246
Irregularity factor Ki	K_i = 0.644 + 0.148 × 16.000 = 3.0120
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 100 × 10,000 × 0.5246 × 3.0120 / (1,920 + 72) = 793.2 V → 793.2 vs limit 921.4 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(4 + 0.5) + (1/4) × (1 − 0.5^{16.000 − 2})] = (1/π) × [1.0000 + 0.2222 + 0.2500] = 0.4686
Step voltage Es	L_S = 0.75 × 1,920 + 0.85 × 72 = 1,501.2 m; E_s = ρ × I_G × K_s × K_i / L_S = 100 × 10,000 × 0.4686 × 3.0120 / 1,501.2 = 940.2 V → 940.2 vs limit 3,019.6 V PASS
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 1,083 °C (exothermic weld: fusing temperature), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(1,317/274)] = 281.84 A·√s/mm²; A = I_G × √t / k = 10,000 × √0.5 / 281.84 = 25.09 mm² → next standard size 35 mm²
Result	2/2 checks pass → PASS; touch 86 % and step 31 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 100/3000)/(2 × 0.1 + 0.09) = 0.7000 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7000 × 3000) × 0.157/√0.5 = 921.4 V (Eq. 33), E_step70 = (1000 + 6 × 0.7000 × 3000) × 0.157/√0.5 = 3019.6 V (Eq. 30); grid 16 × 16 conductors, L_C = 16 × 60 + 16 × 60 = 1920 m, L_R = 24 × 3 = 72 m, L_T = 1992 m, A = 3600 m²; n = n_a·n_b = 16.0000 × 1.0000 = 16.000 (§16.5.1), Ki = 0.644 + 0.148 × 16.000 = 3.0120, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.5246 (Eq. 86), Em = 100 × 10000 × 0.5246 × 3.0120/1992 = 793.2 V (Eq. 85); Ks = 0.4686, L_S = 0.75 × 1920 + 0.85 × 72 = 1501.2 m, Es = 100 × 10000 × 0.4686 × 3.0120/1501.2 = 940.2 V (§16.5.2); R_g = 100 × [1/1992 + (1/√(20 × 3600)) × (1 + 1/(1 + 0.5√(20/3600)))] = 100 × [0.000502 + 0.003727 × 1.9641] = 0.7822 Ω (Eq. 57, Sverak), GPR = 10000 × 0.7822 = 7821.7 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 1083 °C = fusing temperature, exothermic weld joint, Ta 40 °C) A = 10/√[(3.42 × 1e-4/(0.5 × 0.00393 × 1.72)) × ln(1317/274)] = 25.09 mm² (Eq. 37), agreeing with Eq. 41: 10 × Kf 7.0 × √0.5 kcmil × 0.5067 = 25.08 mm² (validation case `c0036-60x60-cu-10ka-exothermic-weld`)

Example 16 — Joint type: brazed annealed copper (450 °C), 10 kA for 0.5 s

A brazed joint limits Tm to 450 °C: ln((234 + 450)/(234 + 40)) = 0.915 against 1.570 at the fusing temperature, so k falls to 215.1 and the minimum rises to 32.87 mm², still within 35 mm².

Given: Soil ρ = 100 Ω·m; grid current I_G = 10 kA for t_s = 0.5 s (no Sf/Df applied). Grid 60 × 60 m, conductors at D = 4 m spacing buried h = 0.5 m, 12.78 mm annealed copper with brazed joints; 24 rods × 3 m (16 mm); surface layer 3,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊60/4⌋ + 1 = 16; n_y = ⌊60/4⌋ + 1 = 16; L_C = 16 × 60 + 16 × 60 = 1,920 m; L_R = 24 × 3 = 72 m; L_T = 1,992 m; A = 60 × 60 = 3,600 m²; L_p = 2 × (60 + 60) = 240 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 100/3,000) / (2 × 0.1 + 0.09) = 0.7000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7000 × 3,000) × 0.157/√0.5 = 921.4 V (Eq. 33); E_step = (1000 + 6 × 0.7000 × 3,000) × 0.157/√0.5 = 3,019.6 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 100 × [1/1,992 + (1/√(20 × 3,600)) × (1 + 1/(1 + 0.5 × √(20/3,600)))] = 100 × [0.000502 + 0.003727 × 1.9641] = 0.7822 Ω; GPR = I_G × R_g = 10,000 × 0.7822 = 7,821.7 V vs E_touch 921.4 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 1,920/240 = 16.0000; n_b = √(L_p/(4√A)) = √(240/(4 × √3,600)) = 1.0000; n_c = n_d = 1 (rectangle); n = 16.0000 × 1.0000 = 16.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(4²/(16 × 0.5 × 0.01278) + (4 + 2 × 0.5)²/(8 × 4 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 16.000 − 1)))] = (1/2π) × [ln(207.84) + 0.8165 × ln(0.0821)] = 0.5246
Irregularity factor Ki	K_i = 0.644 + 0.148 × 16.000 = 3.0120
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 100 × 10,000 × 0.5246 × 3.0120 / (1,920 + 72) = 793.2 V → 793.2 vs limit 921.4 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(4 + 0.5) + (1/4) × (1 − 0.5^{16.000 − 2})] = (1/π) × [1.0000 + 0.2222 + 0.2500] = 0.4686
Step voltage Es	L_S = 0.75 × 1,920 + 0.85 × 72 = 1,501.2 m; E_s = ρ × I_G × K_s × K_i / L_S = 100 × 10,000 × 0.4686 × 3.0120 / 1,501.2 = 940.2 V → 940.2 vs limit 3,019.6 V PASS
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 450 °C (brazed joint: min(450, fusing 1,083)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(684/274)] = 215.14 A·√s/mm²; A = I_G × √t / k = 10,000 × √0.5 / 215.14 = 32.87 mm² → next standard size 35 mm²
Result	2/2 checks pass → PASS; touch 86 % and step 31 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 100/3000)/(2 × 0.1 + 0.09) = 0.7000 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7000 × 3000) × 0.157/√0.5 = 921.4 V (Eq. 33), E_step70 = (1000 + 6 × 0.7000 × 3000) × 0.157/√0.5 = 3019.6 V (Eq. 30); grid 16 × 16 conductors, L_C = 16 × 60 + 16 × 60 = 1920 m, L_R = 24 × 3 = 72 m, L_T = 1992 m, A = 3600 m²; n = n_a·n_b = 16.0000 × 1.0000 = 16.000 (§16.5.1), Ki = 0.644 + 0.148 × 16.000 = 3.0120, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.5246 (Eq. 86), Em = 100 × 10000 × 0.5246 × 3.0120/1992 = 793.2 V (Eq. 85); Ks = 0.4686, L_S = 0.75 × 1920 + 0.85 × 72 = 1501.2 m, Es = 100 × 10000 × 0.4686 × 3.0120/1501.2 = 940.2 V (§16.5.2); R_g = 100 × [1/1992 + (1/√(20 × 3600)) × (1 + 1/(1 + 0.5√(20/3600)))] = 100 × [0.000502 + 0.003727 × 1.9641] = 0.7822 Ω (Eq. 57, Sverak), GPR = 10000 × 0.7822 = 7821.7 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 450 °C = min(450 brazed joint limit, 1083 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 10/√[(3.42 × 1e-4/(0.5 × 0.00393 × 1.72)) × ln(684/274)] = 32.87 mm² (Eq. 37) (validation case `c0036-60x60-cu-10ka-brazed-450c`)

Example 17 — Joint type: bolted annealed copper (250 °C), 10 kA for 0.5 s

Bolted, clamp and compression (pressure-type) connectors limit Tm to 250 °C, the default joint type: k = 169.7 and 41.68 mm² → 50 mm², one size above the welded grid.

Given: Soil ρ = 100 Ω·m; grid current I_G = 10 kA for t_s = 0.5 s (no Sf/Df applied). Grid 60 × 60 m, conductors at D = 4 m spacing buried h = 0.5 m, 12.78 mm annealed copper with bolted / pressure-type joints; 24 rods × 3 m (16 mm); surface layer 3,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊60/4⌋ + 1 = 16; n_y = ⌊60/4⌋ + 1 = 16; L_C = 16 × 60 + 16 × 60 = 1,920 m; L_R = 24 × 3 = 72 m; L_T = 1,992 m; A = 60 × 60 = 3,600 m²; L_p = 2 × (60 + 60) = 240 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 100/3,000) / (2 × 0.1 + 0.09) = 0.7000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7000 × 3,000) × 0.157/√0.5 = 921.4 V (Eq. 33); E_step = (1000 + 6 × 0.7000 × 3,000) × 0.157/√0.5 = 3,019.6 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 100 × [1/1,992 + (1/√(20 × 3,600)) × (1 + 1/(1 + 0.5 × √(20/3,600)))] = 100 × [0.000502 + 0.003727 × 1.9641] = 0.7822 Ω; GPR = I_G × R_g = 10,000 × 0.7822 = 7,821.7 V vs E_touch 921.4 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 1,920/240 = 16.0000; n_b = √(L_p/(4√A)) = √(240/(4 × √3,600)) = 1.0000; n_c = n_d = 1 (rectangle); n = 16.0000 × 1.0000 = 16.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(4²/(16 × 0.5 × 0.01278) + (4 + 2 × 0.5)²/(8 × 4 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 16.000 − 1)))] = (1/2π) × [ln(207.84) + 0.8165 × ln(0.0821)] = 0.5246
Irregularity factor Ki	K_i = 0.644 + 0.148 × 16.000 = 3.0120
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 100 × 10,000 × 0.5246 × 3.0120 / (1,920 + 72) = 793.2 V → 793.2 vs limit 921.4 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(4 + 0.5) + (1/4) × (1 − 0.5^{16.000 − 2})] = (1/π) × [1.0000 + 0.2222 + 0.2500] = 0.4686
Step voltage Es	L_S = 0.75 × 1,920 + 0.85 × 72 = 1,501.2 m; E_s = ρ × I_G × K_s × K_i / L_S = 100 × 10,000 × 0.4686 × 3.0120 / 1,501.2 = 940.2 V → 940.2 vs limit 3,019.6 V PASS
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 250 °C (bolted / pressure-type joint: min(250, fusing 1,083)), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(484/274)] = 169.66 A·√s/mm²; A = I_G × √t / k = 10,000 × √0.5 / 169.66 = 41.68 mm² → next standard size 50 mm²
Result	2/2 checks pass → PASS; touch 86 % and step 31 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 100/3000)/(2 × 0.1 + 0.09) = 0.7000 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7000 × 3000) × 0.157/√0.5 = 921.4 V (Eq. 33), E_step70 = (1000 + 6 × 0.7000 × 3000) × 0.157/√0.5 = 3019.6 V (Eq. 30); grid 16 × 16 conductors, L_C = 16 × 60 + 16 × 60 = 1920 m, L_R = 24 × 3 = 72 m, L_T = 1992 m, A = 3600 m²; n = n_a·n_b = 16.0000 × 1.0000 = 16.000 (§16.5.1), Ki = 0.644 + 0.148 × 16.000 = 3.0120, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.5246 (Eq. 86), Em = 100 × 10000 × 0.5246 × 3.0120/1992 = 793.2 V (Eq. 85); Ks = 0.4686, L_S = 0.75 × 1920 + 0.85 × 72 = 1501.2 m, Es = 100 × 10000 × 0.4686 × 3.0120/1501.2 = 940.2 V (§16.5.2); R_g = 100 × [1/1992 + (1/√(20 × 3600)) × (1 + 1/(1 + 0.5√(20/3600)))] = 100 × [0.000502 + 0.003727 × 1.9641] = 0.7822 Ω (Eq. 57, Sverak), GPR = 10000 × 0.7822 = 7821.7 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 250 °C = min(250 bolted/pressure-type joint limit, 1083 fusing) per IEEE 80 §11.3, Ta 40 °C) A = 10/√[(3.42 × 1e-4/(0.5 × 0.00393 × 1.72)) × ln(484/274)] = 41.68 mm² (Eq. 37) (validation case `c0036-60x60-cu-10ka-bolted-250c`)

Example 18 — Tm override: IEEE 837-qualified compression connectors at 350 °C

A compression connector qualified to IEEE Std 837 for 350 °C is entered as an explicit Tm override, which replaces the 250 °C bolted limit: k = 195.7 and 36.14 mm² → 50 mm².

Given: Soil ρ = 100 Ω·m; grid current I_G = 10 kA for t_s = 0.5 s (no Sf/Df applied). Grid 60 × 60 m, conductors at D = 4 m spacing buried h = 0.5 m, 12.78 mm annealed copper with bolted / pressure-type joints (T_m override 350 °C); 24 rods × 3 m (16 mm); surface layer 3,000 Ω·m, 0.1 m thick; body weight 70 kg.

Geometry	n_x = ⌊60/4⌋ + 1 = 16; n_y = ⌊60/4⌋ + 1 = 16; L_C = 16 × 60 + 16 × 60 = 1,920 m; L_R = 24 × 3 = 72 m; L_T = 1,992 m; A = 60 × 60 = 3,600 m²; L_p = 2 × (60 + 60) = 240 m
Surface layer (Eq. 27)	C_s = 1 − 0.09 × (1 − 100/3,000) / (2 × 0.1 + 0.09) = 0.7000
Tolerable voltages (70 kg)	C_b = 0.157; E_touch = (1000 + 1.5 × 0.7000 × 3,000) × 0.157/√0.5 = 921.4 V (Eq. 33); E_step = (1000 + 6 × 0.7000 × 3,000) × 0.157/√0.5 = 3,019.6 V (Eq. 30)
Grid resistance (Eq. 57) and GPR	R_g = ρ × [1/L_T + (1/√(20A)) × (1 + 1/(1 + h√(20/A)))] = 100 × [1/1,992 + (1/√(20 × 3,600)) × (1 + 1/(1 + 0.5 × √(20/3,600)))] = 100 × [0.000502 + 0.003727 × 1.9641] = 0.7822 Ω; GPR = I_G × R_g = 10,000 × 0.7822 = 7,821.7 V vs E_touch 921.4 V INFO
Effective conductors n	n_a = 2L_C/L_p = 2 × 1,920/240 = 16.0000; n_b = √(L_p/(4√A)) = √(240/(4 × √3,600)) = 1.0000; n_c = n_d = 1 (rectangle); n = 16.0000 × 1.0000 = 16.000
Mesh factor Km (Eq. 86)	K_h = √(1 + 0.5) = 1.2247; K_ii = 1 (rods); K_m = (1/2π) × [ln(4²/(16 × 0.5 × 0.01278) + (4 + 2 × 0.5)²/(8 × 4 × 0.01278) − 0.5/(4 × 0.01278)) + (1.0000/1.2247) × ln(8/(π × (2 × 16.000 − 1)))] = (1/2π) × [ln(207.84) + 0.8165 × ln(0.0821)] = 0.5246
Irregularity factor Ki	K_i = 0.644 + 0.148 × 16.000 = 3.0120
Mesh voltage Em (Eq. 85)	E_m = ρ × I_G × K_m × K_i / (L_C + L_R) = 100 × 10,000 × 0.5246 × 3.0120 / (1,920 + 72) = 793.2 V → 793.2 vs limit 921.4 V PASS
Step factor Ks	K_s = (1/π) × [1/(2 × 0.5) + 1/(4 + 0.5) + (1/4) × (1 − 0.5^{16.000 − 2})] = (1/π) × [1.0000 + 0.2222 + 0.2500] = 0.4686
Step voltage Es	L_S = 0.75 × 1,920 + 0.85 × 72 = 1,501.2 m; E_s = ρ × I_G × K_s × K_i / L_S = 100 × 10,000 × 0.4686 × 3.0120 / 1,501.2 = 940.2 V → 940.2 vs limit 3,019.6 V PASS
Grid conductor (Eq. 37, Table 1)	annealed copper: TCAP = 3.42 J/(cm³·°C), α_r = 0.00393/°C, ρ_r = 1.72 µΩ·cm, K₀ = 234 °C, T_m = 350 °C (user override of the bolted / pressure-type joint limit), T_a = 40 °C; k = 1000 × √[(TCAP × 10⁻⁴/(α_r × ρ_r)) × ln((K₀ + T_m)/(K₀ + T_a))] = 1000 × √[(3.42 × 10⁻⁴/(0.00393 × 1.72)) × ln(584/274)] = 195.67 A·√s/mm²; A = I_G × √t / k = 10,000 × √0.5 / 195.67 = 36.14 mm² → next standard size 50 mm²
Result	2/2 checks pass → PASS; touch 86 % and step 31 % of tolerable; governing: touch voltage
Reference	Hand calculation, IEEE 80-2013: Cs = 1 − 0.09(1 − 100/3000)/(2 × 0.1 + 0.09) = 0.7000 (Eq. 27); E_touch70 = (1000 + 1.5 × 0.7000 × 3000) × 0.157/√0.5 = 921.4 V (Eq. 33), E_step70 = (1000 + 6 × 0.7000 × 3000) × 0.157/√0.5 = 3019.6 V (Eq. 30); grid 16 × 16 conductors, L_C = 16 × 60 + 16 × 60 = 1920 m, L_R = 24 × 3 = 72 m, L_T = 1992 m, A = 3600 m²; n = n_a·n_b = 16.0000 × 1.0000 = 16.000 (§16.5.1), Ki = 0.644 + 0.148 × 16.000 = 3.0120, Kii = 1.0000, Kh = √(1 + 0.5) = 1.2247, Km = 0.5246 (Eq. 86), Em = 100 × 10000 × 0.5246 × 3.0120/1992 = 793.2 V (Eq. 85); Ks = 0.4686, L_S = 0.75 × 1920 + 0.85 × 72 = 1501.2 m, Es = 100 × 10000 × 0.4686 × 3.0120/1501.2 = 940.2 V (§16.5.2); R_g = 100 × [1/1992 + (1/√(20 × 3600)) × (1 + 1/(1 + 0.5√(20/3600)))] = 100 × [0.000502 + 0.003727 × 1.9641] = 0.7822 Ω (Eq. 57, Sverak), GPR = 10000 × 0.7822 = 7821.7 V; grid conductor (annealed soft-drawn copper, Table 1: TCAP 3.42, αr 0.00393, ρr 1.72, K0 234, Tm 350 °C = user override of the bolted/pressure-type joint limit, Ta 40 °C) A = 10/√[(3.42 × 1e-4/(0.5 × 0.00393 × 1.72)) × ln(584/274)] = 36.14 mm² (Eq. 37) (validation case `c0036-60x60-cu-10ka-compression-override-350c`)

Standards & References

IEEE Std 80-2013 — Guide for Safety in AC Substation Grounding. Primary reference for all formulas and safety criteria.
IEEE Std 81-2012 — Guide for Measuring Earth Resistivity, Ground Impedance, and Earth Surface Potentials.
AS 2067:2016 — Substations and high voltage installations exceeding 1 kV a.c.
ENA EG-0:2006 — Power System Earthing Guide (Australia/NZ).

Tips & Limitations

This calculator handles rectangular grids with uniform spacing. For L-shaped, irregular, or multi-zone grids, use the Earth Grid 3D Tool.
Ground rods are assumed evenly distributed around the grid perimeter. In practice, rods at corners and along edges provide the most benefit.
Crushed rock surface layers (3,000–5,000 Ω·m) dramatically increase tolerable touch and step voltages — always include one where installed.
If mesh voltage fails, consider: reducing spacing, adding more ground rods, increasing grid area, or using deeper burial.
Eq. 57 (Sverak) assumes uniform soil and a grid depth of 0.25 – 2.5 m. For two-layer soil, very large or irregular grids, use a numerical (segment or finite element) method.
Soil resistivity varies significantly with moisture, temperature, and depth — use measured values from a Wenner 4-pin test where possible.

Developer Guide — Architecture

This calculator follows the standard calcs component pattern:

components/c0036_earth_grid_design/
├── __init__.py              # Docstring
├── apps.py                  # Django AppConfig
├── calculations.py          # Pure Python — IEEE 80 formulas ported from TS
├── component.py             # Schema, Calculator class, builders, registration
├── equations.py             # LaTeX equation templates
└── templates/.../sidebar_form.html  # Vue 3 sidebar form

The IEEE 80 formulas were ported from the Vue 3 Earth Grid tool at _vuejs/tools/earth_grid/utils/math.ts. The Python implementation uses IEEE 80-2013 Eq. 57 (Sverak) for grid resistance (vs the full segment-based resistance summation in the 3D tool) and Eq. 37 with the Table 1 material constants for the grid conductor.

Developer Guide — Code Paths

Purpose	File	Key Function / Class
Surface derating	`calculations.py`	`calculate_cs()`
Grid geometry	`calculations.py`	`calculate_grid_geometry()`
Grid resistance	`calculations.py`	`calculate_grid_resistance()`
Mesh/step factors	`calculations.py`	`calculate_km()`, `calculate_ks()`, `calculate_ki()`
Grid conductor size	`calculations.py`	`grid_conductor_k()`, `calculate_grid_conductor_size()`, `GRID_CONDUCTOR_MATERIALS`
Tolerable voltages	`calculations.py`	`calculate_tolerable_touch()`, `calculate_tolerable_step()`
Main entry	`calculations.py`	`calculate_earth_grid(data)`
Validation	`component.py`	`EarthGridDesignCalculator.validate_parameters()`
Registration	`component.py`	`register_component(registry)`

Developer Guide — Testing

# Run all tests for this component
pytest calcs/test_c0036_earth_grid_design.py -v

The worked examples above are generated from the validation corpus (mcp_public/validation/cases/fe_calc_earthing/) by python manage.py generate_help_examples --tool fe_calc_earthing; regenerate after any engine change.

Test coverage includes:

TestCalculateCs — surface layer derating factor
TestGridGeometry — conductor count and total length
TestGridResistance — with and without ground rods
TestKmKsKi — IEEE 80 geometric factors
TestTolerableVoltages — touch and step voltage limits
TestSafetyAssessment — full pass/fail logic
TestCalculateEarthGrid — end-to-end integration
TestCalculatorClass — validation, round-trip, registration