High-Voltage Transmission & Substation Duct Banks, NEC 310.15(C) Neher-McGrath Ampacity
Neher-McGrath Underground Cable Ampacity Formula
Calculate AC Skin/Proximity Conductor Resistance Rac (mOhm/m), Available Conductor Thermal Headroom (K), Maximum Continuous Ampacity Imax (A), and Joule Heat Load (W/m).
Dry backfill soil doubling trench thermal resistivity Rca from 0.88 to 1.85 K·m/W slashes 90°C XLPE underground feeder ampacity by 37%!
**NEC Section `310.15(C)` & IEC `60287`** govern underground medium/high-voltage `XLPE`/`EPR` cable ratings via **J. H. Neher & M. H. McGrath’s (`1957`) Thermal Ohm’s Law**: every watt per meter of conductor `I^2 * R_{ac}` Joule heat (`where R_{ac} = R_{dc} * (1 + Y_c)` includes `50/60` Hz skin + proximity increment `Y_c`) plus **Dielectric Loss Temperature Rise `Delta_T_d`** (`°C`) must conduct radially across **Effective Total Thermal Resistance `R_{ca}`** (`K·m/W`, summing insulation, duct air space, concrete encasement, earth burial depth, and mutual heating from adjacent circuits) without exceeding **Maximum Insulation Rating `T_c`** (`90°C` for XLPE/EPR) above **Ambient Earth Temperature `T_a`** (`°C`): **`I_{max} = SQRT((T_c - (T_a + Delta_T_d)) / (R_{ac} * R_{ca}))`** (`Amps`)!
Verify on a `35` kV `500`-kcmil (`240` mm²) Copper XLPE Underground Feeder—Max_Conductor_Temp `T_c` = 90.0°C (B2), Moist_Earth_Ambient `T_a` = 25.0°C (B3), Dielectric_Temp_Rise `Delta_T_d` = 1.4°C (B4), `90°C` DC_Resistance `R_{dc}` = 78.5 uOhm/m (B5), AC_Skin/Proximity_Factor `Y_c` = 0.06 (B6), and Fluidized-Thermal-Backfill `R_{ca}` = 0.88 K·m/W (B7): `R_{ac}` equals `0.0832` mOhm/m (`83.21` uOhm/m, E2), Available Thermal Headroom equals `63.6°C` (E3), Maximum Continuous Ampacity reaches **`932.0` A** (E4), dissipating `72.27` W/m into the duct bank (E5)!
| Neher-McGrath Cable Ampacity KPI (90°C XLPE, Rdc=78.5 uOhm/m, Yc=0.06, dTd=1.4°C) | Formula | Verified Output |
|---|---|---|
| AC Conductor Resistance Rac (mOhm/m) | (Rdc_uOhm_m / 1000) * (1 + Skin_Prox_Yc) | 0.0832 mOhm/m |
| Available Thermal Headroom (°C) | Max_Tc_C - (Soil_Ta_C + Dielectric_dTd_C) | 63.6°C (53.6°C @ 35°C summer soil) |
| Max Continuous Ampacity Imax (A) | SQRT(Headroom_C / ((Rac_mOhm_m/1000) * Rca)) | 932.0 A (590.1 A = -36.7% @ dry soil!) |
| Dissipated Conductor Heat (W/m) | (Max_Ampacity_A ^ 2) * (Rac_mOhm_m / 1000) | 72.27 W/m (28.97 W/m thermal runaway cap) |
NEC 310.15(C) / IEC 60287 Neher-McGrath Thermal-Circuit Ampacity Formula
Validate `Conductor_Temp T_c` in `[60, 105]`°C, `Ambient_Soil T_a` in `[-10, 55]`°C, `Dielectric_Rise dTd >= 0`°C with positive headroom `T_c - (T_a + dTd) >= 5`°C, `R_{dc} > 5` uOhm/m, `Skin/Proximity Y_c` in `[0, 1.0]`, and `Thermal_Resistance R_{ca}` in `[0.2, 5.0]` K·m/W!
=LET(tc_c, B2, ta_c, B3, dt_d, B4, rd_u, B5, yc_f, B6, rc_a, B7, ra_o, (rd_u * 0.000001) * (1 + yc_f), dh_c, tc_c - (ta_c + dt_d), IF(OR(tc_c < 60, tc_c > 105, ta_c < -10, ta_c > 55, dt_d < 0, dh_c < 5, rd_u <= 5, yc_f < 0, yc_f > 1, rc_a < 0.2, rc_a > 5), "Invalid Ampacity Input", ROUND(dh_c / rc_a, 2)))Moisture-Migration Dried Trench Soil (`T_a = 35°C`, `R_ca = 1.85` K·m/W) Derating Check
Bake moisture out of un-engineered native sandy soil during a summer heatwave (`T_a` B3 = 35°C, `R_{ca}` B7 = 1.85 K·m/W): Headroom E3 shrinks to `53.6°C`, Allowed Heat Dissipation E5 drops `60%` (`72.27 -> 28.97` W/m), and Safe Ampacity E4 plunges **`-36.7%` to `590.1` A**—explaining underground XLPE splice blowouts during August peak load!
=IF(ISTEXT(E5), E5, LET(ra_o, (B5 * 0.000001) * (1 + B6), dh_c, B2 - (B3 + B4), ROUND(SQRT(dh_c / (ra_o * B7)), 1)))Try the example
Use the original values or change them to test your understanding.