Arc flash: other methods
DC arc flash: Stokes & Oppenlander arc resistance (iterative arcing current)
- where
- arc resistance (mΩ)
- arc gap (mm)
- arcing current (A)
- d.c. system voltage (V)
- source and circuit resistance (mΩ)
- arc power (kW)
- working distance (cm)
- working distance (mm)
- incident energy (cal/cm²)
- multiplying factor: 1 open air, 3 in a box
- arc duration (s)
Explain this equation
Gives a more realistic d.c. arcing current, arc power and incident energy than the maximum power method, for battery and d.c. distribution systems.
Rarc is the arc resistance from Stokes and Oppenlander: the arc voltage (20 + 0.534 G) × Iarc^0.12 rises with gap G but only slowly with current, so the arc behaves almost like a constant voltage drop. Rsys is the battery internal resistance plus connections and cables.
Because Rarc depends on Iarc, the current is found by iteration until Iarc = Vsys/(Rsys + Rarc) is satisfied. The arc power Iarc² × Rarc for duration Tarc is spread over a sphere of area 4πD²; Mf allows for an enclosure.
Sense check: 250 V, 12.5 mΩ and a 25 mm gap give about 11.8 kA (bolted 20 kA) and an arc voltage of about 100 V.
Watch out: when the arc voltage approaches the supply voltage a sustained arc is unlikely and the result is not meaningful.
A more realistic d.c. estimate than the maximum power method: the arc resistance from Stokes and Oppenlander’s open-air arc tests is solved together with the circuit resistance to give the actual arcing current and power.
Enter your values to see the result.
Good to know
Iarc is found by iteration so that it satisfies both the arc-resistance and the circuit equation; the working shows that converged value in both. The arc voltage (20 + 0.534 G) × Iarc^0.12 rises only slowly with current, so the arc behaves almost like a fixed voltage drop of roughly 100–150 V for 25–50 mm gaps at around 10 kA. The arc energy is spread over a sphere for open air; Ammerman et al. (2010) and NFPA 70E Annex D use this form, with a box factor (e.g. 3, or Wilkins’ factors) for enclosures. Rsys includes the battery internal resistance, cells in series, links and cables to the fault.
Common mistake
Using a gap that is too small (e.g. the clearance between terminals) when the arc in practice stretches; a longer arc means higher arc voltage and lower current, but also possibly a longer duration if the protection is slower.
Limits
Stokes and Oppenlander fitted free-burning arcs in open air (largely a.c. and d.c. laboratory arcs); applying it to d.c. arc flash and enclosures follows Ammerman et al. No standard validates d.c. incident energy by testing across the range. UK duty: under the Electricity at Work Regulations 1989 work on or near live conductors is allowed only where reg 14 is met (unreasonable to make dead, reasonable to work live, suitable precautions); regs 12 and 13 require isolation and precautions for dead working, and reg 4(4) requires suitable protective equipment. HSE HSG85 and your DNO or company safety rules (ENA-based distribution safety rules) take precedence. Apply the hierarchy of controls: de-energise first, then engineering controls (faster protection, maintenance switches, remote racking/operation, arc-resistant switchgear), then distance and procedures; arc-rated PPE is the last line of defence.
Sources
- A. D. Stokes, W. T. Oppenlander, Electric arcs in open air, Journal of Physics D: Applied Physics 24(1), 1991
- R. F. Ammerman, T. Gammon, P. K. Sen, J. P. Nelson, DC-arc models and incident-energy calculations, IEEE Trans. Industry Applications 46(5), 2010
- NFPA 70E-2024 Standard for Electrical Safety in the Workplace: Informative Annex D (incident energy and arc flash boundary methods) and Table 130.7(C)(15)(c)
- Battery Design: DC arc flash hazard calculations (Stokes & Oppenlander iteration, open-air and box energy)
- The Electricity at Work Regulations 1989 (SI 1989/635), regulations 4, 12, 13 and 14
For competent high-voltage engineers. Results are calculations from the published methods shown and depend entirely on the data you enter. They are not a substitute for a system study, manufacturer data, the applicable standards and Engineering Recommendations, or your organisation’s safety rules. Never use a result to justify working on or near live HV equipment.