800 V DC Arc Flash: Capacitor Discharge and Incident Energy

Kirchhoff's laws give $C_j\dot V_j=-I_j$ and $L_j\dot I_j=V_j-R_jI_j-V_a$ for each accessible capacitor. Ideal series diodes block reverse branch current. The converter satisfies $I_s=\min[I_{lim},\max(0,(V_{dc}-V_a)/R_s)]$. A safeguarded Newton/bisection solver selects the larger-current stable operating point, enforces voltage compliance and accepts an unconstrained root only below a 10⁻⁷ V residual. No feasible positive root means no sustained source contribution; energy conservation alone is not an operating-point test.

Stokes–Oppenlander (1991), equations 1–2, gives $P_a=(20+0.534g)I_a^{1.12}$ above $I_t=10+0.2g$ and $P_a=P_a(I_t)$ below it for horizontal arcs. Thus the horizontal low-current voltage rises as $P_a(I_t)/I_a$, rather than remaining at a voltage floor. Gap is in mm and current in A. The high-current relation is shared by both tested arrangements. The separate vertical low-current equation has not been verified here; affected vertical scenarios are withheld.

Adaptive Dormand–Prince integration resolves the two nonlinear RLC branches. Stiff histories use second-order L-stable SDIRK integration. Its implicit stages bracket the shared high-current branch; on the horizontal low-current branch they solve the piecewise quadratic load-line/power intersection and choose the lower-voltage, larger-current solution. Every accepted stage satisfies its arc-voltage residual within 10⁻⁶ V. Loss of a feasible intersection causes extinction. The continuous integrator stops at 0.1 A, the bottom of the measured range; the remaining magnetic energy is assigned to the arc. No automatic restriking is simulated. The former constant-voltage analytic discharge tail has been removed.

An initiated arc needs a finite current rise from zero. A short transition-voltage trigger phase spans the RL rise to $1.05I_t$, estimated from the starting capacitor voltages; the full capacitor RLC equations still run during it. The calculation withholds capacitor cases whose resolved peak never reaches $I_t$. It does not predict breakdown or ignition energy. Separate states integrate capacitor and source arc power, resistive losses and converter output energy $(V_a+R_sI_s)I_s$.

Ammerman (2010), equations 12–13: $H_{air}=W_a/(4\pi d^2)$ and $H_{box}=kW_a/(a^2+d^2)$. Dimensions are mm, then converted to J/cm². Panel: $a=100$ mm, $k=0.127$. LV switchgear: $a=400$ mm, $k=0.312$. One cal/cm² equals 4.184 J/cm².

Source-only clearing bounds include each evaluated nonlinear capacitor tail. All-path bounds use the uninterrupted energy crossing. The current-range audit covers the protected history, the uninterrupted exposure curve and every evaluated search tail. The Doan comparison holds the initial prospective bolted current constant, uses $P_{max}=V_{dc}I_{bf}/4$ and applies the selected exposure geometry. Capacitors contribute initial voltage divided by loop resistance; the converter contributes its limited bolted current. This screen is not an upper bound for every current-limited source. The separate Annex D expression $IE=0.01V_{dc}(I_{bf}/2)t/d_{cm}^2$ remains for the published reference cases.

One initiated, free-burning arc with fixed gap; both capacitors initially at bus voltage and loop currents initially zero. Constant R, L and C. Ideal series blocking diodes in both capacitor branches prevent reverse current and recharge from the converter. Match this topology to the fault location; unblocked capacitor branches and antiparallel voltage clamps require a different circuit model. Zero capacitance removes a branch.

The converter current limit is immediate, with no overshoot, output inductance or control dynamics. Source isolation is instantaneous at the entered total clearing time. All-path interruption diverts loop magnetic energy away from this fault. All electrical arc energy enters the thermal spreading model. Enclosure class affects exposure, not arc voltage. The thermal threshold is fixed at 1.2 cal/cm².

Scenario analysis, not a validated 800 V rack test or PPE selection. The arc data cover gaps of 5–500 mm and currents of 0.1 A to about 20 kA. Any protected, uninterrupted or clearing-search history above 20 kA flags both the exposure and the clearing bound as extrapolated. Electrode jets, barriers and shelf vent geometry need test-supported models. Enclosure factors came from AC data and were applied to DC by Ammerman; they are not shelf-specific calibrations.

Horizontal low-current power follows the published constant-power relationship. Vertical calculations are available only for a sustained high-current source with no accessible capacitors; other vertical scenarios are blocked because equation 3 has not been verified. Do not substitute a horizontal electrode arrangement for physical vertical electrodes. The initiated-arc trigger phase is a numerical closure, not a breakdown model. Real extinction, erosion, plasma persistence and restriking can change the exposure after source isolation.

ODE relative accuracy is 10⁻⁶. Stiff source-on histories use 600 and source-off histories 400 logarithmically spaced SDIRK steps to keep default evaluation near three seconds. This retains the RLC equations but adds second-order damping and finite extinction-event timing error. Against 1200 source-off steps, incident energy differed by about 0.003% at the defaults and 0.012% in the 1 F stress case. Review the energy residual and use a refined independent study when close to the thermal limit.

Pulse-chart duration follows the LC scale and the resolved peak time. A smooth 601-point axis is merged with all adaptive or implicit solver samples in that interval, retaining peaks and switching events; curves and playback interpolate the solved history with shape-preserving interpolation. The result and clearing search do not depend on this drawing grid. Playback reuses solved histories.

The source-only search uses at most eight bracketed secant evaluations and returns the last evaluated time below threshold; this can leave unused margin. The displayed bound rounds down to 0.01 ms. A bound at the search horizon is not a maximum deadline. Settled tails are reused only when capacitor voltages agree within 10⁻⁷ of bus voltage and currents within 1 µA.

NFPA 70E 2024 Annex D supplies informative context: check against the standard in force. Schneider WP219 and Uptime Update 529 motivate architecture-aware transient analysis; this tool does not reproduce their system models. Only their public abstracts were accessible here, not the complete 2026 white paper. Cal/cm² values carry their units in labels; J/cm² supplies the convertible energy-density output.