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Votage drop *Varc* across the equivalent arc resistance which is introduced by the arc plasma in typical AC circuit depicted on Figure 1 can be expressed as:

where *I* is current in Amps, *Vs* stands for an open source voltage in Volts, *Rs* and *Xs* are system resistance and reactance respectively, including source and feeders.

*Figure 1. Single & three phase AC arc flash equivalent circuit.*

It was demonstrated that there is a minimum voltage needed to maintain an arc. That minimum depends on the current magnitude, gap width, and orientation of the electrodes. This transitional point can be calculated by ^{[1]}:

where

*G* - conductor gap distance in millimeters

*It* - the transitional point current measured in amperes

Above that minimum, the arc V-I characteristic can be expressed as ^{[1]}:

or alternatively as:

where

*Varc* - arcing voltage in Volts

*Iarc* - arcing current in Amps

Equations (1) through (3) can be solved for the maximum conductor gap where the arc can likely be sustained under the specified system voltage, the equivalent source X/R ratio and available short circuit fault current.

To find the point where the arc V-I characteristic crosses the circuit load line, solve equations (1) and (4) using the iterative method. As the first approximation, assume *Iarc* is equal to half of the available short circuit fault current. Then, follow the steps below:

- determine
*Varc*from equation (1) - substitute
*Varc*into equation (4) to determine new*Iarc*

Cycle through the steps 1 and 2 above until the answers for *Varc* converge.

Table 1 shows arcing current values measured by two laboratories^{[2]} along with accompanying system parameters, and the current values calculated solving equations 1 through 4 using the same parameters. Calculated arcing current values nearly match the actual test data. The calculations have been performed using ARCMASTER V1.0 software program.

Test | Line Voltage, V | Bolted Fault Current, A | Fault P.F., % | Gap, in. | Measured Arcing Current, A | Calculated Arcing Current, A | Variation, % |
---|---|---|---|---|---|---|---|

TI-4 | 276 | 996 | 7.9 | 1 | 900 | 940 | 4% |

TI-7 | 277 | 980 | Low | 1/2 | 900 | 900 | 0% |

TI-8a | 120 | 940 | Low | 1/2 | 750 | 750 | 0% |

TI-9 | 146 | 1150 | Low | 1/2 | 1000 | 990 | 1% |

TI-12 | 278 | 650 | 41.8 | 1/2 | 600 | 590 | 2% |

TII-6 | 263 | 2910 | 44 | 1 | 2570 | 2390 | 3% |

TII-8 | 263 | 7400 | 44 | 1 | 6000 | 5830 | 3% |

TII-9 | 263 | 10580 | 40 | 1 | 8450 | 8360 | 1% |

TII-15 | 263 | 25000 | 22 | 2-1/4 | 16600 | 16590 | 0% |

TII-19 | 263 | 41600 | 22 | 3-1/4 | 20640 | 19530 | 5% |

The energy released during an arc flash event is roughly proportional to the arc duration. The upstream protective device operation controls the arc flash duration. A fuse or properly maintained over-current protection device has a predictable time to open the circuit with a specific arc current value. Hence, arcing current impacts the released energy directly through the current itself and through interaction with the protection device.

The power in the arc can be calculated by:

where the *Parc* is measured in watts.

The energy in the arc is a function of power and time:

where the *Earc* is measured in Joules.

Incident energy exposure for an open-air arc where the heat transfer depends on the spherical energy density is then expressed as:

where

*Eiair* - incident energy from an open air arc at distance *D* in Joules/cm^{2}

*D* - distance from the arc in centimeters

This formula assumes the radiant heat transfer. Not all of the arc energy will be transferred as radiant heat as a portion of the energy is transformed into pressure rise and in heat conducted to the electrode material. Therefore, the equation (9) will produce a conservative but safe estimate of incident energy exposure.

For the arc in a box, the enclosure has a focusing effect on the incident energy. For the selected enclosure type and test distance, the incident energy from an arc flash in a box can be calculated by ^{[3]}:

where

*Eibox* - the incident energy from an arc flash in a box in Joules/cm^{2}

*D* - distance from the arc in centimeters

*A* and *K* are obtained from optimal values listed in table below:

Enclosure Type | A, cm | K |
---|---|---|

Panelboard | 10 | 0.127 |

LV Switchgear | 40 | 0.312 |

MV Switchgear | 95 | 0.416 |

Equation (9) written in terms of arc flash boundary, becomes:

where

*AFB* - arc flash boundary measured in centimeters

*Et* - threshold incident energy to second degree burn in J/cm^{2} calculated by ^{[4]}:

where *t* stands for exposure time in seconds.

Likewise, equation (10) written in terms of arc flash boundary, becomes:

The above procedure may also be applied for calculating of the released arc energy and arc flash protection boundary in three-phase system. Please keep in mind that in such a case the equivalent circuit impedance values in equation (1) should be calculated based on the available three-phase short circuit fault current values and line-to-line voltage. Also, the arcing energy in equation (8) need be multiplied by 3^{1/2} before the 3-phase fault incident energy and arc flash boundary values can be determined from equations (10) and (11) respectively.

- A.D. Stokes and W.T. Oppenlander, "Electric arcs in open air", Journal of Physics D: Applied Physics, 1991, pp. 26-35
- L. Fisher, "Resistance of Low-Voltage AC Arcs", IEEE Transactions on Industry and General Applications, vol. IGA-6, No.6, Nov/Dec 1970
- R. Wilkins, "Simple improved equations for arc flash hazard analysis", IEEE Electrical Safety Forum, August 2004
- M. Furtak and L. Silecky, "Evaluation of onset to second degree burn energy in arc flash hazard analysis", IAEI News, July-August 2012

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