Know Your Arc: DC arc flash calculations

More than a century ago, two giants in the fledgling electrical power industry battled it out for supremacy. The conflict, sometimes referred to as “The War of the Currents,” would define whether electric power systems would use alternating current (AC) or direct current (DC). Thomas Edison was well-known for his Pearl Street generating station in New York City, which was based on DC. George Westinghouse, with help from Nicola Tesla, was a proponent of AC.


Westinghouse ultimately won the war, and AC is used to power most of the world’s electrical loads. However, even though they are not as pervasive, DC systems are also common. Examples include rectifiers, traction power systems, adjustable frequency drives, photovoltaic systems, battery banks and more.

AC and DC—electrical hazards


The electric shock hazard from both AC and DC power systems has been well documented and understood for decades, thanks to the research of people such as Charles Dalziel. Even our understanding of the arc flash hazard has greatly improved, thanks to years of research by many individuals and the 2002 introduction of IEEE 1584—IEEE Guide for Arc Flash Hazard Calculations. However, when performing arc flash calculations, IEEE 1584 only addresses the AC arc flash hazards. Currently, there are no standards for calculating the arc flash hazard for DC power systems. DC arc flash is the proverbial elephant in the room.


A work in progress


Two landmark technical papers changed the understanding of DC arc flash. D.R. Doan’s “Arc Flash Calculations for Exposures to DC Systems” helped elevate the discussion of DC arc flash calculations. It was published in IEEE Transactions on Industry Applications, Vol. 46, No. 6. This paper provides a theoretical approach to DC incident-energy calculations based on the concept that the maximum possible power in a DC arc flash occurs when the arcing voltage is 50 percent of the system voltage. Ultimately, the equations from this paper were included in the informative annex of the 2012 Edition of NFPA 70E and remain in Annex D of the 2015 edition.


A subsequent paper, “DC-Arc Models and Incident-Energy Calculations,” by R.F. Ammerman, T. Gammon, J.P. Nelson and P.K. Sen, provides a comparison study of the existing body of research into DC arcs and arc flash modeling that has been conducted over the years. It also provides a series of calculation methods for determining the incident energy from a DC arc flash in open air as well as in a box. This second paper is the basis for DC arc flash calculations that many industry professionals and software packages currently use.


Calculating the incident energy for a DC arc flash begins with a simple application of Ohm’s Law:


I = V ÷ R


Where:


I = Current in amperes (A)


V = Voltage in volts (V)


R = Resistance in ohms


By including the DC arc resistance as part of the DC circuit model illustrated in Figure 1, you can easily determine the arcing current. This circuit diagram of a battery string includes the DC voltage, DC battery resistance, conductor resistance and DC arc resistance. As part of the overall process, you must also calculate the DC arc resistance because it is usually not known. Once all of the resistance values have been determined, you can calculate the DC arcing current and IDC arc with the following:


IDC arc = VDC ÷ (Rbattery + Rconductor + Rarc)


The DC arc models paper also refers to another important document, “Electric Arcs in Open Air,” published in the Journal of Physics D: Applied Physics in 1991 by W.T. Oppenlander and A.D. Stokes. The research included in this document led to the development of the following equation for arc resistance:


Rarc = [20 + (0.534 × G)] ÷ (IDC arc 0.88)


Where:


Rarc = resistance of the arc in ohms


G = conductor gap distance in millimeters


IDC arc = DC arcing current


To calculate the arc resistance using this equation, you must know the conductor gap distance and the DC arcing current. The gap distance is specified by the user. However, in order to determine the DC arcing current, you must know the arc resistance. This creates an interesting dilemma since you need the arcing current to calculate the arc resistance and you need the arc resistance to calculate the arcing current.


An iterative method can be used to solve this problem, but it requires an initial assumption of the DC arcing current. It’s reasonable to assume that the DC arcing short-circuit current is 50 percent of the DC bolted short-circuit current. Assuming this, you can calculate DC arc resistance and use it to recalculate the DC arcing current. Then, you can use the “new” DC arcing current to recalculate the DC arc resistance. Continue this process until the DC resistance and DC arcing current values no longer change significantly and converge to a final answer.


DC arc resistance and DC arcing 
current calculations—iterative solution


Figure 2 illustrates the circuit that is used as an example for calculating the DC arc resistance and the DC arcing current. The calculation process begins by determining the DC bolted short-circuit current. This requires taking the DC voltage (V DC) and dividing by the known impedances of the conductor and battery string.


Begin by solving for the bolted DC short-circuit current using the values in Figure 2. For the bolted case, ignore arc resistance and conductor gap distance; use only the resistance of the battery string and conductor.


IDC bolted = VDC ÷ (Rbattery + Rconductor)


IDC bolted = 256V ÷ (0.01150Ω + 0.00194Ω) = 19,048A


As a first approximation of the DC arcing current, IDC arc :


IDC arc = 0.5 × IDC bolted


Therefore:


IDC arc = 0.5 × 19,048A


IDC arc = 9,524A


DC arc resistance worksheet


There are so many calculation steps to keep track of that I developed a series of worksheets in 2010 for my arc flash training program. These can be used to simplify the calculation process. The worksheets, as well as the examples that follow, are from my book, “Complete Guide to Arc Flash Hazard Calculation Studies,” published by Brainfiller Inc. in 2010.


The DC arc resistance worksheet in Figure 3 is used for calculating the DC arc resistance of this example. It provides a step-by-step method for calculating the DC arc resistance based on the Stokes/Oppenlander equation. To use the worksheet, the following data is required:


• Conductor gap distance in millimeters (mm)


• DC arcing current


Step 1: Enter the conductor gap distance in millimeters and multiply by 0.534. The user must define the gap distance. IEEE 1584 provides a table of “typical” gap distances.


Step 2: Add the constant 20 to the result found in step 1.


Step 3: Enter the arcing short-circuit current (IDC arc) and raise it to the power of 0.88. Since the arcing short-circuit current is not usually known, a typical first approximation is to assume that IDC arc = 50 percent of IDC bolted.


Step 4: To obtain the DC arc resistance in ohms, divide step 2 by step 3. 


The following example illustrates how to calculate the value of the DC arc resistance based on the initial assumption of the DC arcing short-circuit current. After the DC arc resistance has been calculated, iterative solutions can be used.


For this example, an arc gap of 25 mm was used, which is one of the “typical” values given in IEEE 1584. Using the first approximation of 9,524A, which was calculated previously for the arcing short circuit current, the arc resistance (Rarc) is calculated as 0.01051Ω, as shown in Figure 3.


The next step in this process requires a series of iterations. You can add the calculated value of arc resistance to the original circuit, and you can recalculate the DC short-circuit current as follows:


IDC arc = VDC ÷ (Rbattery + Rconductor + Rarc)


IDC arc = 256V ÷ (0.01150Ω + 0.00194Ω + Rarc)


IDC arc = 256V ÷ (0.01150Ω + 0.00194Ω + 0.01051Ω)


IDC arc = 10,688.9A


Once you have calculated the new value of IDC arc, you can substitute it back into the DC arc resistance worksheet and calculate a new value of Rarc. The iteration process continues until the values of IDC arc and Rarc do not change significantly from the previous values and converge to the final answers of 11,433.7A for IDC arc and 0.00895 for Rarc, as illustrated in Table 1 and Figure 4.


Power and energy in the arc


Once you have determined the DC arcing current and DC arc resistance, the power in the arc can be calculated by:


Parc = IDC arc2 × Rarc


Parc = power in the arc in watts


IDC arc = DC arcing circuit current in amperes


Rarc = DC arc resistance in ohms


The energy in the arc is a function of power and time. Therefore, the energy in the arc can be calculated with the following:


Earc = Parc × tarc


Where:


Earc = arc energy in watt seconds or Joules (J)


tarc = arc duration in seconds


The arc flash duration will either be dependent on the clearing time of an upstream protective device operating or the reaction time of a person jumping away from the hazard. IEEE 1584 currently suggests that a maximum time of 2 seconds may be used based on the reaction time and assuming there are reasonable conditions for a person to escape.


DC incident energy calculations—open air


Similar to the IEEE 1584 calculation methods, consideration must be given to whether the DC arc flash occurs in open air or in an enclosure/box. If the DC arc flash occurs in open air, the energy will radiate spherically in all directions and the person would be exposed to a smaller portion of the energy. If the event occurs in an enclosure, the incident energy exposure will be greater, since it is focused out of the box opening.


According to the DC arc models paper, the incident energy for an arc flash in open air at a specific distance can be calculated based on the following equation:


Ei air = Earc ÷ (4π × d2)


The worksheet in Figure 5 is based on this equation and used to solve the arc flash in open air example problem. It breaks the calculation process down into individual steps, and a final step converts the units from Joules per mm2 (J/mm2) to the more commonly used units of cal/cm2.


To use this worksheet, the following data is required:


• DC arcing current in amperes, IDC arc


• Arc resistance in ohms, Rarc


• Arc duration in seconds, tarc


• Distance from the arc in mm, d


Step 1: Enter IDC arc, Rarc and tarc obtained from the previous iterative calculations. Square the IDC arc value and multiply by Rarc and tarc to determine the energy in the arc, Earc in terms of watt seconds or Joules.


Step 2: Enter the distance from the arc (working distance) in mm. Multiply d by 4 × π, or 12.56637.


Step 3: Calculate Ei air by dividing step 1 by step 2. The result will be in J/mm2.


Step 4: Convert the answer obtained in step 3 from J/mm2 to cal/cm2 by multiplying by 23.9.


Using the DC arcing short-circuit current and the previously calculated arc resistance, calculate the incident energy. This requires knowing the working distance from the prospective arcing location to the worker as well as knowing the duration of the arc flash. 


For this calculation, a maximum arc duration of 0.3 seconds was used. The characteristic of an upstream protective device would normally define this value. Use a working distance of 18 inches (457 mm), which is a “typical” value obtained from IEEE 1584.


Where:


Earc = arc energy in watt seconds or Joules


Ei air = incident energy from an open air arc 
at distance d in J/mm2


DC arc flash in an enclosure/box


If the DC arc flash occurs in an equipment enclosure, the energy will be directed out of the box’s open end. For this calculation, the DC arc models paper refers to another technical paper, “Simple Improved Equations for Arc Flash Hazard Analysis,” IEEE Electrical Safety Forum, Aug. 30, 2004, by R. Wilkins. According to this paper, the following is the equation for determining the incident energy from a DC arc flash being focused out of an enclosure:


Ei box = k × Earc ÷ (a2 + d2)


Where:


Ei box = incident energy from an arc flash in 
a box at distance (d) in J/mm2


Earc = arc energy in watt seconds or Joules


d = distance from the arc source in mm


The Wilkins paper defines optimal values for a and k, listed here in Table 2.


A worksheet was developed for calculating the DC incident energy for an arc flash in an enclosure/box. This worksheet is based on the box equation and reduces the calculation into a series of simple steps.


To use this worksheet, the following data is required:


• DC arcing current in amperes, IDC arc


• Arc resistance in ohms, Rarc


• Arc duration in seconds, tarc


• a and k from Table 2


• Distance from the arc in mm, d


Step 1: Enter IDC arc, Rarc and tarc obtained from the previous iterative calculations. Square the IDC arc value and multiply by Rarc and tarc to determine the energy in the arc in terms of watt seconds or Joules.


Step 2: The value of “a” must be obtained from Table 2. The value of the distance from the arc (working distance), d in mm, must also be defined. Enter each value in the appropriate space in step 2. Square each value, and add the two terms together.


Step 3: Look up the value of k from Table 2. Multiply k and Earc from step 1.


Step 4: Divide step 3 by step 2. The result will be the incident energy in terms of J/mm2 at working distance (d).


Step 5: To convert the units from J/mm2 to the more commonly used cal/cm2, multiply the answer in step 4 by 23.9.


Using values that were previously calculated for IDC arc and Rarc, calculate the incident energy based on the arc flash occurring in a box/enclosure. Assume the enclosure to be a panelboard and use the same working distance and arc duration from the earlier example.


To begin this problem, the values of a and k must be determined. Obtaining these values from Table 2 for a panelboard indicates the value of a is 100 and k is 0.127. The previous calculations indicate that IDC arc = 11,433A and Rarc = 0.00895Ω. The working distance (d) is 457 mm and the duration (tarc) is 0.3 seconds. These values can be used with the DC arc flash worksheet for calculating the incident energy in an enclosure as illustrated in Figure 6. The result for this calculation is 4.9 cal/cm2.


DC arc flash calculations and standards


Except for two technical papers referenced in the annex of NFPA 70E, DC arc flash equations and calculation methods are not yet part of any standard. As research on DC arc flash and DC source modeling continues, DC arc flash calculation methods will likely become part of a standard someday. 


Until then, since no standard exists for DC arc flash calculations, why would anyone perform them? You don’t have to look too far back in the history of arc flash to find the answer. Even though IEEE 1584 was first published in 2002, some people were performing arc flash studies and calculations for AC systems long before then. How could that be? They did it by using the best known methods and equations that were available at the time—and the same thing is happening once again today with DC arc flash. History often repeats itself.


As with any analytical calculations or engineering study, only qualified people should perform them.


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