DC ARC FLASH CALCULATIONS – ARC-IN-OPEN-AIR & ARC-IN-A-BOX –USING A SIMPLIFIED APPROACH (MULTIPLICATION FACTOR METHOD)
Copyright Material IEEE Paper No. ESW2012-25 Paper with figures and tables is available from IEEE
Michael D. Fontaine, National Fire Protection Assoc. 1 Batterymarch Park, Quincy, MA 02169
Peter Walsh, P.E. Member IEEE, Mersen USA Newburyport-MA, LLC, 01950
Abstract – This paper suggests a method for calculating the incident energy and the arc flash boundary (AFB) distance for dc systems when an arc-in-a-box situation is involved. The method uses the dc maximum power method and a multiplying factor instead of using distance exponents. The method may also be used for an arc-in-open-air by using a multiplying factor of 1. It is based on the basic assumption that the spherical energy density can be increased to a new
value to account for the additional reflected heat radiation from a box situation and that the ratio of the new value to the spherical energy density equals a multiplying factor for correcting – from the energy density for an arc-in-open-air situation to an arc-in-a-box situation, based on a simplified equation or methodology. When calculating the AFB, the method requires the use of an iterative process , since the multiplying factor is a nonlinear variable and is based on the distance from the arc. This paper should provide a good starting point for further discussion and development regarding dc arc-in-a-box situations.
Index Terms — arc-in-a-box, dc maximum power method, multiplying factor.
Following is a method for calculating the incident energy and the arc flash boundary (AFB) distance for dc systems using the dc maximum power method as
developed by D. R. Doan  and a multiplying factor method as developed by R. Wilkins  and furthered by R. F. Ammerman, et al  in lieu of the use of
distance exponents. The method is based on the assumption that the spherical energy density IEoa can be increased to a value of IEb to account for the additional reflected heat radiation from a box situation and that IEb/IEoa equals a multiplying factor Mf for correcting from the energy density for an arc-in-openair situation to an arc-in-a-box situation, based on a simplified equation or methodology. A similar method was developed by R. Wilkins  and furthered by R. F. Ammerman, et al . Equations provided here for determining the multiplying factor were derived from Equations 5 and 6 developed by R. Wilkins .
The multiplying factor approach is based on the multiplying factor equation(s) derived below and is only valid for the radiated component of heat flux – that radiation that is radiated directly out of the box – and for that radiation striking or re-striking the inside of the box and is then radiated out. If a plasma cloud is driven by plasma jets or other such phenomena, an additional heating term would need to be included. The effect of high circuit inductance relative to the arc resistance was not explored.
The following definitions are given to ensure that the meaning of the terms used in this paper is clear to the reader.
_ Arc flash boundary equals the distance at which the incident energy density equals 1.2-cal/cm2 (5.0-J/cm2).
_ 1 Joule equals 0.2390005736 thermal calories, cal(th), or 1 Joule is approximately equal to 0.239 cal(th) and 1 Joules equals 4.184 calories.
_ Parc equals the power of an arc, or approximately equals the arc current, Iarc times the arc voltage, Varc, or approximately equals the arc current squared Iarc2 times the arc resistance Rarc, in watts – since arcs create transient conditions and the currents and voltages are not static.
_ Warc equals the energy of an arc in joules or Parc times Tarc (watt-seconds).
_ Tarc equals the arc duration, the total clearing time of the arc in seconds.
_ Earc equals the energy density of an arc in Joules/cm2.
_ IEarc equals the energy density of an arc in cal/cm2.
_ IEoa equals the energy density of an open-air-arc in cal/cm2.
_ IEb equals the energy density of an arc-in-a-box in cal/cm2.
_ Dcm equals working distance from the arc flash source, centimeters.
_ Din equals working distance from the arc flash source, inches.
_ Dabf_cm equals the arc flash boundary distance, centimeters.
_ Dabf_in equals the arc flash boundary distance, inches.
_ Dabf_ft equals the arc flash boundary distance, feet.
III. MULTIPLYING FACTOR
A. Multiplying Factor Equations for dc Arc Flash Calculations:
1) Dimensions in Millimeters: Equation 3.1 uses dimensions in millimeters.
Eoa = Warc/(4_d2) 
From R. Wilkins  Equation 5
Eb = (k Warc)/(a2 + d2) [2[
From R. Wilkins  Equation 6
Mf = IEb/IEoa = (k4_)/[1 + (a/d)2] [3.1]
IEb = incident energy level at a specific distance from a box, cal/cm2
IEoa = incident energy level at a specific distance from an arc-in-open-air, cal/cm2
d = distance from the arc flash source, mm
a = a characteristic dimension that lets us represent the arc-plus-box as a single heat source, mm. The values of a were determined by R. Wilkins  and are given in Table A-I in Appendix A of this report for specific equipment categories from IEEE 1584TM. The value of a depends upon the type of equipment used.
k = a dimensionless correction factor determined by R. Wilkins2. The values are given in Table A-I in Appendix A of this report for specific equipment categories from IEEE 1584TM. The value of k depends on the situation (arc-in-open-air or arc-ina-box) and on the type of equipment used.
2) Dimensions in Inches: Equation 3.2 below has been developed so that the distance is in inches instead of in millimeters.
Mf = (k4_)/(1 + [Ain/Din]2) [3.2]
Din = distance from the arc flash source, inches
Ain = 0.03937a, inches
Other values are as stated in equation 1.1 above
B. Optimum Values of “a” and “k” as Determined by R. Wilkins :
Table A-I in Appendix A gives the optimum values of a and k that were determined by R. Wilkins  for the three specific classes of equipment in IEEE 1584TM, and they are related to the dimensions of the boxes used to represent the specific equipment classes. Using Figures 4 and 5 in R. Wilkins , it is possible to estimate the values of a and k, and therefore also of Ain for box sizes other than those specified in IEEE 1584TM. This can also be done by using Figures 1 and 2
below which are similar to Figures 4 and 5 in R. Wilkins . R. Wilkins  developed the optimum values of k and a as the values, which gave the least-squares best fit to the test data. The results are shown in Table A-I in Appendix A. Figures 1 and 2 below illustrate that the optimum values of k and a correlate strongly with box dimensions.
Fig. 1 Dependence of “k” on Equipment Size
1. Similar to Figure 4 in Wilkins  plotted using a second order polynomial trend line.
Fig. 2 Dependence of “a” on Equipment Size
1. Similar to Figure 5 in Wilkins . However, plotted using a linear trend line verses a second order polynomial trend line.
C. Tables Giving Multiplying Factors vs. Distance for Different Equipment Categories in IEEE 1584TM:
The following three tables are derived from the information contained above and give the multiplying factors for various distances for the equipment categories given in IEEE 1584TM. In NFPA 70E®-2012 Section D.8.1.1  a multiplying factor of 3 is suggested. For the three equipment categories given in IEEE 1584TM, it can be seen that a multiplying factor of 3 is conservative where panelboards are used; is only conservative up to 24 inches for LV Switchgear and may be appropriate up to a distance of around 30 inches; and for MV Switchgear is conservative up to around 42 inches. The test data used in Doughty, et al  was taken at 24 inches. In general based on the preceding, it may be said that a multiplying factor of 3 is conservative up to 24 inches (for all IEEE 1584TM equipment categories as defined above). See Tables I, II, and III below.
MULTIPLYING FACTORS PANELBOARDS VS. DISTANCE
Table I Multiplying Factor vs. Distance for PNLB(s)
MULTIPLYING FACTORS LV SWITCHGEAR VS. DISTANCE
Table II Multiplying Factor vs. Distance for LV SWG
MULTIPLYING FACTORS MV SWITCHGEAR VS. DISTANCE
Table III Multiplying Factor vs. Distance for MV SWG
1. The actual value calculates to 0.98. However, a value lower than 1 should never be used forcalculating incident energy. Figure 3 graphically shows how the multiplying factors for the different equipment categories in IEEE 1584TM change with distance. As can be seen for panelboards, the curve is fairly flat and for all practical purposes a multiplying factor of 1.6 can be used. At around 9 ft, the multiplying factor for low voltage switchgear levels off at a value of 3.85. However for medium voltage switchgear, the multiplying factor at 10 feet is still increasing, though at a slower rate. This multiplying factor peaks out around 5.23 somewhere at a distance greater than 500 feet. The test data on which IEEE 1584TM was based was taken over a range of 12 to 72 inches. It should be noted that a multiplying factor of less than 1 should never be used, as it is the multiplying factor for an arc-in open-air.
Fig. 3 Multiplying Factors vs. Distance
IV. MODIFIED DC MAXIMUM POWER METHOD EQUATIONS
For the steady state condition, maxim power in a dc circuit arc occurs when the arc resistance equals the system resistance or the watts of the arc equals 0.25
(¼) times the bolted-fault watts, 0.25 Wbf (0.25 x Vsys xIbf). There are 4.184 calories (th) per Joule – therefore a factor of 0.239 is used for converting from Joules to calories (th) and 1/4_ equals (0.79577). Therefore, the multiplier before equation 4.1 should be 0.00951(0.5 x 0.239 x 0.079577) and the multiplier before equation 6.2 should be 0.004755. In this paper, we will use the more exact numbers as indicated in equations 4.2 and 6.2. The maximum power method equations have also been modified by including the multiplying factor variable Mf in the equations as shown below.
IEm = 0.01 x Vsys x Iarc x Mf x Tarc/Dcm
IEm = 0.00951 x Vsys x Iarc x Mf x Tarc/Dcm
(Not rounded off to 0.01)
Iarc = 0.5 x Ibf 
IEm = 0.005 x Vsys x Ibf x Mf x Tarc/Dcm
IEm = 0.004755 x Vsys x Ibf x Mf x Tarc/Dcm
(Not rounded off)
Where: IEm = estimated dc arc flash energy at the maximum power point, cal/cm2 Iarc = dc arcing current, Amperes Ibf = dc system bolted fault current, Amperes
Vsys = dc system voltage, Volts
Tarc = total clearing time for dc arcing current, arcing time, sec.
Dcm = working distance, cm
Mf = multiplying factor
1.0 for arc-in-open-air Value from equation in Section III above or 3.0 for arc-in-a-box effect where more conservative (less than 24 inches)
V. DERIVATION OF DC ARC FLASH BOUNDARY DISTANCE EQUATION
A) Derivation from Arc-in-Open-Air Equation:
Equation 7 below is Equation 6.2 this report without the multiplying factor Mf. The multiplying factor has been removed to show the logic of how the arc flash boundary (AFB) for an arc-in-a-box can be derived from the arc-inopen- air dc maximum power method equation.
IEm = 0.004755 x Vsys x Ibf x Tarc/Dcm
By changing around the above maximum power equation, we can look at the distance squared for a particular incident energy level.
2 = 0.004755 x Vsys x Ibf x Tarc/IEm
From the definition above for the arc flash boundary, we are interested at the point where the incident energy level is 1.2 cal/cm2. In terms of generalizing this statement when an arc-in-openair is 1.2/Mf cal/cm2 an arc-in-a-box will be 1.2- cal/cm2. Substituting 1.2/Mf into the above equation for IEm, we obtain:
2 = (0.004755) x Vsys x Ibf x Tarc/(1.2/Mf)
2 = (0.004755/1.2) x Mf x Vsys x Ibf x Tarc
2 = (0.004755/1.2) x Mf x Vsys x Ibf x Tarc
2 = (0.003963) x Mf x Vsys x Ibf x Tarc
We can now substitute Dafb_cm 2 for Dcm 2
2 = (0.003963) x Mf x Vsys x Ibf x Tarc
In order to determine the arc flash boundary in cm, we take the square root of each side of the equation.
= [(0.003963) x Mf x Vsys x Ibf x Tarc]0.5
Changing from centimeters to inches, we multiply both sides by (1/[2.54 x 2.54]) in.2/cm2 or 0.155 in.2/cm2.
0.155 x Dcm
2 = (0.003963) x Mf x Vsys x Ibf x Tarc x
0.155 x Dcm
2 = Dafb_in
2 (arc flash boundary squared in inches)
2 = 0.155 x Dcm
2 = (0.003963 x 0.155) x Mf x Vsys x Ibf x Tarc
2 = 0.000614 x Mf x Vsys x Ibf x Tarc
In order to determine the arc flash boundary in
inches, we take the square root of each side of the equation.
Dafb_in = (0.000614 x Mf x Vsys x Ibf x Tarc)0.5
If the values given in Equations 4.1 and 6.2 were used instead of the values given in Equations 4.2 and 6.2, the constant in Equations 8.2A would be 0.000646 and the constant in Equation 8.2C would be 646 – an increase of around 5%. Therefore, the equations for determining the dc arc flash boundary are as follows:
Dafb_in = (0.000614 x Mf x Vsys x Ibf x Tarc)0.5 [8.2A]
Dafb_ft = (4.264 X 10-6 x Mf x Wbf x Tarc)1/2 [8.2B]
Dafb_in = (614 x Mf x MWbf x Tarc)1/2 [8.2C]
Dafb_ft = (4.264 x Mf x MWbf x Tarc)1/2 [8.2D]
Dafb_in = arc flash boundary, inches
Dafb_ft = arc flash boundary, feet
Mf = multiplying factor 1.0 for arc-in-open-air Value from equation in Section III above or 3.0 for arc-in-a-box effect where more conservative (less than 24 inches)
Vsys = dc system voltage, Volts
Ibf = dc system bolted-fault current to the location of fault, Amps
Wbf = Vsys x Ibf
MWbf = 106 x Vsys x Ibf
Tarc = total clearing time for dc arcing current, sec.
VI. ITERATIVE APPROACH REQUIRED WHERE VARIABLE FACTOR IS USED
Since the multiplying factor Mf is a nonlinear variable, calculating the arc flash boundary requires the use a special method. R. Ammerman et al  pointed out the way by using an iterative process to calculate the incident energy values for dc arcs using a variable arc resistance model and then by suggesting the use of a multiplying factor of 2.2 for LV switchgear based on the work of R. Wilkins2 . Neither, R. Ammerman, et al  or R. Wilkins  provided the equation for determining the multiplying factor in their papers.
However, R. Ammerman, et al  quoting R. Wilkins  and R. Wilkins  provided the equations necessary for determining the equation we use here to find the multiplying factor (equations 5 and 6). R. Wilkins  and R. Ammerman el al  also provided the optimal values of a and k for three specific equipment categories in IEEE 1584TM to be used in determining the multiplying factor, Mf as detailed in Table A-I in Annex A. However, only R. Wilkins  provided in his
paper figures (Figures 4 and 5) that can be used to determine the optimum values of a and k for other equipment categories with different box dimensions.
As an alternate to the method provided in this paper the arcing current could be found using the iterative method and variable arc resistance model, such as the model given in Ammerman, et al as Equation (8)  and identified as shown in Equation 9 below:
Rarc = (20 + 0.534Dg_mm)/Iarc 0.88  Rarc equals the resistance of the arc, ohms
Dg_mm equals the gap distance, millimeters
Iarc equals the arc current, amperes
Then another iterative process would be used to find the arc flash boundary.
It should be noted that the multiplying factor, Mf is dependent upon the type of distribution equipment used, and the distance from the arc. Therefore, an iterative process has to be used to determine the arc flash boundary distance. An initial value of Mf must be set. An initial value of 3 is suggested. Then based on this initial value an arc flash boundary distance is calculated, using an equation given in section V above for dc systems. This value is then plugged into a multiplying factor equation given in section III above using values of a and k for the appropriate class of equipment given in Table A-I in Annex A below. The process is continued until the value converges sufficiently. An example of this process is given Figure 4 and in Table IV below using parameters taken from an example in D. R Doan.  and values of a and k for LV Switchgear.
Fig. 4 dc Circuit for Example of Iterative Process
Table IV Example of Determination of dc AFB
This paper provides an early stage model for calculating the incident energy values and arc flash boundaries values for dc arcs based on the work of others for arc-in-open-air and arc-in-a-box situations. It is hoped that this paper contributes something to the process: much work is yet to be done.
Special thanks are due to many such as Ravel Ammerman, Chet Davis, Daniel Doan, T. Gammon, Mike Lang, Ralph Lee , Robert Lou, Conrad St. Pierre and R.
Wilkins and others for their previous work, which provided the information necessary for the development of this paper, or for reviewing drafts of this paper. Dr. Ravel F. Ammerman of the Colorado School of Mines contributed by teaching me how to derive the multiplying factor equation; by providing the idea for the use of an iterative process of making calculations where non-linear variables are used; and for his reminder that additional testing is needed to develop accurate V-I characteristics and better dc arc resistance models. Chet Davis and R. Lou contributed by reviewing previous drafts of this paper and by correcting a mathematical error in one of the sections of the drafts of this paper. That section was deleted from the paper so as to sharpen the paper’s focus but that section and another deleted section may form the basis for a new paper. Daniel R. Doan of DuPont Engineering contributed by developing the dc maximum power method; by his reminder that arcs do not tend to follow our ideas of simple equations; by his reminder that the test data in IEEE-1584TM-2002 was taken over a working distance range of 12 to 72 inches with most of it at 24 inches and by his reminder that Doughty’s  date was taken at 24 inches.
Tammy Gammon of John Mathews and Associates, Inc. contributed by reviewing previous drafts of this paper and by providing related information. Mike Lang of Mersen contributed by providing information on Dr. Robert Wilkins’ basic assumption that the spherical energy density component can be increased to a value that accounts for additional reflected heat radiation and can be accounted for by the use of a simple method suitable for use in calculations – the use of a multiplying factor . Ralph Lee  contributed by starting off all of our journeys in this area. Conrad St. Pierre of Electric Power Consultants, LLC
contributed by suggesting that the multiplying factor be brought into the equations earlier in the paper. This paper could not have been written without Dr. Robert Wilkins contribution and we acknowledge the eminent contribution his work provided toward this paper. Grateful acknowledgment is made to other reviewers of the drafts of this paper and for their constructive criticisms even though they may not be mentioned by name.
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 Ammerman, R. F., Gammon, T., Sen, P.K., and Nelson, J.P., “DC arc models and incident energy calculations”, IEEE Transactions on Industry Applications, Vol. 46, No.5, Sept/Oct 2010, pages 1810-1819.
 NFPA 70E®, 2012, Standard for Electrical Safety in the Workplace, D.8.1.1, page 70E-71.
 Doughty, R. L., Neal, T. E., Dear, T. A., and Bingham, A. H., “Testing update on protective clothing and equipment for electric arc exposure”, IEEE Industry
Applications Mag., Vol. 5, Issue 1, Jan/Feb 1999, pages 37–49.
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Michael D. Fontaine is a Senior Electrical Engineer with NFPA. He has a BSEE, BSBA in Accounting, MSEE, MST, and a JD. He has been in the industry for over 35 years and has been a Registered Professional Electrical Engineer for over 30 years. He has held various positions in the electrical industry including electrical engineer, senior electrical engineer, electrical engineering manager, chief electrical engineer, instructor, teacher, training development manager, and president of an MEP consulting engineering firm. Michael has developed and presented seminars on OSHA electrical safety requirements. He is an editor and
published author on electrical safety issues. He wrote and published webinars on NFPA 70® and NFPA 70E®. He developed the certification examination for NFPA’s 70E training program and he is the NFPA staff liaison for NFPA 70A, 70B, and 70E. He provides advisory services for NFPA 70, 70E, and 72. Michael has been president or vicepresident of several local sections or regional engineering societies.
Peter Walsh is a Senior Field Engineer for Mersen. He has a BSEE from WPI and an MBA from Suffolk University. His present responsibilities include providing electrical services such as engineering and safety training. Many projects focus on electrical overcurrent and surge protection conforming to the
NFPA NEC-70, 70E, 79, and other codes and standards. Prior to joining Mersen (formerly named Ferraz Shawmut). Peter held the positions of Independent
Electrical Consulting Engineer, and Electrical Engineer at such companies as GE and Cooper Industries. Peter is currently:
_ Technical Committee Member of NFPA 110, Standard for Emergency and Standby Power Systems and also NFPA 111.
_ Member IEEE Industry Applications Society for 30 years and participating on several committees
_ Technical Delegate to NEMA 8SG for Mersen
_ OSHA Authorized Trainer to give 10-hour and 30-hour General Industry Safety Course for OSHA issued Card
_ Author writing numerous articles on electrical protection
_ Registered Professional Engineer for 30 years