I keep seeing a technical paper by Wilkins referenced in other DC arc flash calculation papers when discussing the multiplying effect of a DC arc-in-a-box.

NFPA 70E uses a "prudent" 3x multiplying factor regardless of the enclosure type/size but it looks like the Wilkins method takes those into account.

I can't seem to find that paper anywhere. Any ideas? It's referenced in almost every single DC arc flash calculation paper I've come across but I can't find the Wilkins "Simple Improved Equations" paper itself. IEEE Xplore links to the Wilkins paper when its referenced but the link leads to another paper by Wilkins but with a different name.

Here's an example:
http://140.98.202.196/xpl/abstractRefer ... %3D6165544

I was hoping to use it to determine the k and a values for different combiner box sizes. If this is something you're more experienced with, I'd love to hear more about it.

Here's the paper I'm looking for:

Wilkins, R., “Simple improved equations for arc flash hazard analysis�, IEEE Electrical Safety Forum, posted August 30, 2004.

Wilkins posted it on an IEEE bulletin board.

I believe personally that the amount of energy produced by an arc is well characterized. However there are two problems with this. First there is a very thin actual "arc column". Temperatures within the arc column are correctly reported with crazy high values such as "20,000 K". However almost of the energy radiated from the central arc column is absorbed into the air surrounding this (in the first 1-2 mm). This of course elevates the temperatures at those levels which is then reradiated ad infinitum. At some point this process slows down and we get what we are interested in as far as radiation goes but it's actually coming from the heated air around the arc column, not the arc column itself. Various attempts have been made to estimate this and they seem to vary between 10 and 40% of the thermal energy produced by the arc. So right away even if we can accurately estimate DC arc voltage (or alternatively the resistance) and DC arc current, which is fairly easy to measure and has been well documented from several sources, that leaves the absorption/rreradiation going on.

Furthermore the next step is that we are focussing the radiation to some degree. It is either absorbed or reflected from the enclosure walls and emitted in a roughly cone or hemispherical shaped zone. The exact way of estimating this multiplier is via "radiative view factors" and there is open source software to do this. I believe that this is what you are referring to when you mentioned part of what Wilkins has published and it has been echoed elsewhere. I have a copy of the program but haven't sat down yet to try to plug in some values and see what I get for myself.

Either way enclosure shape within the current IEEE 1584 model is represented as a constant plus an exponent on the distance. Wilkins model is more correctly based on radiative view factors and the math is slightly different but again depends on two factors a and b. Depending on which paper you read, the constants are either a and k or a and b. Ultimately I like the full radiative view factor approach especially when we get into enclosure shapes and sizes that IEEE 1584 was just never really intended to address.

I believe Tammy Gammon did a pretty good job overall of totally discrediting the "multiply by 3" number as well as discussing these model discrepancies and others in the recent paper: "A Review of Commonly Used DC Arc Models", IEEE Trans on Ind. Appl., Vol 51., No. 2, March/April 2015, pp. 1398-1407, and is a pretty thorough review overall including references and graphs comparing the various models to the scant amount of published measured data that is available. Based on Tammy's paper if you had to pick a number out of thin air, 1.0 is much closer than 3.0, but if you go further I think Fontaine and some others (published openly...free) have some things published at Battcon where they just outright use the Wilkins distance model for the radiative view factor and either plug in the Neal (Lee modified for AC) model or the more complicated Ammerman model. Either way at low voltages compared to Kinetrics data (also published on Battcon and shown in much greater detail in Gammon's paper) even with all these modifications it still overestimates the real world by a factor of 1.4 to 3.

So...following up, Wilkins published two "models" that I'm aware of. The one is on the bullletin board if you're an IEEE IAS member so you have the exact reference correct. The other one is available for free off Mersen's site. The basic difference between the two is that the Mersen paper is slightly incomplete in a few spots but the model it describes is a time-series model so essentially it simulates the actual voltages and currents during an arc. It is to put it mildly, complicated, to get a value but it is the most accurate model to date. The other model, the "simple improved" model, is based more on the same theoretical basis as the time series model but takes a much more simple IEEE 1584 style approach and improves on the accuracy compared to IEEE 1584 which is essentially 100% empirical (observe chart, pick a simple math formula such as power law to fit the chart).

The radiative view factors model that Wilkins proposed essentially results in a two-factor model but again the equations are based on a more theoretical basis compared to the two factors in the IEEE 1584 model (exponent x and a constant) that are completely empirically based and based on only 3 or 4 reference models that were laboratory tested over about 300 tests. In theory you could use Wilkins two factor model to model any arbitrary enclosure but he fit the IEEE 1584 data to the calculation and thus we have two empirically derived values for a and b (or a and k or m or whatever the variable name is). Wilkins did NOT use a software package to calculate actual radiative view factors and then just tune a constant to make the phsics-based results fit the IEEE 1584 data set so it cannot be used for arbitrary shaped enclosures, at least the version I've seen. I think the only reason it has become popular is because it produces a somewhat more accurate fit to the data than the IEEE 1584 formulas do.

Paul,

I cannot thank you enough for taking the time to provide me with such a detailed and informative response. All the references in your post will come in very handy while I’m doing my research.

I started out with the most obvious path of following the DC arc flash calculation shown in NFPA 70E but I quickly realized how inflated those results could be. My first flag was using the maximum power transfer theorem and assuming that my arc voltage is equal to 50% of my system voltage and then using that voltage to calculate power AND assuming that 100% of that power is radiated heat. I came across then Ammerman paper and the Stokes/Oppenlander model which I think will be helpful in estimating arc power.

As to absorption/irradiation issue, I completely see your point but I haven't come across anything I can references for a more realistic calculation. My guess was that we're going to have to wait for an empirical model like 1584 to get a better answer. Do you know of any of the references show the 10 to 40% estimate?


That's exactly what I was looking, thanks. I was hoping the Wilkins paper I'm looking for gave me equations for calculating the a and k factors for any enclosure given dimensions. I found a few online calculators for radiative view factors. I’m going to look into those calculators and the Tammy Gammon paper next to better understand the 3X multiplier counter-arguments.

I’m going to join IAS so I can look through their bulletin board. That Wilkins paper is referenced in literally dozens other technical papers, books and even software but finding it online, even for purchase, has been impossible. Thanks for the tip, I’ll look in the bulletin board.

Do I need the Wilkins missing paper to calculate the two factors for an enclosure that doesn’t match the 3 listed in 1584? I’m guessing that’s where I could use radiative view factor software, right?

Thanks again, Paul. You post has been huge help.

what I find interesting when you start looking at the theoretical models for DC arcs is that there is a parabolic curve and the arc can exist in one of two states on that curve but due to some "stability" type hand waving, only one state can actually exist. The very tip of the parabola is the Lee/Doan maximum power transfer formula.



Yes, Tammy Gammon summarized them, but the paper that she referenced is pretty darned sketchy if you ask me. The pressure estimates given in the same paper are crazy. Actual published measurements of arc pressures peak at around 5-10 PSI for a sealed box scenario and somewhat less for other cases. The paper Tammy is quoting has a box with two very large openings at the top meant to simulate a switchgear cell and claims measurements of 40 PSI. That doesn't mean that the rest of the math is wrong too, just that it makes it suspect.

There are a lot of variables by the way. Just the basic radiative view factors turns into a "radiosity" problem. To get a flavor for the math, google "ray tracing" and "radiosity" until you can wrap your head around what's going on because radiation is the same whether it's thermal or visible light.



The best one is actually an open source package called View3D.



Theoretically, yes. I'm going out on a limb here myself. Wilkins models the view factors as a pair of discs and then essentially does a curve fit of the IEEE 1584 data back to the model. You can read a summary version of this in the Battcon papers that are free (no subscription needed). So yes, he basically starts with the IEEE 1584 models and calculates two parameters for an alternative equation that is not the same as the IEEE 1584 empirical one but in the end ends up being empirical in my opinion anyway. Since it's just taking known data and "fitting" it and not actually using physical parameters such as the box dimensions, and there isn't even a "big constant" that makes the IEEE data match the theoretical model, it doesn't really quite turn into a truly satisfying solution. One the other hand, Wilkins is using a very trivial model that doesn't really work the same as say View3D. So all he creates is an alternative model that fits the data better especially as distance changes. It's a step up from IEEE 1584 which is for the most part purely empirical but doesn't allow us for instance to predict the incident energy for an enclosure that doesn't match the existing data set. We're probably going in the right direction here...just not there yet.

This will be what I dig into next. While I like the idea of using open source software to calculate an arc-in-a-box multiplying factor, I would feel more confident in my calculation if I could show the calculation done by hand. That would also help me document my calculations in the future, saying that I used software I found online can raise flags when my calculation is reviewed.

Paul, do you know of a paper or calculation that shows a PV DC arc flash calculation from start to finish using the Ammerman and Wilkins models?

This is my preliminary calculation methodology after reading your feedback, the Ammerman paper, the Wilkins paper and going through the example Jim Phillips explained in his "Know Your Arc: DC arc flash calculations" post:

1. Collect module data:
--- V_MaxPower: I figured V_MaxPower would be more realistic than using V_OpenCircuit since current flow will not be zero.
--- I_ShortCircuit: This should be more conservative than I_MaxPower since the load has been bypassed with a lower resistance arc.

2. Find fault values:
--- I_BoltedFault = TotalStringsInParallel x I_ShortCircuit
--- I feel like using a constant 125% multiplier here to account for higher Solar Insolation days is overkill. Looking at Solar Insolation graphs, 1060 w/m^2 is that maximum value for the Northern Hemisphere and that's at 12 noon on a bright clear summer solstice day.
--- I saw some calculations that used another 125% multiplier to comply with NEC's requirement for sizing output circuit conductors and OCPD for continuous loads. I don't see how that justifies the multiplier.

3. Draw equivalent circuit
--- Circuit will consist of V_System, R_System, R_Arc in series.
--- V_System: Use V_MaxPower and the number of series modules in a string to find V_System (V_MaxPower x ModulesInSeries)
--- R_System = V_System/I_BoltedFault. I could use some feedback here. I opted to ignore cables to keep the calculation simpler. I see this as being more conservative as well.
--- In the examples I went through, I usually end up with an R_system of 0.3 to 0.5 ohms.

4. Apply Stokes/Oppenlander equations (Ammerman Model)
--- Use I_arc = 50% of I_BoltedFault for the first iteration.
--- Obtain gap distance between positive and negative terminal in working location (in mm).
--- Find R_arc = (20+0.534*Zg)/(I_arc^0.88)
--- Use R_arc back in equivalent circuit and calculate new I_arc for next iteration I_arc = V_system/(R_system + R_arc)
--- Find R_arc for second iteration and so on until I_arc no longer changes significantly.
--- For the few examples that I have done, I ended up with I_arc = 0.75 * I_bolted
--- Find V_arc = (20 + 0.534*Zg)*I_arc^0.12 or simply I_arc * R_arc

5. Calculate DC arc power
--- P_arc = I_arc * V_arc (watts)

6. Calculate DC arc energy
--- E_arc = P_arc * FCT (Joules)
--- I am considering either using a 2 second fixed FCT value or looking up fuse TCCs.
--- I have yet to come across an example where the fuse burns in less than 2 seconds. Any experience here?

7. Convert energy to calories:
--- E_arc / 4.18

8. Find energy distributed across surface area of sphere:
--- E_sphere = E_arc/(4*PI*D^2)
--- D is typically 18" (45.7 cm )

9. Find arc-in-a-box multiplying factor:
--- Still not sure here.
--- One option is to use the Wilkins equation with the 'k' and 'a' of a panel (per IEEE equipment types).
--- To me, a combiner box is much closer to a panel than LV switchgear because the depth is only 8" or so.
--- Looking at the Wilkins k values for IEEE equipment types, k seems to scale linearly with depth.
--- That would result in a MF of 1.5 from sphere to box.
--- Or, look into a better way to account for the box effect.

I'm still making the very conservative assumption that 100% of the electrical energy released will be heat energy but the final values are significantly lower than using the Maximum Power Transfer theorem and the 300% multiplier.

Any thoughts?

I understand your frustration but look at what doing radiative view factors looks like. First we break up all the surfaces within the enclosure (and components if we model those) into small surfaces to discretize what is a continuous function using a method called tesselation. This results in N surfaces. Then we have to consider N+1 radiators (+1=arc source) impinging onto N surfaces. We iterate the calculation a few times until it stabilizes because each surface is both an absorber and an emitter (reflector) of energy that strikes every other surface unless the two surfaces are not in view of each other (ray tracing needed to check for interference). For VERY simple models such as two discs there are analytical solutions but for the real world this is not realistic. This is similar to doing finite element or finite volume models...the equations are very simple but the shear number of them makes it physically impossible to do them by hand.



In terms of Wilkins, no and both the time domain and simplified models are AC, not DC. In terms of Ammerman, yes.

Fontaine has modified the simplified model for DC (using the Neal/Lee maximum power transfer method) here:
http://www.battcon.com/PapersFinal2014/ ... 0Final.pdf

In terms of Ammerman "examples" for testing purposes, see this one which also compares both the maximum power transfer and Ammerman models to real world lab measured data:
http://www.battcon.com/PapersFinal2012/ ... 0Flash.pdf



The real arcing current (and arcing power) is not ever the maximum power transfer case. Ammerman's model attempts to use the huge body of data on DC arcs that does exist to estimate the arc current or equivalently resistance. Also reactance does matter and affects the current but not very many models or practitioners for that matter take this into account due to the extra complexity. Overall though we always start with the maximum power transfer model using available fault current and then move into a more realistic model. Thus what we need as starting material is the Thevenin equivalent model (voltage plus a resistor) so for batteries this is internal resistance and battery voltage, two values that are pretty well known or can be estimated pretty well.



The issue is FAULT CURRENT. NEC multipliers have nothing to do with it. What you want in the end is the Thevenin equivalent values; the system voltage and the system resistance. So I'll buy into the multipliers for solar insolation for solar panels but not any kind of NEC adjustments for ampacity. In the case here (<2 seconds) copper doesn't have enough time to heat up to matter much.



Sounds about right. Another consideration you may want to look at is when you get into power converters. These have a published short circuit rating. IGBT's themselves are rated on RMS values so if the short term/circuit values aren't given, 186% of maximum rated output is pretty close. Most maximum output specs are between 150% and 250% when a peak value is given. A semiconductor fails VERY quickly (milliseconds) under short circuit loads. Hence the reason for semiconductor grade fuses that trip in a few milliseconds at most. So in practice these give a way to estimate short circuit available through a power converter (inverter, VFD, UPS, etc.). If you have capacitors, you can also use the stored energy to arrive at a Joule value that can be used in the same way as the maximum power transfer method that eventually ends up with an amount of energy in calories as a result. Then dividing by the area of a sphere at the working distance in centimeters gives the expected result (cal/cm^2). Just as with maximum power transfer methods, only a small fraction of the energy will be thermal radiation but again we have no way to estimate what fraction this is.



Yep, that's how it's done. Note that the "2 second rule" is really more of just something that was mentioned in IEEE 1584 for AC 3 phase systems and doesn't really have any scientific basis at all behind it. But...it seems to work in practice based on data collected by Doan and Hoagland on real world incidents. I have had ridiculously fast trip times when it comes to semiconductor fusing in power converters. Elsewhere there is a tendency to oversize fuses (and circuit breakers) in most systems because in the past the goal was just to interrupt power before the conductors turn into slag. Tripping at the minimum value was avoided at all costs because power system analysis is still somewhat guess work and nobody wanted to cause nuisance trips. In reality typically once you come this far it becomes obvious where to set the fuse and whether these fears were justified or not.



This is where Fontaine went so look at his paper on the subject. It's all thermal energy. Again there is no actual test data to speak of out there so it's all estimating here. The step "8" above really isn't necessary because there is both an exponent and a constant involved for the IEEE 1584 version of the model, or just an exponent (as described in step 8 above) for the open air/spherical assumption, or a slightly different form for the Wilkins (Fontaine) version. This is really just one step...we have a normalized value and we need to convert it to the denormalized case. IEEE 1584 includes time as part of the denormalizing process as well.

Did Wilkins assume that all arc energy irradiating the inner enclosure surface was either reflected or absorbed/emitted back on the calorimeter? That was the assumption I was making when doing some practice work on 1 surface at a time, that the surfaces did nothing but redirect energy (reflect and emit) and nothing was absorbed. The only arc energy that doesn't make it to the calorimeter was due to a zero view factor. Also that the calorimeter doesn't emit anything back to the enclosure.

I attached a 3D image of my model so far. There's a "grey body" setting in View3D that I figured needs to be on for our scenario. Any idea what would be optimal dimensions for my receiving plate? I was thinking a 10"X10" area right where the sphere is tangent to the calorimeter (or person standing 18" away).

Wilkins did no such thing. He modelled the source as a disk with radius R and simply scaled the disc size and I think a proportional constant (calibration factor) until the two parameters perfectly matched the experimental data from the IEEE 1584 test data set. You are already way ahead of this. If you keep the proportionality constant and the rest of your model "works", whereas Wilkins just "fixed" IEEE 1584's purely exponential model of enclosures, yours can simulate any enclosure.

As to reflection, all real world objects are "grey bodies". A black body is a theoretical object that absorbs energy regardless of frequency or angle of incidence. A white body is a theoretical object that has a rough surface and reflects energy regardless of frequency or angle of incidence. Black body radiators emit radiation according to Planck's law in all directions uniformly at all frequencies. A black body has an emissivity of 1.0. A grey body is basically a black body except that the amount of radiation (and reflection) is a fraction of the black body case, modelled with the emissivity constant. Note that emissivity is frequently treated as a constant but it is actually a function of material, temperature, and frequency of the emitter, and incident angle really does matter. Don't try to model this though...just accept that emissivity is not a nice simple number. This is one of the reasons that "IR scans" are not as simple as it appears.

Does anyone know if Wilkins post is still available today?

I've been trying to track down this content myself, by chasing through the Internet Archive.

As best I can tell, the "IEEE Online Communities" content from the "IEEE Electrical Safety Forum" ought to have been migrated to "IEEE Collabratec".

Refer https://web.archive.org/web/20240224033 ... /a_id/3103 .

However I'm logged into IEEE Collabratec and the content seems like it's nowhere to be found.

I suspect that unless someone has a PDF printout of the page (as it appeared back in 2004!) it might be lost off the open internet forever.