Adiabatic switching time of a filament lamp

[PGRacer]

Hello all.

Firstly I am sorry if this is in the wrong place but I was unsure exactly where it fits in as it involves physics, integrated equations, and computer programming.

I am trying to write an emulator for a machine which is running 256 filament lamps in a 16*16 matrix.
I want to simulate the brightness levels in near real time.

I am working from this pdf file http://physik.uibk.ac.at/04-05/erde/spezial/aufgaben/Gluehbirne.pdf

My question relates specifically to equation at the end which gives the Adiabatic SwitchingTime (8).

I am trying to replicate the function in computer code using the example given in the pdf as inputs, in the hope that I can get the outputs to match the example to ensure the maths is correct. Then I can substitute the values of the bulbs I am using and use that for calculating the current temperature of the filament, which in turn can be converted to a brightness level to be displayed on the screen.

Using the inputs given in the example I can correctly calculate the resistance, maximum inrush current, & steady state temperature. But when using the final function to calculate the switching time, the value I get back is 0.0142 and according to the pdf should be 0.06.

I've been through the code and checked that I am doing things in the correct order (BODMAS) and that seems to be correct. However because I am not a mathematician, or a physicist, I am not even sure whether the equation is correct. There could be a glaring error and I would never know because I am not experienced in working with this level of maths. This is not homework, If I had a physics or maths professor, I would ask them for help.

So my opening questions are..
1) Does the equation (8) in the pdf look correct?
2) If I post C++ code here will it be of any use?
3) Please can anybody help?

 

[tech99]

I think the "on" and "off" times will be different.
Have you allowed for heat losses?

 

[PGRacer]

Yes the heating time will be quite a bit quicker than the cooling time. The cooling time follows the inverse square law I think. So the bigger the difference between the filament temperature and the ambient temperature the faster it will cool.

I am only dealing with the heating side at the moment, as if I can get this working as expected the cooling side is quite a lot easier to calculate.

I do not allow for heat loss due to convection / radiation but this is expected. Calculating the heating time and not allowing for losses will do for my purposes as all the bulbs I am using are the same rating 12v 1.2w, so proportinally the values will be correct.

The main issue is that the matrix sometimes underdrives the lamps to dim them or overdrives the lamps for a brief bright flash. So I cannot use a static table as the input rms voltage varies wildly.

 

[Babadag]

In my opinion the (8) formula is correct. If Rg=45 you'll get 0.061 sec. However you may simplify the calculation considering C constant for T=Ts=2887 oK and Rg=45 and put H=V^2*To^1.215/Ro formula(1) you'll get:
M*C/H*T^1.215*dT=dt and integrating:
ta=M*C/H*(Ts^2.125-To^2.125)/2.125=0.075 sec

 

[Babadag]

At first the formula I said (1) it is (5), of course. From:
Stephen W H and Wang Chun Tungsten Sources, Metallurgy, Properties and Applications, formula 6.7 Rg is 45.25 indeed.
Since the units are cal/g/oK in order to compare with formula (6) you have to multiply the coefficients by 4186.8 [in order to translate it in J/kg/oK].
By the way, in my opinion, no cooling in “adiabatic” exist. You have to evacuate the heat outside.

 

[PGRacer]

in order to compare with formula (6) you have to multiply the coefficients by 4186.8 [in order to translate it in J/kg/oK].

This is the specific bit I was unaware of and was causing the bad values, thanks for your help.

You seem to have a good understanding of this. Would you be willing to discuss it over skype or similar?

 

[Babadag]

Thank you, PGRacer for your invitation. I cannot schedule the Skype conversation.
The timing it’s a problem. For instance, my son lives in Florida and we can discuss only
once per month [as an average] namely after midnight.

 

[PaulWDent]

As the filament approaches its final temperature, the radiation tends to equal the power input. That means the adiabatic assumption is not accurate, nor is it necessary. It is is just as easy to include the Stefan-law radiation in a time-step simulation as not.

 

[Tom.G]

Just a hardware implementation tip:

To avoid the initial surge current, bias the lamps for an idle current (using resistors) that is below the incandescent level. This keeps the filaments warm to reduce the cold surge. For a starting value to experiment with try 10% - 20% of rated current.

This approach also allows lower power driver electronics and power supply. But if the device is battery operated, the extra standby current may be counter-productive.

Cheers,
Tom