On Feb 15, 1:01 am, (Dave Martindale) wrote:
"Redbelly" writes:
Lots of factors are at play here. Changing the supply voltage will
change the filament temperature, and hence the filament resistance.
So cutting the voltage in half will not cut the current in half.
I made a spreadsheet which accounts for the changes in temperature and
resistance, and here's what I found:

Where did you get the formulas for these? As you say, an incandescent
lamp filament is far from a fixed resistor, but how does resistance and
temperature change with voltage?

Dave

A decent approximation is that the resistance is proportional to the
absolute temperature. A more accurate relationship is given at the
end of this post.

For my calculation, I used an iterative approach which goes as follows
(somewhat lengthy description, skip if not interested. NOT FOR
MATHOPHOBES!):

Step "0". the design values:

First, assume three conditions at the filament's designed operating
point, eg. power, voltage, and filament temperature (3000 Kelvin is a
reasonable number). Current and resistance are then determined from
power and voltage.

Step 1. 1st iteration for new operating point:
Next, change the voltage but do the calculation for the same
resistance, as if it were fixed. This gives a new power and current,
which will be wrong, but our iterative approach will converge to the
correct values. We also need to calculate a new temperature using the
blackbody radiation law:

Power is proportional to T^4
or in other words
T is proportional to P^0.25

The way we REALLY use this relation is:

T_1 = T_0 * (P_1 / P_0)^0.25
where "0" refers to the original operating point, and "1" refers to
the updated temperature and power.

Step 2. 2nd iteration of values:
Recalculate the resistance, R_2, using:

R_2 = R_0 * (T_1 / T_0)

Since voltage is fixed, we can recalculate power and current from

I_2 = V / R_2
P_2 = V * I_2

And then a new temperature from
T_2 = T_0 * (P_2 / P_0)^0.25

Step 3.
Recalculate the resistance:
R_3 = R_0 * (T_2 / T_0)

Then recalculate I_3, P_3, and T_3 as in Step 2.

Steps 4, 5, etc.
Keep repeating the process until the power, temperature, and
resistance converge to steady values that don't change significantly.

That's it, really. A somewhat more accurate description (for
tungsten, anyway) is that resistance is proportional to T raised to
the power of 1.2. Accounting for this did not change the results by
very much.

Regards,

Mark