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Just in case anyone is interested. I found an article http://www.ee.bgu.ac.il/~pel/pdf-files/conf104.pdf on modeling tungsten filament heaters. His model goes into a lot more detail, heat loss, etc.. As with a lot of simulation models from academia, it requires parameters like the pupil dilation ratio of the prepubescent madagascar dung beetle and other parameters not readily available. I just extracted some of the fundamentals for playing with heater limiting circuits and inrush. As you can see from the current limited case below, limiting inrush to 3A makes it take longer to warm up. (Yay!)
The fundamentals are:
Filament resistance is dependent on temperature.
Temperature is a function of power integrated over time (minus losses. I bypass the losses and just integrate power over time until it reaches steady state temperature. As long as the Ihot, Icold, rc are correct, it's good enough for power control circuit modeling.
Here's my greatly simplified model. I decided it was easier to specify parameters in things more easily measured, so it's current with cold filament, hot filament, and a time constant.
This is based on an EL-34, time constant is WAG (no S). Hot current comes from the datasheet, Cold current comes from measuring DC resistance of a cold tube. Vnom scales the power to the accumulator so it winds up in the right place.
.SUBCKT FILAMENT 1 3 Icold=15 Ihot=1.5 Vnom=6.3 rc=0.5
Vm 1 2 DC 0
Rl 2 3 R={Vnom / Ihot * V(acc) + Vnom / Icold * (1-V(acc))}
B1 4 0 V=abs(i(Vm) * V(1,3)) / ({Vnom * Ihot})
Rt 4 acc 1k
Ct acc 0 {rc/1k} ic=0
.ENDS
Here's a graph showing 1) heater current draw from a constant voltage source (6.3v) in red, 2) heater draw limited to 3 amps (blue), 3) the voltage across the heater for the current limited case (yellow)
Here's the test circuit:
V1 1 0 DC 6.3
X1 0 1 FILAMENT Icold=15 Ihot=1.5 Vnom=6.3 rc=0.5
I2 0 2 DC 3
D2 0 2 DMOD
V2 2 3 DC 0
X2 0 3 FILAMENT Icold=15 Ihot=1.5 Vnom=6.3 rc=0.5
.MODEL DMOD D(Is=1e-6 N=0.1 Bv=6.3)
.control
tran 250u 1 uic
plot -i(v1), i(v2), v(2) ylimit 0 15
.endc
.end
And here's a snapshot of what's going on in the power accumulator:
1) Constant voltage source in red, 2) Current limited in blue.
Now go throttle some heaters.
[edit - fixed some inaccurate verbage on calculating the temperature.]
Also, I think my cold current figure is high here, I misplaced my numbers, but I'm pretty certain that the 15A figure was more than one EL-34. Most sources on internet, that article included place the cold current at 10-15x. My measurements were more like 7-8x IIRC. If it matters precisely, you should probably dig further into the source cited above. I believe the model is accurate enough to model warmup behavior in current limited filaments, including switched/duty cycle modulated heaters.
The fundamentals are:
Filament resistance is dependent on temperature.
Temperature is a function of power integrated over time (minus losses. I bypass the losses and just integrate power over time until it reaches steady state temperature. As long as the Ihot, Icold, rc are correct, it's good enough for power control circuit modeling.
Here's my greatly simplified model. I decided it was easier to specify parameters in things more easily measured, so it's current with cold filament, hot filament, and a time constant.
This is based on an EL-34, time constant is WAG (no S). Hot current comes from the datasheet, Cold current comes from measuring DC resistance of a cold tube. Vnom scales the power to the accumulator so it winds up in the right place.
.SUBCKT FILAMENT 1 3 Icold=15 Ihot=1.5 Vnom=6.3 rc=0.5
Vm 1 2 DC 0
Rl 2 3 R={Vnom / Ihot * V(acc) + Vnom / Icold * (1-V(acc))}
B1 4 0 V=abs(i(Vm) * V(1,3)) / ({Vnom * Ihot})
Rt 4 acc 1k
Ct acc 0 {rc/1k} ic=0
.ENDS
Here's a graph showing 1) heater current draw from a constant voltage source (6.3v) in red, 2) heater draw limited to 3 amps (blue), 3) the voltage across the heater for the current limited case (yellow)
Here's the test circuit:
V1 1 0 DC 6.3
X1 0 1 FILAMENT Icold=15 Ihot=1.5 Vnom=6.3 rc=0.5
I2 0 2 DC 3
D2 0 2 DMOD
V2 2 3 DC 0
X2 0 3 FILAMENT Icold=15 Ihot=1.5 Vnom=6.3 rc=0.5
.MODEL DMOD D(Is=1e-6 N=0.1 Bv=6.3)
.control
tran 250u 1 uic
plot -i(v1), i(v2), v(2) ylimit 0 15
.endc
.end
And here's a snapshot of what's going on in the power accumulator:
1) Constant voltage source in red, 2) Current limited in blue.
Now go throttle some heaters.
[edit - fixed some inaccurate verbage on calculating the temperature.]
Also, I think my cold current figure is high here, I misplaced my numbers, but I'm pretty certain that the 15A figure was more than one EL-34. Most sources on internet, that article included place the cold current at 10-15x. My measurements were more like 7-8x IIRC. If it matters precisely, you should probably dig further into the source cited above. I believe the model is accurate enough to model warmup behavior in current limited filaments, including switched/duty cycle modulated heaters.