I assume that your current waveform is something like an amplitude modulated waveform. The carrier frequency is at 50hz or higher. The envelope is slowly varying - steady in places, step change in places, ramps in places.
Since your thermal time constant (90 sec) is so much higher than the period of the carrier frequency (1/50hz or shorter), we can simplify the problem by ignoring the carrier frequency component and replacing the waveform with envelope (divided by sqrt2 to convert to 1-cycle-rms). That is of course the similar to what you already did to begin with when you replaced the ramp sinusoid with a ramp to calculate the factor 1/3. This would simplify any analytical solution and I think would also reduce the number of iterations required for numerical solution (numerical which would be my preference since it easily allows us to accmodate non-linearities such as temperature coefficient of resistance.... could also easily be adjusted to account for non-linearities in heat transfer characteristic if they are known).
If I get a chance this weekend, I will adjust the spreadsheet to add a tab where the user can easily specify an arbitrary input current waveform (which could be the full am waveform or the envelope /sqrt2).
There is now enough info to solve the thermal model (simply plug 90 sec into Tau). If you wanted to do some rough validation of the thermal model there are some some double-checks which can be done:
1 - provide the info on mass of copper and steel and calculate thermal capacitance from that to see if it matches the thermal capacitance provided from Cth = Tau/Rth
2 - if you happen to have plot of temperature vs time during your dc test (could be constructed from resistance vs time accounting for temp coefficient), it should be roughly a straight line on a log-log scale if you have a 1-degree of freedom thermal system (one mass and one temperature). If 2dof system, we might see two different slopes in different parts of the curve.
3 - Multiple dc tests of course provide more detailed info on the thermal model. The form of heat transfer out of the device is most likely Ith = dT^m / Rth
where Ith is heat transfer out (watts)
dT is temperature rise above ambinet
R is thermal parameter
m is unknown coefficient. would be near 1 for conduction heat transfer but can be in the range 0.1 - 0.2 for radiation heat transfer near room temperature, approximately 0.25 for laminar convective heat transfer and 0.33 turbulent convective heat transfer. If multiple heat transfer mechanisms exist thatn the overall effective value of m would be something like a weighted average of those numbers Comparing the final equilibrium temperautres for different values of heat input = heat output = Ith would allow us to estimate the exponent m.
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