Electrical – How to calculate temperature in Wire

I don't believe you can do this with any degree of precision : it depends too much on the wire and its environment which determines its rate of cooling.

This 450 page book would give you a brief introduction to the difficulty; the TL/DR version is: Don't.

Which leaves you with the alternative problems of:

• determine the wire's temperature and control the voltage to achieve the correct temperature. You could measure the drive current and voltage, to determine its resistance, and compute the temperature from that
• measure the actual motion and control the voltage accordingly (programming a PID controller is the best way, but the response time of the wire makes this tricky)
• Experimenting to determine what voltage is "good enough" in your environment. This is by far the simplest and may be good enough for your application; you haven't told us enough to tell.

Maybe, maybe not, but I'd ask why you are not correlating hot chips with power supply currents, and why you're not putting a temperature sensor on the heatsink. If the thermal path from the die to the heatsink is impaired you'll get a different temperature differential between the die and the heatsink. Likewise, if the chip is drawing more current you should be able to predict the final temperature of the die based on normal thermal behavior. And measuring the heatsink temp doesn't require a dedicated contact sensor: a temporary one will do, or a non-contact IR unit should work, since the emissivity of the heat sinks should be pretty uniform.

As to why the maybes, consider the following model:

^{simulate this circuit – Schematic created using CircuitLab}

If the thermal resistance from the die to the heatsink is much larger than the thermal resistance of the heatsink to ambient, and the thermal capacity of the die is much less than the capacity of the heat sink (and I would guess both to be true), the latter is the dominant factor in determining the thermal time constant of the heatsink, and thus of the die. In this case, increases in the die/HS thermal resistance will have only small effects on the time constant of the die, but will cause the die to get hotter. You'll have to figure the values for your board to see if this is the case.