Recently I have undergone a project involving a Thermo-Electric Cooler (TEC). I successfully recreated the manufacturer's datasheet curves by simulating the TEC in LTSpice using thermal circuit modelling techniques. In doing so, I discovered a question that, while irrelevant, distracted me for hours.
Consider a big block of aluminum of some mass such that its thermal capacitance is 1000 (J/K). Imagine it is connected to a smaller block with thermal capacitance of 500 (J/K) through a thermal strap with thermal resistance of 0.1 (K/W). The thermal circuit equivalent is below (yes, GND should be zero Kelvin, but shush the end result is the same shifted by 273.15C in this case):
Now, here's where things get knocked into twelfth gear. I set the big block's initial temperature to 20C by doing a ctrl+rmb over the capacitor and typing "IC=20" in the "SpiceLine" field. In a similar fashion I set the small blocks temperature to 80C.
My question is: how does one find an equation (closed-form or otherwise) in the time domain AND the Laplace domain for currents (heat flows in this case) and voltages (temperatures in this case) in circuits in which Ls and Cs have initial conditions? That is, what are the lumped Laplace models for Ls and Cs with initial currents and voltages, respectively? I also humbly request some intuition as to why these equations make sense!!
LTSpice has no problem telling me the system settles to around 40C, as seen in the figure above. To help me answer this conceptual question of modeling initial conditions, I looked for practice problems solving Thévenin Equivalent Circuits. I can find countless examples involving Ls and Cs in textbooks and on the internet. However, in EVERY SINGLE ONE they assume zero initial conditions!
I will attempt to answer this question below using Laplace expressions, however if someone could point me in the direction of a time-domain expression for two capacitors sharing charge, it would be much appreciated! I've been racking my brain at this for days.
Impedance of Capacitor C in Laplace domain--> 1/(sC)
Laplace model of Capacitor C with initial charge --> 1/(sC) + Vi/s (?)
How about Inductors with some initial currents through them? For this we must leave the thermal stuff behind, because as far as I'm aware there are no thermal energy storage phenomena in life that lend themselves to inductors:
Impedance of Inductor L in Laplace domain--> sL