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instrumentation - How to get information within the time constant of an instrument that has a time change source environment?

I have encountered a question during extracting data from an instrument that has a time constant of around 18 seconds. As we know that after around 4-5 time-constant, we can reach the exact signal. However, my instrument has an environment of a source that is changing with time and I was trying to get the best estimate out of it. To explain this mathematically, my thermocouple time constant calibration (which was placed in boiling water then moved to almost frozen water) has an exponential equation as exp(-0.06 t), where t is time. On the other hand, when I placed it in the environment with the source which is changing with time, my exponential equation is exp(-0.02 t), as expected is slower than the first one.

How can I get the information within these curves? Thank you

question from:https://stackoverflow.com/questions/65926298/how-to-get-information-within-the-time-constant-of-an-instrument-that-has-a-time

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