# Response time and time constant of Mercury thermometer

I searched to find differences between response time and time constant of a thermometer. So I found that those two are not same and we have to get 99.3% of the total step change to find response time and we have to get 63.2% of its total step change to find time constant of thermometer... Is this correct or not and are those percentages are correct or not? Can you explain why that is... Thanks in advance...

• So a mercury thermometer? May 16, 2021 at 19:21
• Yes it is a mercury thermometer. May 16, 2021 at 19:23

Response time, more precisely called settling time is time after the system output settles to the new value after change in the input (in your case the output is reading of the thermometer and input is actual value) - more precisely time after the error between target value and actual doesn't exceed certain threshold, because the value will never reach that targen in finite time, it will only asymptoticly approaching it. How big is that threshold (how wide is error band), depends on arbitrary decision - in your case it seems like it's 0.7% error, which is quite small one. More often it's something like 2 or 5%. But to settling time be meaningful that error band need to be known.

Time constant on the other hand is just constant in equations/transfer function describing the system. For first-order system (which thermometer probably can be approximated with) it just happens to be equal time after 63.2% of that change is made. That's because for first-order system it describes time after output would reach target level if initial rate of change was maintained (see initial rate of movement on the picture below). And since the value changes are exponential it turns out to be 63.2%.

(First order system step response)

The typical first order unit step response is an exponential decay, $$1-e^{-t/T}$$ where $$T$$ is the time constant. The value of the expression after $$t=T$$ is $$1-e^{-1}$$ or 0.632=63.2%. That's where that comes from.
The 99.3% is either writing the same specification as another multiple of T (i.e. $$t=5T$$, thus $$1-e^{-5}$$ = 0.993 =99.3%), and/or that the system is not first order, which shouldn't be assumed.