= Temperature sensor response time
{title2=$\tau$}
A \b[temperature sensor response time] specifies how long a <temperature sensor> takes to complete a stated fraction of its response to a change in the surrounding <temperature>. The fraction and the test conditions are part of the specification: a 90% response time and a 63.2% response time describe different points on the same record. Rapid immersion in a different-temperature liquid approximates the required step input.<Response Time and Temperature Sensors by Minco Staff>[Opening two paragraphs: response criterion and immersion test]{ref}
For a <lumped thermal model> with sensor <heat capacity> $C$, constant <thermal conductance> $G$ to a bath at $T_b$, and negligible internally generated <heat>, the <energy conservation> equation and its solution are
$$
C\frac{dT_s}{dt}=G(T_b-T_s),\qquad
\frac{T_s(t)-T_b}{T_s(0)-T_b}=e^{-t/\tau},\qquad \tau=\frac CG.
$$
Thus the <time constant> $\tau$ is the time to complete $1-e^{-1}\approx63.2\%$ of the change. The time to complete a fraction $f$ is $t_f=-\tau\ln(1-f)$, so $t_{90}=\tau\ln10$, whereas the elapsed time between 10% and 90% is $\tau\ln9$. For example, $\tau=3\,\mathrm{s}$ gives $t_{90}\approx6.91\,\mathrm{s}$ and $t_{99}\approx13.82\,\mathrm{s}$.
A <two-bath step test of a temperature sensor> tests this <exponential function> model rather than assuming it. Several internal <heat capacity> elements, a protective sheath, or electronic filtering can produce several response stages. The thermometry guide reports fast and slow components in tested platinum thermometer assemblies; a single quoted <time constant> need not describe their entire approach to equilibrium.<Guide on Secondary Thermometry: Industrial Platinum Resistance Thermometers by Jonathan Pearce and others>[§4.3, printed pp. 28–30, equations 4.3.1–4.3.3 and the two-component response]{ref}
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