= Two-bath step test of a temperature sensor
A \b[two-bath step test of a temperature sensor] estimates the dynamic response of a <temperature sensor> by transferring it between two approximately uniform, stable baths at different <temperatures>. The apparatus consists of the two baths, independent reference thermometers, a reproducible immersion arrangement, and a recorder fast enough to resolve the <temperature sensor response time>. Rapid insertion into a liquid at a different <temperature> is a practical approximation to an instantaneous change in the sensor's surroundings.<Response Time and Temperature Sensors by Minco Staff>[Second paragraph: rapid insertion into heated water or oil]{ref}
Let the initial equilibrium be $T_1$ and the second bath be $T_2$. Convert the recorded sensor output to <temperature>, then form the dimensionless residual
$$
r(t)=\frac{T_s(t)-T_2}{T_1-T_2}.
$$
For a single <lumped thermal model>, $r=e^{-t/\tau}$, so a plot of $\ln|r|$ against elapsed time is a straight line with slope $-1/\tau$. This provides a direct check on the assumed <time constant>; a curved plot suggests additional response stages or an imperfect input step. The conversion to <temperature> matters when sensor output is nonlinear.<Response Time and Temperature Sensors by Minco Staff>[Paragraph beginning “For a rigorous approach”]{ref}
For a constructed example, transfer a sensor from $20\,{}^{\circ}\mathrm{C}$ into a $60\,{}^{\circ}\mathrm{C}$ bath. Readings of $40$, $50$, and $55\,{}^{\circ}\mathrm{C}$ after $2$, $4$, and $6\,\mathrm{s}$ give residuals $1/2$, $1/4$, and $1/8$. Each successive interval halves the residual, giving $\tau=2/\ln2\approx2.89\,\mathrm{s}$ and $t_{90}\approx6.64\,\mathrm{s}$.
Record the transfer delay, immersion depth, stirring conditions, probe sheath and readout filtering with the result. These affect <heat transfer> or the measured transient. A <temperature sensor response time> measured in one medium need not apply in another.<Guide on Secondary Thermometry: Industrial Platinum Resistance Thermometers by Jonathan Pearce and others>[§4.3, printed p. 28: response depends on thermometer and surrounding medium]{ref}
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