Sensor

A sensor converts a physical quantity into an observable signal. A temperature sensor might produce a voltage or resistance change that a measuring system converts into a temperature estimate. A useful model separates sensitivity, offset, noise, calibration and dynamic response: a large output change does not guarantee accuracy if the calibration drifts. Sensors also load the system they measure, for example by drawing heat or electrical current. The complete measurement chain includes the sensor, readout electronics and the model mapping the signal back to the intended quantity.

A temperature sensor is a sensor whose physical response is used to estimate temperature. A resistance thermometer and a thermistor use changes in electrical resistance; a thermocouple uses the Seebeck effect. The sensor element, its thermal contact and its electrical readout together determine what a temperature measurement means.^[ref]

A contact sensor first responds to its own temperature, which need not immediately equal that of the object or fluid. A useful original lumped thermal model writes the first law of thermodynamics for a sensor with heat capacity $C$, thermal conductance $G$ to an environment at $T_e$, and internal heating power $P$:For constant inputs, its thermal time constant and equilibrium are $\tau =C/G$ and $T_{s,\infty }=T_e+P/G$. The temperature sensor response time and the bias from heating are different properties. Better thermal coupling both shortens $\tau$ and reduces $P/G$ in this model.

Take stipulated values $C=0.020J/K$ and $G=0.010W/K$. Then $\tau =2\mathrm{s}$. With $P=0$ and an environment step from $20$ to $30{}^{\circ }\mathrm{C}$,At $t=2\mathrm{s}$ the sensor reads about $26.32{}^{\circ }\mathrm{C}$, despite a perfect electrical readout. Reaching $90\%$ of the step takes $\tau \ln 10=4.605\mathrm{s}$. A steady $1\mathrm{mW}$ heating input would instead add $0.10\mathrm{K}$ at equilibrium. These are model calculations, not universal sensor specifications.

A proposed two-bath step test of a temperature sensor transfers a small probe between two well-stirred baths with independently established temperatures and records its readout against time. Repeating with a protective sleeve tests the thermal part of the measurement chain. The DOE handbook illustrates insulated sensor elements inside metal sheaths and protective wells; electrical accuracy alone does not describe that complete assembly.^[ref]

Thermal loading by a contact temperature sensor is another limit: a probe exchanges heat with the object and can perturb the quantity it is meant to measure. A small object may require a coupled thermal model rather than a fixed $T_e$. For a resistance thermometer, the excitation also causes Joule heating; checking several currents can distinguish that effect from a fixed readout offset.^[ref]

A 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.^[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 residualFor 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.^[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 $\tau =2/\ln 2\approx 2.89\mathrm{s}$ and $t_{90}\approx 6.64\mathrm{s}$.

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.^[ref]

Thermal loading by a contact temperature sensor is the disturbance of an object's temperature caused by exchanging heat with the temperature sensor used to measure it. A sensor initially colder than a small object draws heat from the object: the final reading can be accurate for the disturbed system while differing from the object's original temperature.

An illuminating idealization treats the object and sensor as two uniform bodies with constant heat capacities $C_o$ and $C_s$, joined by thermal conductance $G$ and isolated from everything else. Their energy conservation equations areAdding the equations shows that $C_o{T}_o+C_s{T}_s$ is constant. Consequently,For $C_o=0.20J/K$, $C_s=0.020J/K$, $T_{o0}=80{}^{\circ }\mathrm{C}$ and $T_{s0}=20{}^{\circ }\mathrm{C}$, the common equilibrium is $74.55{}^{\circ }\mathrm{C}$. The measurement has cooled the object by about $5.45\mathrm{K}$.

Subtracting the two differential equations gives an exponentially decaying temperature difference with time constantIncreasing the contact thermal conductance makes the disturbance happen faster without changing this isolated model's final equilibrium. Reducing the sensor heat capacity reduces the disturbance. A real object also exchanges heat with its surroundings, while sensor leads can provide another heat transfer path.

An electrically read sensor can additionally generate Joule heating. That is a separate loading mechanism: the thermometry guide measures resistance-thermometer self-heating by changing the measuring electric current and extrapolating toward zero dissipated electric power.^[ref]

A 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.^[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 areThus the time constant $\tau$ is the time to complete $1-e^{-1}\approx 63.2\%$ of the change. The time to complete a fraction $f$ is $t_f=-\tau \ln (1-f)$, so $t_{90}=\tau \ln 10$, whereas the elapsed time between 10% and 90% is $\tau \ln 9$. For example, $\tau =3\mathrm{s}$ gives $t_{90}\approx 6.91\mathrm{s}$ and $t_{99}\approx 13.82\mathrm{s}$.

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.^[ref]

A tachometer measures rotational speed, commonly displayed in revolutions per minute. A non-contact optical tachometer illuminates a reflective mark on a shaft or rotor and detects its passage as a pulse. HIOKI's fan-speed experiment uses reflective tape and counts reflected-light pulses.^[ref]

If the readout detects $p$ pulses per revolution and counts $N$ pulses during a gate of duration $T$, its average speed estimate isHere $\omega$ is the angular velocity magnitude in radians per second. For an original example, $p=2$, $N=120$ and $T=0.5\mathrm{s}$ give $r=7200$ revolutions per minute. Mistakenly configuring one pulse per revolution would double the displayed value without changing the rotor.

The counting step corresponds to $60/(pT)$ revolutions per minute per pulse: $60$ in this example. Increasing gate time improves that counting resolution but averages over a longer interval, potentially hiding speed changes. At low speeds, measuring the time between successive marks can be more useful than waiting for many counts. A rotary encoder provides a related pulse-based measurement and can also encode angular position or direction.

Missed marks, extra reflections and an incorrect pulses-per-revolution setting are distinct from uncertainty in the timing clock. A single repeated mark normally measures speed magnitude; it does not establish the direction of rotation. A convincing check specifies the mark pattern, counts per revolution and timing convention rather than treating every electrical pulse as a complete turn.

An accelerometer measures proper acceleration, the acceleration associated with nongravitational support and applied forces on its sensing mass. A common design suspends a proof mass on springs inside a housing. Housing motion displaces the mass relative to the support; a capacitive accelerometer reads the resulting change in capacitance. At frequencies well below mechanical resonance, Hooke's law and Newton's second law give displacement magnitude $|x|\simeq m|a|/k$, with spring constant $k$.^[ref]

A simple calibration rotates a stationary sensor so one axis points up, sideways and down. Near Earth's surface, the axis reading changes approximately from $+g$ through zero to $-g$, where $g\simeq 9.81\mathrm{m}{\mathrm{s}}^{-2}$. A supported stationary device therefore has a nonzero reading even though its position is constant; in ideal free fall, the mass and housing fall together and the reading approaches zero. This experimentally separates the measured support acceleration from coordinate acceleration.^[ref]

For a horizontal ideal sensor with $m=10^{-6}\mathrm{kg}$ and $k=1\mathrm{N}{\mathrm{m}}^{-1}$, an acceleration of $2\mathrm{m}{\mathrm{s}}^{-2}$ gives a displacement magnitude of $2\mu \mathrm{m}$. The readout converts a tiny mechanical displacement into an electrical signal; its scale, bias and frequency response must be calibrated.

Specific force is the nongravitational force per unit mass experienced by a body. In a Newtonian local model, neglecting Earth's rotation, let $\mathbf{a}$ be its coordinate acceleration and $\mathbf{g}$ the local gravitational acceleration vector. Newton's second law givesSpecific force has units $\mathrm{m}{\mathrm{s}}^{-2}$. An ideal accelerometer senses this quantity along its own axes, rather than coordinate acceleration alone. The Android sensor documentation illustrates the distinction with a nonzero stationary-table reading and approximately zero reading during free fall.^[ref]

Choose an upward unit vector $\hat{\mathbf{z}}$ and $\mathbf{g}=-g\hat{\mathbf{z}}$. A stationary supported instrument has $\mathbf{a}=0$ and therefore $\mathbf{f}=+g\hat{\mathbf{z}}$: support supplies the upward nongravitational force. In ideal free fall, $\mathbf{a}=\mathbf{g}$ and $\mathbf{f}=0$. Gravity has not disappeared; the instrument and its proof mass accelerate together.

For an original elevator example with $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, an upward coordinate acceleration of $2.00\mathrm{m}{\mathrm{s}}^{-2}$ gives upward specific force $11.81\mathrm{m}{\mathrm{s}}^{-2}$. Moving upward at constant velocity instead gives $9.81\mathrm{m}{\mathrm{s}}^{-2}$. The reading depends on acceleration and support, not the direction of velocity.

If a rotation matrix $R$ maps instrument-axis components to world-axis components, an ideal body-axis readout is $R^{\mathsf{T}}(\mathbf{a}-\mathbf{g})$. Reconstructing $\mathbf{a}$ therefore also requires orientation and a gravity model. A sustained horizontal acceleration can resemble a tilted gravity-related reading; an accelerometer alone does not generally separate those causes. In this local slow-motion setting the sensed support acceleration connects to proper acceleration.

A device combining sensing, local processing, a radio and a power subsystem. An energy budget must account for sleep leakage, active operation, startup and peak communication current; stored energy can bridge intermittent harvesting.