Sensors, transducers, and measurement chains

Identify sensing and transducing roles

Start by specifying the quantity you want to measure: for example, the temperature of an identified body at a stated location and under stated conditions. The sensor is the measuring-system element directly affected by the body or phenomenon carrying that quantity. A measuring transducer is a device that provides an output quantity with a specified relation to its input. These definitions describe roles; the same element can perform both.

A hypothetical temperature-dependent resistor illustrates the distinction. Its sensing element responds to its own temperature, and its resistance is the output quantity used by the next stage. To use that response for the body's temperature, the model must justify the relation between body and sensor temperatures. Our first case stipulates equilibrium and equal temperatures at the specified location; it supplies no real thermometer assessment.

A transducer's output need not be an electrical voltage, and transduction need not change the kind of quantity. Mechanical and electrical devices can both take part in measurement. Identify the actual input and output relation rather than treating every sensor as a complete instrument or requiring a separate box for each role.

Follow one signal path

A measuring chain follows one signal path from a sensor to an output element. The measuring system is broader: it includes the instruments and other necessary devices, supplies or reagents assembled for the measurement. It can contain more than one signal path or even consist of a single instrument. Our examples focus on one path at a time.

For the temperature example, the chosen path is body temperature → sensing-element resistance → powered readout voltage → a voltage indication. The readout stage is stipulated to have the energy supply needed for its operation; a varying resistance is not assumed to generate a voltage by itself. A conditioning stage changes a signal into a form useful to the following stage, and its input/output relation belongs in the chain model.

An indication is the value supplied by a measuring instrument or system. It need not have the same quantity kind as the measurand: a voltage indication can carry information used to infer a temperature. The signal chain is also distinct from the preceding lesson's traceability chain of standards and calibrations. Mapping the signal path alone does not establish metrological traceability.

Declare static stage models and units

Supply this original ideal static model only for 273.15 K ≤ T ≤ 333.15 K: R = 100 Ω + (0.4 Ω/K)(T − 273.15 K). T is the equal body/sensor temperature, and R is the sensing element's resistance. The chosen reference temperature is 273.15 K, not zero kelvin. The offset and coefficient are exact teaching stipulations, not specifications for a real platinum resistance thermometer.

The powered readout is separately supplied as V = (0.02 V/Ω)(R − 100 Ω), where V is its output voltage relative to the stated common electrical reference. Assume both stages have reached their static values and that no loading, noise or output limiting is present in this case. Substitution gives V = (0.008 V/K)(T − 273.15 K). The resistance unit cancels between the two stage coefficients.

At T = 293.15 K, the temperature difference is 20 K, the resistance is 108 Ω and the voltage is 0.16 V. Within the declared model domain, the nonzero slope also permits inversion: T = 273.15 K + V/(0.008 V/K). This is an inverse of the supplied ideal model; it is not a performed calibration or a real uncertainty statement.

Combine sensitivities with their units

A measuring system's sensitivity compares a change in its indication with the corresponding change in the measured quantity. It may depend on the input value; practical evaluation also needs changes that can be resolved. For our exact linear static relations, every nonzero temperature change staying within the supplied domain has the same voltage-to-temperature ratio.

The first stage has dR/dT = 0.4 Ω/K, and the second has dV/dR = 0.02 V/Ω. For this serial static model, the chain rule gives dV/dT = (dV/dR)(dR/dT) = 0.008 V/K. Multiplying bare numbers without their units would hide what is being converted. Sensitivity is not the same quantity as accuracy or uncertainty.

For example, raising T by 5 K increases R by 2 Ω and V by 0.04 V. Changing a readout's scale factor changes the voltage response but does not by itself establish better knowledge of T. The combined relation remains valid only under the assumptions used to connect the stages; the next two sections add separate ideal models of interactions and limits.

Include the load at a voltage interface

Connecting a measuring device can affect the signal being measured. The MIT laboratory material explicitly models a voltmeter's finite resistance and investigates the resulting change in a resistor network. For an independent original example, represent one DC voltage-output port by an ideal source V_s = 2 V in series with resistance R_s = 1 kΩ. Connect a receiver of input resistance R_in from the output terminal to the common reference; its indicated V_in is the voltage across R_in.

All resistances are positive, constant and exact in this hypothetical circuit. The same current flows through R_s and R_in: I = V_s/(R_s + R_in). Ohm's law then gives V_in = V_s R_in/(R_s + R_in). With R_in = 9 kΩ, I = 0.2 mA and V_in = 1.8 V. The signed change from the unloaded 2 V is −0.2 V, or −10% of V_s. The ideal source itself remains 2 V; the terminal voltage is reduced by the drop across R_s.

With R_in = 99 kΩ, I = 0.02 mA and V_in = 1.98 V: the signed change is −0.02 V, or −1%. Increasing R_in reduces loading in this particular voltage-measurement model. This conclusion is not a rule for every current-output sensor or time-varying circuit. Neither percentage is a standard uncertainty; each is a determined change within the supplied circuit model.

Check where information is lost

An output stage can reach a limit before the sensing element stops responding. Add a separate hypothetical limiter to the temperature chain: V_m equals the ideal V between 0 V and 0.4 V, but is held at the nearest endpoint outside that interval. V_m is the limited voltage indication. The original resistance relation still applies only on 273.15–333.15 K; this limiter was absent from the earlier static case.

At T = 323.15 K, the ideal resistance is 120 Ω and ideal V is 0.4 V. At 333.15 K they are 124 Ω and 0.48 V, yet V_m is again 0.4 V. Thus distinct temperatures have the same limited indication. Applying the original inverse to 0.4 V cannot distinguish them; multiplying that already limited value by a later gain cannot restore the lost distinction. This is saturation in the stated ideal limiter model.

The relevant limits are conditions of use, not merely the endpoints printed on a display. VIM's measuring interval requires defined conditions and specified instrumental uncertainty. A bounded Japanese NMIJ research abstract reports highly sensitive accelerometers that saturate at an ordinary larger vibration-calibration level and therefore need much smaller vibrations. It illustrates a real range constraint; it does not supply a general clipping law or validate our temperature model.

Distinguish static response from delay

A static relation does not specify how quickly a system responds to a changed input. VIM's step response time refers to an abrupt change between specified constant inputs and to the indication settling within specified limits around its final steady value. The change, limits and operating conditions therefore matter.

For a separate original lag example, an ideal voltage stage's target changes from 0 V to 4 V at t = 0 and stays at 4 V. Specify only its output at check instants t_n = nΔt, with Δt = 1 s and integer n ≥ 0: y_0 = 0 V and y_(n+1) = y_n + (1/2)(4 V − y_n). The factor 1/2 is dimensionless. This supplied recurrence is not inferred from the static temperature chain or asserted for all sensors; it defines no output between the check instants.

Each update halves the remaining difference: 4 V − y_n = (4 V)2^(−n). The table follows directly. At n = 5 the difference is 0.125 V, and at n = 6 it is 0.0625 V. Hence 6 s is the earliest specified check instant within 0.1 V of the target, and all later specified checks remain within that limit. Because the between-check behavior is unspecified, these calculations alone do not establish a complete continuous-time step response or its settling time.

Keep the chain model and configuration

A useful chain description names the measurand, sensor, stages, input/output quantities, units and relations. Retain the stage settings, supplied energy, signal-reference convention and stated operating conditions. A resistance-to-voltage slope is meaningful only with its offset and input domain; a voltage-loading calculation also needs the source and receiver resistances.

Keep static response, interactions, saturation and delay distinct. The static example's 0.008 V/K sensitivity does not determine the separate circuit's loading or the separate stage's lag. The explicit models let you calculate each effect, but they do not establish the accuracy, uncertainty, calibration or usable measuring interval of a real assembled system.

Before interpreting recorded indications, ask whether the chain still retains the distinction needed between candidate inputs and whether the chosen relation applies to the actual configuration. This lesson followed conversion and interface behavior. The next subject, Sampling, digitization, aliasing, and the Nyquist limit, examines how recording at selected times and representing signals digitally introduce further questions.

Worked examples

Try it yourself

1. Use the original static temperature chain at T = 298.15 K. Find the temperature difference, resistance and output voltage. Then infer the temperature corresponding to an ideal 0.24 V output, within the same model domain.

   Show the answer and reasoning

   The difference is 25 K, R = 110 Ω and V = 0.20 V. An ideal 0.24 V corresponds to 303.15 K.

   The 25 K change produces 10 Ω and then 0.20 V. Inversion of 0.24 V gives a 30 K difference from the 273.15 K reference. The intermediate quantities have different kinds and units.

2. Keep the same sensing-element relation but replace the powered readout coefficient by 0.01 V/Ω, with the same 100 Ω offset. Find the new combined sensitivity and output at T = 293.15 K. Does this change alone establish a different temperature accuracy?

   Show the answer and reasoning

   The combined sensitivity is 0.004 V/K and the output is 0.08 V. The scale change alone establishes no temperature-accuracy comparison.

   The sensing element still has R = 108 Ω. Multiplying 0.01 V/Ω by 0.4 Ω/K gives 0.004 V/K, and the 8 Ω difference produces 0.08 V. No actual errors, uncertainties or calibration evidence were supplied.

3. Keep V_s = 2 V and R_s = 1 kΩ in the DC voltage model. What minimum positive R_in makes the magnitude of the loading change at most 2% of V_s? Give the voltage at that boundary.

   Show the answer and reasoning

   R_in must be at least 49 kΩ. At 49 kΩ, V_in = 1.96 V and the signed change is −2%.

   The fractional reduction is (V_s − V_in)/V_s = R_s/(R_s + R_in). Requiring it to be at most 0.02 gives R_in ≥ 49 R_s. The 50 kΩ total gives I = 0.04 mA and V_in = 1.96 V. This bound belongs to the stated exact DC model.

4. With the separate 0–0.4 V limiter on the original static chain, compare T = 323.15 K and T = 333.15 K. Can an indication of 0.4 V identify a unique temperature within the declared 273.15–333.15 K domain?

   Show the answer and reasoning

   Both temperatures give V_m = 0.4 V. That indication does not identify a unique temperature; every T from 323.15 K through 333.15 K has that limited indication.

   The ideal voltages are 0.4 V and 0.48 V, but the limiter holds all values at or above 0.4 V to that endpoint. The sensing-element resistances differ, 120 Ω and 124 Ω, while the final indication has lost that distinction.

5. In the original half-difference lag sequence, replace the permitted remaining difference by 0.05 V. What is the earliest specified check instant meeting the new limit, and what is its output?

   Show the answer and reasoning

   The earliest specified check is 7 s, with y_7 = 3.96875 V and remaining difference 0.03125 V.

   Require 2^n ≥ 4/0.05 = 80. The values 2^6 = 64 and 2^7 = 128 show n = 7 is first. At n = 6 the difference 0.0625 V exceeds 0.05 V. Only the supplied check instants are assessed.

6. For a new sequence with the same target, check instants and y_0, replace the factor 1/2 by 1/4: y_(n+1) = y_n + (1/4)(4 V − y_n). Find the first three outputs and compare them with the original sequence. Do equal eventual targets imply equal response speeds?

   Show the answer and reasoning

   The new outputs are 1 V, 1.75 V and 2.3125 V. They approach the same 4 V target more slowly at the specified checks. Equal eventual targets do not imply equal response speeds.

   The remaining difference is multiplied by 3/4 at each update instead of 1/2, so it remains larger at every positive n for this same starting difference. Both geometric factors lie between 0 and 1 and their remaining differences tend to zero. This is a comparison of two supplied sequences, not a universal sensor law.

What to carry forward

• A sensor directly responds to the phenomenon carrying the measurand; a transducer supplies an output quantity through a specified relation. Their roles can overlap.
• A measuring chain is one signal path within a measuring system; it is distinct from a metrological traceability chain.
• The hypothetical resistance-to-voltage temperature chain has combined static sensitivity 0.008 V/K, under its declared domain and assumptions.
• Finite receiver resistance causes loading in the supplied DC voltage circuit: the two separate receiver voltages are 1.8 V and 1.98 V.
• Saturation can give different inputs the same indication; a later gain cannot recover that lost distinction.
• Static response and dynamic behavior require separate models. Check-instant calculations do not establish unspecified between-check behavior or a real instrument's performance.