Engineering & Technology · Instrumentation, Sensors & Embedded Integration

Sensor principles, measurands and environmental effects

A force display changes when the room warms, even though the supported load appears unchanged. The electronics may be faithfully reporting a voltage that responds to two physical quantities. Understanding the sensor means identifying that relationship before interpreting the display as force.

Use Measurement, Units & Engineering Evidence; Electrical Circuits & Network Behaviour; and Electronics, Signals & Devices. This chapter defines its conversion slopes, timing terms and byte conventions. Force-voltage models, reference pairs, clocks, motor parameters and all plotted data are original constructed teaching inputs. They are not hardware selections, wiring instructions, operating limits or qualified calibration records.

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After reading, you can

Relate a sensor signal to a defined measurand and distinguish environmental response from the intended observation.

Define the quantity and the physical interaction

The measurand is the quantity intended to be measured, with its location, direction, time and conditions specified. “Force” is incomplete when several contacts or directions exist. A force on a fixture, strain on its surface and stress inferred from a material model are different quantities. Their relationship needs a mechanical model and evidence about the load path.

A transducer converts one physical effect into another observable signal. A bonded resistance gauge responds to local deformation through a resistance change. A bridge circuit can turn a small resistance difference into voltage. Inferring an applied force from that voltage adds assumptions about geometry, stiffness, mounting and temperature. The display unit does not remove those assumptions.

Follow a local deformation into a bridge voltage

Longitudinal strain ε is change in length divided by the original length along the chosen direction: ε=ΔL/L0. It is dimensionless; positive means extension and negative contraction. A bonded metallic gauge follows the local surface deformation to the extent that its backing and bond transfer it. In a simple uniform-conductor model, R=ρL/A: resistance R depends on material resistivity ρ, conducting length L and cross-sectional area A. Stretching changes the geometry and may also change resistivity. It is therefore the complete resistance response, not length alone, that needs characterisation.

Define gauge factor GF as fractional resistance change divided by strain: GF=(ΔR/R0)/ε for a nonzero strain in the stated local relationship, with R0 the reference resistance. A supplied constant-GF small-strain model is ΔR/R0=GF ε. GF is dimensionless and depends on the gauge and conditions; no universal value is implied. Inferring force requires a further relationship between that local strain and the force/load path. The bridge example below is a separate transduction illustration, not a derivation of the later supplied force-voltage coefficients.

Our original mathematical bridge has two parallel voltage dividers across excitation Vex, with their lower ends at reference 0 V. In the left divider, R1 is the upper sensing arm and R2 the lower arm; midpoint L is between them. In the right divider, R3 is upper and R4 lower; midpoint R is between them. Assume ideal constant excitation, negligible lead resistance and a voltage reader drawing negligible current from either midpoint. Then VL=Vex R2/(R1+R2) and VR=Vex R4/(R3+R4). Declare the output order Vdiff=VR−VL; changing that order changes its sign.

With all four arms equal, both midpoints are Vex/2, so the differential output is zero although each midpoint has a nonzero potential relative to the lower rail. Increasing R1 alone lowers the left divider fraction and VL; VR remains fixed, making our Vdiff positive. This is how a resistance change becomes a voltage difference without changing excitation. The circuit responds to ratios, not to the label “strain” on one resistor.

Real strain, temperature, unequal leads, bridge excitation and the readout can change those ratios or the inferred strain. Uniformly multiplying all four ideal resistances by the same factor would leave the divider fractions unchanged, but that special cancellation is not proof of temperature compensation in an actual installation. Gauge-factor evidence, local strain transfer and the installed response still matter.

Original ideal two-divider bridge: a 2 V excitation is between upper rail P and lower reference rail N. The left upper arm R1 is the sensing resistance 100.20 ohm and lower R2 is 100 ohm. The right upper R3 and lower R4 are 100 ohm each. Negligible-current midpoint readings give VL=0.999001 V and VR=1.000000 V; the defined output VR−VL is positive 0.999 mV. All-equal arms would give 1 V at both midpoints and zero differential output. Values are a constructed small-strain example, not hardware settings or a force calibration.

Use sensitivity with its units and range

A local linear response can be written V=b+sF, where b is offset in volts, s sensitivity in volts per newton and F force in newtons. Sensitivity is a slope, not a statement of accuracy or maximum load. Real responses may be nonlinear, direction dependent or history dependent. Specify the interval over which the approximation was checked.

Environmental quantities can enter the same response. Temperature can change a sensing element, its support and the connecting circuit. Mounting can alter which deformation reaches it. A sensor can also disturb the object through its stiffness, contact or power dissipation. Ask how the installed measurement differs from the intended undisturbed quantity.

Model V=0.50+0.020F+0.002(T−20), with V in volts, F in N and T in °C. At fixed F=40 N, ignoring temperature gives indication error 0.1(T−20) N: −2 at 0 °C, 0 at 20 °C, 2 at 40 °C and 4 at 60 °C. Subtracting the supplied exact temperature term gives zero error in this ideal example. Real compensation retains uncertainty.

The sensor has a time and place

Spatial averaging and response time matter. A small gauge reports its local orientation and region; it does not reveal an entire structural field. A sensor that follows changes slowly can understate a short peak even when its steady calibration is good. The relevant bandwidth belongs to the complete installation and acquisition path.

A compensation model uses additional information to estimate an unwanted influence. It must be supported over the relevant conditions and retains uncertainty in its coefficients and environmental reading. Avoid treating compensation as proof that every influence vanished. Preserve the raw signal and the transformation so a changed interpretation can be examined later.

Separate force from a supplied temperature effect

Use the original model V=.50+.020F+.002(T−20), with volts, newtons and degrees Celsius. At F=40 N and T=30 °C, V=.50+.80+.02=1.32 V. An inverse calibrated only at 20 °C gives (1.32−.50)/.020=41 N. The extra 1 N comes from ignoring the supplied temperature term.

With the exact model and exact temperature in this constructed case, subtracting .002×10 V gives 40 N. The figure varies temperature while keeping force 40 N. It does not say that a real device has these coefficients or that a temperature reading can correct unrelated mounting effects.

For the separate bridge illustration, stipulate R0=100 Ω, GF=2 and local ε=0.001. Fractional resistance change is 2×0.001=0.002, so ΔR=0.20 Ω and R1=100.20 Ω. Keep the other arms 100 Ω and mathematical Vex=2 V. Then VL=2×100/200.2≈0.999001 V and VR=1.000000 V, giving Vdiff≈+0.000999 V, or +0.999 mV. These values are constructed model inputs, not gauge specifications or excitation recommendations.

Change only the supplied strain to −0.001. The same local GF model gives R1=99.80 Ω, VL=2×100/199.8≈1.001001 V and Vdiff≈−1.001 mV. The output sign reverses for the stated upper sensing arm and output order. Its magnitude is not exactly symmetric because the exact divider denominator also changes. A small-strain linear bridge approximation would hide that small difference; neither calculation yet supplies force without the mechanical relationship.

Try the reasoning

Does a voltage labelled in newtons directly observe force without a model?
Show the answer and reasoning
No. The voltage-to-force relationship includes transduction, installation and calibration assumptions.
Is sensitivity the same as accuracy?
Show the answer and reasoning
No. Sensitivity describes response per input change; accuracy and uncertainty concern agreement and evidence.
Why retain raw voltage after compensation?
Show the answer and reasoning
It allows the correction, coefficients and alternative explanations to be checked without losing the original observation.
Move the sensing resistance to the lower left arm R2. Keep R1, R3 and R4 at 100 Ω, raise R2 to 100.20 Ω and retain Vex=2 V and Vdiff=VR−VL. What changes?
Show the answer and reasoning
Now VL=2×100.2/200.2≈1.000999 V, while VR=1 V. Vdiff≈−0.999 mV. The resistor change is positive in both cases, but its position in the divider reverses the output sign. The declared topology and voltage order are essential.
All four ideal bridge arms increase from 100 to 101 Ω by the same factor. Does the output establish that a real force measurement is immune to temperature?
Show the answer and reasoning
The two ideal divider fractions remain one-half, so the differential output stays zero. This only establishes cancellation for the supplied equal-ratio change. Real arms, leads, bonding, strain transfer and gauge factor need not respond identically; equal output does not prove that every environmental influence or force-conversion error vanished.

What to carry forward

Specify the measurand and the installed response before trusting a displayed quantity.

References

Teaching resources and research supporting this lesson’s explanations and examples.