Sensors and Industrial Instrumentation

Strain gauge gauge factor

The gauge factor G_F of a metallic strain gauge is the ratio of the fractional change in resistance to the strain that causes it, so G_F = (\Delta R/R)/\varepsilon. It is dimensionless, and most metal-foil gauges have a value close to 2. Strain is often expressed in microstrain, where 1 µm/m equals 10^{-6}, so the resistance change is small and must be read with a bridge rather than a plain ohmmeter.

To find the new resistance, compute \Delta R = G_F\,\varepsilon\,R first and then add it to the nominal value R, keeping the sign in mind: tension increases the resistance and compression decreases it. For a 500 Ω gauge with G_F = 2 under a strain of 200 µm/m, the change is 0.2 Ω.

Temperature also changes the resistance of a gauge. With a temperature coefficient of resistance \alpha and a temperature rise \Delta T, the gauge resistance can be written as R = R_0(1+\alpha\Delta T)(1+G_F\varepsilon). When all arms of a bridge share the same temperature, the common thermal factor scales every arm equally, so the output depends only on the arm-to-arm difference, which is set by the strain.

When a bridge is balanced at zero strain, a small change \Delta R in one or more arms unbalances it. A balancing potentiometer connected in the bridge can restore the null, and the resistance it must supply depends on the ratio of \Delta R to the nominal arm resistance R.

Thermocouple cold junction compensation

A thermocouple produces an electromotive force (EMF) through the Seebeck effect, which arises when two dissimilar metals are joined and their two junctions are held at different temperatures. The EMF therefore encodes the temperature difference between the measuring junction and the reference junction, not the absolute temperature of the measuring junction. The Seebeck EMF is nonlinear in temperature and is only roughly proportional to the temperature difference over a limited range. Thermocouples do not depend on a resistance change or on Joule heating.

Cold junction compensation is the correction that removes the effect of the reference junction, whose temperature is rarely zero in practice. The reference junction temperature is measured with a separate sensor such as a thermistor or an RTD, and the equivalent EMF is added to the reading. For example, a local sensitivity of 40 µV/°C with a hot junction 60 °C above the reference junction gives 1.6 mV before any amplification.

In an instrumentation amplifier, compensation is often done differentially. Two thermocouple legs are connected in opposition, so a constant offset common to both legs cancels, and the amplifier sees only the difference in junction EMF. Its output is then the gain multiplied by that difference.

Inserting a third metal, such as copper leads, into a thermocouple circuit does not change the net EMF provided both new junctions are at the same temperature. This is the law of intermediate metals, and it is why extension wires and voltmeter leads can be added without error when the junctions are kept isothermal.

Transducer type matching

Matching a sensor to a quantity requires knowing the transduction principle of the device. Each device converts one physical quantity into an electrical signal through a specific mechanism, so the pairing is judged on that mechanism rather than on the form of the output.

• LVDT — converts linear position into a voltage through the variation of mutual inductance between coils, so it measures linear displacement.
• Thermistor — a resistor whose resistance changes strongly with temperature, so it is a temperature sensor.
• Strain gauge — converts strain into a change of resistance, so it can be used for force and torque when the strain in the body is known.
• Diaphragm — deflects under an applied pressure, so it is used to measure pressure.
• Optical pyrometer — infers temperature from the wavelength and brightness of the radiated energy.
• Thermocouple — generates an EMF at the junction of two dissimilar metals, so it measures temperature.
• RVDT — rotary variable differential transformer, which gives an output proportional to angular displacement.
• Ionization gauge — measures very low gas pressures, so it is used for vacuum measurement.

Strain gauge bridge arrangement

A Wheatstone bridge converts small arm resistance changes into a voltage. For small changes, the output is a sum of the arm changes, each divided by R and weighted by the excitation V_s/4, with the sign of each term set by the arm position. Adjacent arms that see strains of the same sign cancel, whereas adjacent arms that see strains of opposite sign add. Gauges in diagonally opposite arms that see the same sign add to the output.

The output depends on how many arms are active. With one active gauge the output is V_o = V_s G_F\varepsilon/4; with two active gauges in opposite arms it is V_s G_F\varepsilon/2; with four active gauges it is V_s G_F\varepsilon. A cantilever with gauges on opposite faces gives the greatest sensitivity when the top-surface and bottom-surface gauges occupy arms that satisfy the sign rule, so that the diagonal pairs share the same strain sign.

Lead wire resistance adds to the measured arm and can introduce apparent strain. In a three-wire connection, one lead is placed in the adjacent arm so that its effect subtracts from the lead in the active arm. With matched leads the residual error is the mismatch between the leads, which appears as a false strain reading.

For a half-bridge with two active gauges, V_o = \frac{V_s}{2}G_F\varepsilon. The strain is found by inverting this relation. For example, with a 5 V supply and G_F = 2, an output of 0.5 mV corresponds to a strain of 100 µm/m.

Thermistor characteristics

A thermistor is a semiconductor resistor with a large temperature coefficient. A negative temperature coefficient (NTC) thermistor, the common type for measurement, follows the exponential law R = R_0\exp\!\left[\beta\left(\frac{1}{T}-\frac{1}{T_0}\right)\right] with T in kelvin, so its input-output characteristic is exponential and strongly nonlinear. This feature distinguishes it from thermocouples, RTDs and liquid-in-glass thermometers, whose responses are approximately linear over their working ranges.

The material constant \beta is found from two resistance-temperature points, because the ratio of two resistances eliminates R_0. The relation is \ln(R_1/R_2) = \beta(1/T_1 - 1/T_2), and temperatures must be in kelvin. For example, a thermistor reading 10 kΩ at 25 °C and about 3.6 kΩ at 50 °C has \beta close to 3950 K.

Hall effect sensing principle

The Hall effect is the generation of a transverse voltage when a current-carrying conductor or semiconductor is placed in a magnetic field. Its physical basis is the Lorentz force \vec F = q(\vec v\times\vec B), which deflects the moving charge carriers to one side until the electric field they create balances the magnetic force. The resulting Hall voltage is proportional to the magnetic flux density, so a Hall sensor measures a magnetic field, not a temperature or a light intensity.

For a slab of thickness t carrying current I in a field B, the Hall voltage is V_H = R_H B I/t, where R_H is the Hall coefficient. Rearranged, the field is B = V_H t/(R_H I). All quantities must be in SI units, with thickness in metres, current in amperes and R_H in m^3/C.

Eddy current effect on inductance

Eddy currents are circulating currents induced in a conductor by a changing magnetic flux. When a conducting, non-magnetic object such as a copper sheet is placed near an alternating-current coil, the induced eddy currents produce a field that opposes the coil flux, in accordance with Lenz's law. The net flux linking the coil per unit current falls, so the measured inductance decreases. The effect is strongest at high excitation frequency and for high conductivity.

The same principle underlies a hand-held metal detector. Its search coil carries an alternating current, and a metallic object in range has eddy currents induced in it. Their reaction field changes the coil impedance, which the electronics detect. The mechanism is eddy current generation in the target, not a change in the reluctance of an iron core, because the sensing does not depend on a ferromagnetic core.

Potentiometric displacement sensor resolution

A wire-wound potentiometer produces a voltage step at each turn of wire. With N turns connected end to end across a supply V_s, there are N-1 equal intervals between the first and last turns, so the voltage increment per turn is V_s/(N-1). This step sets the basic resolution of the wiper output, and a larger number of turns gives finer resolution. For example, 11 turns across a 5 V supply give 0.5 V per turn.