Q4Power System Instrumentation

(a) Strain Gauge Transducer and Gauge Factor Derivation

A strain gauge is a transducer whose electrical resistance changes in proportion to the mechanical strain (fractional deformation) applied to it, most commonly constructed as a thin metallic foil (or fine wire) grid pattern bonded to a flexible insulating backing, which is in turn firmly cemented to the surface of the structural member whose strain is to be measured, ensuring the gauge deforms identically together with the underlying surface.

Derivation of Gauge Factor: the electrical resistance of a conductor of length L, cross-sectional area A, and resistivity rho is R=rho*L/A. When the conductor is strained (stretched) by an axial force, its length increases by a small amount dL, its cross-sectional area decreases correspondingly (due to the Poisson effect, since stretching in the axial direction causes a corresponding lateral contraction), and its resistivity may also change slightly due to the piezoresistive effect. Taking the total differential of R and expressing the result in terms of the fractional changes:

Using the Poisson's ratio relationship for the lateral strain (dA/A = -2nudL/L, where nu is Poisson's ratio, since area A=pid^2/4 for a circular cross-section wire, and a fractional diameter change of -nudL/L produces a fractional area change of -2nudL/L), and defining the axial strain as e=dL/L, this becomes:

The gauge factor (GF) is defined as the ratio of the fractional change in resistance to the applied strain:

For most common metallic strain-gauge materials (such as constantan or nichrome alloys), Poisson's ratio nu is typically around 0.3, giving the geometric contribution (1+2*nu) a value of approximately 1.6, while the additional piezoresistive contribution (the change-of-resistivity-with-strain term) for these alloys contributes a further amount bringing the overall practical gauge factor to approximately 2.0 to 2.2 for typical metallic foil strain gauges — semiconductor (silicon) strain gauges, by contrast, exhibit a much stronger piezoresistive effect, giving gauge factors that can be an order of magnitude higher (commonly 50-200), at the cost of a more strongly temperature-dependent and less linear resistance-strain characteristic compared to metallic foil gauges.

Practical use: because the fractional resistance change for a typical metallic strain gauge under normal working strain levels is very small (often well under 1%), strain gauges are normally connected in a Wheatstone bridge configuration (often using two or four active gauges in adjacent/opposite bridge arms for improved sensitivity and automatic temperature compensation), with the resulting small bridge output voltage imbalance amplified by a suitable instrumentation amplifier (as discussed in a related answer elsewhere in this paper) to obtain a usable measurement signal proportional to the applied strain.

Metallic Versus Semiconductor Strain Gauges

Metallic foil strain gauges (constructed from alloys such as constantan or nichrome, etched into a thin zigzag foil grid pattern) offer a modest gauge factor (typically around 2 to 2.2, as derived above), but this is accompanied by good linearity over a wide strain range, low sensitivity to temperature variation (particularly when self-temperature-compensated alloy formulations are used), and good long-term stability, making metallic gauges the standard choice for general-purpose structural strain measurement, load cells, and force/torque transducers. Semiconductor (piezoresistive) strain gauges, typically fabricated from a thin strip or diffused element of doped silicon, exploit a much stronger intrinsic piezoresistive effect (the resistivity term in the gauge factor derivation dominates strongly over the geometric term), yielding gauge factors on the order of 50 to 200 — one to two orders of magnitude greater than metallic gauges — which allows a much larger bridge output signal for a given applied strain, simplifying signal-conditioning amplifier requirements. However, semiconductor gauges exhibit a markedly non-linear resistance-versus-strain characteristic and a strong temperature dependence of both their zero-strain resistance and their gauge factor itself, generally requiring more elaborate temperature-compensation and linearization circuitry than metallic gauges, and are also typically more brittle and costlier, restricting their use to applications specifically demanding very high sensitivity (such as miniature pressure transducers and low-force load cells) where the metallic gauge's much smaller output would be impractical to measure reliably.

Wheatstone Bridge Configurations for Strain Gauges

• Quarter bridge: uses a single active strain gauge in one arm of the Wheatstone bridge, with the remaining three arms made up of fixed precision resistors — the simplest and least costly configuration, but offering the lowest sensitivity of the three arrangements, and providing no inherent temperature compensation, since any resistance change in the single active gauge due to ambient temperature variation is indistinguishable from a genuine strain-induced change.
• Half bridge: uses two active strain gauges, typically mounted so that one experiences tensile strain and the other compressive strain for the same applied load (for example, one gauge bonded to the top surface and the other to the bottom surface of a bending beam), placed in adjacent arms of the bridge — this configuration doubles the bridge output sensitivity compared to a quarter bridge for the same applied strain, and additionally provides automatic temperature compensation, since a uniform temperature change affects both active gauges identically and its effect cancels out in the bridge output, leaving only the genuine differential strain signal.
• Full bridge: uses four active strain gauges, one in each arm of the bridge, typically arranged so that two experience tensile strain and two experience compressive strain for the applied load — this configuration provides the maximum possible bridge sensitivity (four times that of a quarter bridge for the same strain) together with full temperature compensation, and is the preferred arrangement for precision load cells, torque transducers, and other high-accuracy force-measurement instruments where maximum sensitivity and stability are required, at the cost of greater complexity and cost due to the need for four matched, precisely positioned strain gauges.

(b) Differentiation of Transducer Types

(i) Active and Passive Transducer: an active transducer generates its own electrical output signal directly from the energy of the physical quantity being measured, without requiring any external power/excitation source to produce this output signal (though signal conditioning/amplification circuitry may still require external power) — examples include a thermocouple (generating an EMF directly from a temperature difference via the Seebeck effect) and a piezoelectric transducer (generating a charge/voltage directly from applied mechanical force or pressure). A passive transducer, by contrast, does not generate its own output signal energy, but instead modulates some electrical circuit parameter (resistance, capacitance, or inductance) in response to the measured quantity, requiring an external electrical excitation source to produce a usable output signal — examples include a strain gauge (modulating resistance, requiring an external bridge excitation voltage, as discussed in part (a)) and an LVDT (modulating mutual inductance/coupling, requiring an external AC excitation to its primary winding, as discussed in the preceding answer).

(ii) Primary and Secondary Transducer: a primary transducer (sometimes called a sensing element) directly senses the physical quantity of interest and converts it into an intermediate mechanical (or other non-electrical) form, without itself producing an electrical output — for example, a Bourdon tube directly senses applied pressure and converts it into a mechanical displacement of its free end, or a bimetallic strip directly senses temperature and converts it into mechanical bending displacement. A secondary transducer then takes this intermediate mechanical output from the primary transducer and converts it further into a usable electrical output signal — for example, an LVDT (as the secondary transducer) might be mechanically coupled to a Bourdon tube's (the primary transducer's) displacement output, converting the Bourdon tube's mechanical pressure-induced displacement into a proportional electrical voltage signal. Many complete practical measurement systems for physical quantities such as pressure, force, or flow are therefore built as a two-stage arrangement of a primary transducer (sensing the physical quantity and producing a mechanical intermediate output) followed by a secondary transducer (converting this intermediate mechanical output into the final electrical signal used for indication, recording, or further processing).

Further illustrative examples: a diaphragm-type pressure gauge (the primary transducer, converting applied pressure into a proportional mechanical deflection of the diaphragm) coupled to a strain gauge bonded onto that diaphragm (the secondary transducer, converting the diaphragm's mechanical deflection/strain into a proportional electrical resistance change) together form a complete pressure-to-electrical-signal measurement chain, illustrating the primary/secondary distinction in a widely used industrial pressure transmitter design. Similarly, a Bourdon tube or bellows element sensing pressure and mechanically driving the core of an LVDT (as the secondary transducer) is another common combination in process instrumentation. For the active/passive distinction, additional examples include a photovoltaic solar cell (an active transducer, generating an output voltage directly from incident light energy without needing external excitation) as against a photoresistor/LDR (a passive transducer, merely modulating its own resistance in response to incident light and requiring an external circuit voltage to produce a usable output signal), and a piezoelectric microphone or accelerometer (active, generating charge directly from applied mechanical stress) as against a capacitive microphone or capacitive-type displacement sensor (passive, modulating capacitance and requiring external bias/excitation circuitry). Recognizing whether a given transducer is active or passive, and whether it is a primary or secondary element within a larger measurement chain, is important in practical instrumentation system design because it directly determines what excitation source, signal-conditioning circuitry, and calibration procedure must be provided around the sensing element to obtain a usable, accurate measurement.