14 Bridge Linearity and Instrumentation

14.1 Start With the Measurement Story

This section: Start With the Measurement Story — Picture a strain gauge that changes slightly while its display stays at zero.

Make a Tiny Bridge Change Visible

Picture a strain gauge that changes slightly while its display stays at zero. An analog-to-digital converter is the circuit that turns a measured voltage into a digital number. It is also called an ADC.

Name the bridge supply, resting output, expected change, amplifier gain, converter range, and noise limit. Test zero load, a known load, reversed wiring, and a warm cable.

Keep raw and amplified voltages, digital counts, settings, time, and reference load. This proves one measurement chain and range, not every installation; the deeper sections cover bridge balance, linearity, gain, common-mode limits, and calibration.

A bridge circuit can reveal a tiny physical change, but only if the weak differential signal survives imbalance, noise, and loading. Start by following the signal from bridge output through instrumentation gain to an ADC-ready value.

14.2 Learning Objectives

This section: Learning Objectives

After this page, you should be able to:

• Explain why bridge circuits cancel large sensor baselines before amplification.
• Estimate quarter-bridge output voltage from gauge factor, strain, and excitation voltage.
• Choose instrumentation-amplifier gain from the worst-case differential signal and ADC range.
• Use common-mode rejection and dummy gauges to suppress cable noise and thermal drift.

14.3 After Signal Conditioning

This section: After Signal Conditioning — Signal Conditioning for Sensors introduces the full chain from physical sensor output to ADC-ready voltage.

Signal Conditioning for Sensors introduces the full chain from physical sensor output to ADC-ready voltage. This page narrows in on bridge sensors, where the useful signal is often a few millivolts riding on a much larger baseline.

Use it when a strain gauge, load cell, pressure transducer, torque sensor, or other bridge sensor needs a readout that will not clip, waste ADC range, or treat temperature and cable noise as real motion.

The mathematical gist. Gauge factor splits into geometry and material response: $GF=(1+2\nu)+(d\rho/\rho)/\varepsilon$. With the chapter’s $GF=2.0$ and $\nu=0.30$, geometry contributes 1.60 (80.0%) and changing resistivity contributes 0.40 (20.0%). At 1,000 microstrain, $\Delta R/R=0.00200$ and the chapter’s 5.0 V quarter bridge produces 2.50 mV.

Math Bridge · guided foundationsWhere does a metal-foil gauge factor of 2.0 come from?Let Phoebe separate length, thinning, and piezoresistivity before rebuilding the bridge output. Learn the maths

14.4 Tiny Signals on Large Baselines

This section: Tiny Signals on Large Baselines — Some of the most useful sensors move only a whisker.

Some of the most useful sensors move only a whisker. A metal-foil strain gauge changes its resistance by roughly one part in a thousand at full mechanical load. If you drop that gauge into a plain voltage divider, the tiny wanted change rides on top of a large, steady baseline voltage — like trying to see a ripple on the surface of a full bathtub. The ADC spends almost all its range on the baseline and almost none on the signal.

Before acting on tiny signals on large baselines, inspect Figure 14.2 because it condenses the evidence path: the long serpentine foil track turns a minute stretch of the bonded surface into a measurable resistance change; a bridge circuit is needed because that change is tiny beside the gauge’s baseline resistance. Photo: Cristian V., CC BY 4.0.

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Read Figure 14.2 from the soldered leads toward the central serpentine foil. The long folded track is the active element: bonding it to the test surface makes strain lengthen or compress that path, producing a very small resistance change. That physical scale explains why the next stage is a bridge rather than a direct ADC connection—the circuit must reject the large resting resistance and preserve its small change.

Two classic building blocks solve this together. The Wheatstone bridge subtracts a matched reference so that, at rest, the output is zero and only the change appears as a small differential voltage. The instrumentation amplifier then amplifies that small differential voltage by a large, precise factor while rejecting interference that is common to both wires.

For a concrete feel, compare two ways to read a 350 Ω strain gauge excited from 5 V. In a single divider with another 350 Ω resistor, the midpoint sits near 2.500 V. A 0.1% resistance change moves that midpoint by only about 1.25 mV, so a 12-bit, 3.3 V ADC sees roughly 1.6 counts of movement on top of a 3100-count baseline. The bridge produces the same millivolt-scale change as a signed differential output around zero, which an instrumentation amplifier can safely multiply before the ADC. With gain 200, that 1.25 mV becomes 250 mV, or about 310 ADC counts. The measurement did not become more physically sensitive; the circuit simply stopped wasting ADC range on the baseline.

The bridge removes much of the baseline, but its divider law is still curved. Figure 14.1 shows where matching the reference arm makes a local linear approximation useful and where that approximation stops being safe.

Shows: Figure 14.1: Half-bridge circuit with a resistive sensor and a transfer-characteristic curve showing the most linear measurement region near matched bridge resistance.

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Start with the Half Bridge Circuit in Figure 14.1. Fixed R and changing Rsense set Vout through Vout = Vin × Rs/(Rs + R). Next read the transfer curve near the matched point. It is close to a straight line only over a small range. Farther away, the curve flattens. Equal changes in resistance then make unequal changes in voltage. This is the circuit view of the series below. The first term gives useful gain near R0. Later terms describe error that the design must bound or remove by calibration.

Vout = Vin * Rs / (Rs + R)
Rs = R0 + DeltaR

V(Rs) ~= V(R0)
        + DeltaR * [dV/dRs at R0]
+ (DeltaR^2 / 2) * [d2V/dRs2 at R0] + ...

14.5 Quarter-Bridge Amplifier Sizing

This section: Quarter-Bridge Amplifier Sizing — Read Figure 14.3 from the metal package to its two rows of pins.

With one active gauge (a quarter-bridge) and the other three arms fixed at the nominal resistance, the bridge output for a small change is approximately:

Before acting on quarter-bridge amplifier sizing, inspect Figure 14.3 because it condenses the evidence path: an instrumentation amplifier integrates the matched high-gain differential stages needed to lift a bridge’s millivolt output while rejecting voltage common to both inputs; the AD520J shown here is an early single-chip example of that component class. Photo: Mister rf, CC BY-SA 4.0.

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Read Figure 14.3 from the metal package to its two rows of pins. The photograph establishes the component class, but it does not supply the gain, input range, supply limits, or common-mode limits needed for a design; those still come from the selected part’s datasheet. That distinction connects the physical amplifier to the calculation below: choose gain from the bridge’s worst-case differential signal and the ADC headroom, not from package appearance.

Vout / Vex ≈ (1/4) × (ΔR / R)

and for a strain gauge:   ΔR / R = GF × ε
  GF = gauge factor (≈ 2.0 for metal foil)
ε  = strain (dimensionless, often quoted in microstrain)

Given: strain ε = 1000 µε = 0.001
       gauge factor GF = 2.0
       excitation Vex = 5.0 V

Fractional change:  ΔR/R = 2.0 × 0.001 = 0.002 (0.2%)
Bridge output:      Vout = 5.0 × 0.002 / 4 = 2.5 mV

2.5 mV is far too small for a 3.3 V ADC to resolve well.
Amplify with an instrumentation amp at gain 100:
Vout' = 2.5 mV × 100 = 250 mV   (now comfortably readable)

Keep the amplifier range honest before choosing the gain resistor. If the same load cell can see 1500 microstrain in an overload test, the raw output is 5.0 V × (2.0 × 0.0015) / 4 = 3.75 mV. Gain 100 produces 375 mV, leaving plenty of headroom. Gain 1000 produces 3.75 V, which clips a 3.3 V ADC and hides the overload exactly when you need evidence. A practical design therefore picks a gain from the worst-case signal, not just the nominal signal, then uses calibration coefficients to convert ADC counts back to force or pressure.

14.6 CMRR and Thermal Cancellation

This section: CMRR and Thermal Cancellation — The bridge-plus-instrumentation-amp pairing is not just about gain.

The bridge-plus-instrumentation-amp pairing is not just about gain. It is about rejecting everything you did not mean to measure.

A quick noise budget shows why common-mode rejection matters. Suppose the real bridge signal is 2 mV, but a nearby motor couples 100 mV of 50 Hz noise equally onto both sensor wires. With a plain single-ended amplifier at gain 100, the noise would try to become 10 V and would slam the output into a rail. With an instrumentation amplifier whose common-mode rejection ratio is 100 dB, the common-mode component is reduced by a factor of 100,000 before the differential gain matters: 100 mV becomes about 1 µV equivalent input. After gain 100, that is only about 0.1 mV at the output, while the wanted 2 mV differential signal becomes 200 mV.

Thermal effects need the same differential thinking. A foil gauge might change resistance from both strain and temperature. If the active gauge rises by 0.20% from strain and both the active and dummy gauges rise by 0.05% from temperature, the bridge subtracts the shared thermal term and leaves the strain term as the dominant differential change. If the dummy gauge is mounted far from the active gauge, that assumption fails; the bridge then reports temperature gradients as fake mechanical load.

This is why load cells, pressure transducers, and torque sensors almost universally use a bridge feeding an instrumentation amplifier: the topology cancels the baseline, cancels temperature drift through matched arms, rejects common-mode interference, and keeps high source impedance from being disturbed, leaving a clean amplified copy of the mechanical signal.

14.7 Release Checklist

This section: Release Checklist — Start by the bridge excitation voltage is within the sensor manufacturer’s self-heating and accuracy limits.

Before relying on a bridge-sensor readout, confirm these points:

Start by the bridge excitation voltage is within the sensor manufacturer’s self-heating and accuracy limits. Then the expected maximum strain, load, or pressure is converted into a worst-case millivolt bridge output. Next instrumentation-amplifier gain leaves ADC headroom for overloads, offset, startup transients, and noise. After that the ADC reference and bridge excitation are intentionally ratiometric or independently stabilized. Continue by cable shielding, twisted pairs, and amplifier layout preserve the common-mode rejection assumed in the calculation. Finally dummy or matched bridge arms see the same temperature as the active sensing element when thermal cancellation is expected.

14.8 See Also

This section: See Also — Start by Signal Conditioning for Sensors for the full amplification, filtering, and ADC chain.

Start by Signal Conditioning for Sensors for the full amplification, filtering, and ADC chain. Then Resistive Dividers and ADC Loading for single-ended resistive sensors before bridge circuits are needed. Next Sensor Data Processing for filtering and calibration after the bridge signal reaches firmware. Finally Accuracy, Precision, Drift, and Hysteresis for reporting remaining uncertainty after conditioning.

14.9 Next

This section: Next — Return to Signal Conditioning for Sensors once the bridge gain and noise budget are clear, then continue to Sensor Data Processing for digital filtering and calibration.

Return to Signal Conditioning for Sensors once the bridge gain and noise budget are clear, then continue to Sensor Data Processing for digital filtering and calibration.