Strain gauge bridges and load cells

Quarter, half and full Wheatstone bridges for strain gauges, temperature and bending compensation, and strain-gauge load cells with rated output in mV/V.

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Why it matters

A strain gauge changes resistance by about 0.1 %, far too little to read directly. The Wheatstone bridge turns that change into a millivolt signal, cancels temperature effects and, with the right gauge placement, rejects unwanted loads. Load cells — in weighbridges, process tanks, testing machines and kitchen scales — are just an elastic element plus a carefully arranged bridge.

Key ideas

The bridge. Four arms R1–R4 are excited by Vex across one diagonal and the output Vo is taken across the other. With all arms equal the bridge is balanced and Vo = 0. A small resistance change in any arm unbalances it. For small changes, the output is proportional to (ΔR1/R − ΔR2/R + ΔR3/R − ΔR4/R): adjacent arms subtract, opposite arms add. This single rule explains every configuration.

  • Quarter bridge (one active gauge): Vo ≈ Vex·GF·ε/4. Slightly non-linear because the active arm's change also changes the current in its half; the non-linearity is about GF·ε/2 of the reading (0.1 % at 1000 µε for GF = 2). A dummy gauge — identical, on the same material, unstrained — in the adjacent arm cancels temperature-induced resistance change.
  • Half bridge (two active gauges in adjacent arms, one in tension and one in compression, e.g. top and bottom of a cantilever): Vo = Vex·GF·ε/2, exactly linear, temperature compensated. If the second gauge is mounted transversely on an axially loaded member it sees −ν·ε (a "Poisson gauge"): Vo ≈ Vex·GF·ε(1 + ν)/4.
  • Full bridge (four active gauges): with two in +ε and two in −ε on opposite arm pairs, Vo = Vex·GF·ε, exactly linear, four times the quarter-bridge output, fully temperature compensated. On a column load cell the "negative" gauges are transverse (−ν·ε), giving Vo ≈ Vex·GF·ε(1 + ν)/2.
  • Rejecting unwanted loads: on a column, gauges on opposite faces see equal and opposite bending strains; putting them in opposite arms (which add) cancels bending and keeps the axial signal.
  • Lead-wire resistance adds to the gauge arm and changes with temperature; the three-wire connection puts one lead in each of two adjacent arms so they cancel.
  • Excitation and readout: 2–10 V DC (or AC carrier) excitation, an instrumentation amplifier for the millivolt output, and often remote sensing of the excitation. Output is usually quoted ratiometrically in mV/V.

Load cells convert force into strain in an elastic element whose shape sets capacity and sensitivity: column/canister (high capacity, compression), S-beam (tension/compression), bending and shear beams, ring and diaphragm types. The specification gives the rated output in mV/V at rated capacity (typically 1–3 mV/V), so Vo = (rated output) × Vex × (F/F_rated). Other specifications: combined error (non-linearity + hysteresis), creep, zero and span temperature coefficients, safe overload. Hydraulic and pneumatic load cells exist too, but the strain-gauge type dominates.

Formulas

Vo = Vex·[R1/(R1 + R2) − R4/(R3 + R4)] (general bridge; R1, R2 in one half, R4, R3 in the other) Vex = excitation (V); Vo = output (V); R in Ω.

Vo ≈ (Vex/4)·(ΔR1/R − ΔR2/R + ΔR3/R − ΔR4/R) (small changes, equal arms)

Vo = Vex·x / (4 + 2x), x = GF·ε (quarter bridge, exact)

Vo = Vex·GF·ε / 2 (half bridge, ±ε, adjacent arms)

Vo = Vex·GF·ε (full bridge, ±ε)

Vo = Vex·x(1 + ν) / [2 + x(1 − ν)] (column load cell, two axial + two transverse gauges, exact) ν = Poisson's ratio of the column.

Vo = RO·Vex·F/F_rated RO = rated output (V/V); F = applied force (N); F_rated = capacity (N).

Worked examples

Example 1 — quarter-bridge non-linearity. A 120 Ω gauge, GF = 2.0, ε = 1000 µε, in a quarter bridge with Vex = 10 V. Find the exact output and the error of the linear approximation.

  1. x = GF·ε = 2.0 × 1000 × 10⁻⁶ = 0.002.
  2. Approximate: Vo ≈ Vex·x/4 = 10 × 0.002/4 = 5.000 mV.
  3. Exact: Vo = Vex·x/(4 + 2x) = 0.02/4.004 = 4.995 mV.
  4. Error of approximation = (5.000 − 4.995)/4.995 = 0.10 %.

Answer: Vo = 4.995 mV; the linear formula overestimates by 0.1 %.

Example 2 — column load cell (GATE level). A steel column (E = 200 GPa, ν = 0.3) of cross-section 20 mm × 20 mm carries an axial load of 40 kN. Two axial gauges are in opposite arms and two transverse gauges in the other opposite arms; GF = 2.0, Vex = 10 V. Find the output.

  1. A = 0.02 × 0.02 = 400 × 10⁻⁶ m²; σ = F/A = 40 × 10³ / 400 × 10⁻⁶ = 100 MPa.
  2. ε = σ/E = 100 × 10⁶ / 200 × 10⁹ = 500 µε; x = GF·ε = 0.001.
  3. Approximate: Vo ≈ Vex·x(1 + ν)/2 = 10 × 0.001 × 1.3/2 = 6.50 mV.
  4. Exact: Vo = 10 × 0.001 × 1.3/(2 + 0.001 × 0.7) = 6.498 mV.
  5. If 40 kN is the rated capacity, the rated output is 6.50/10 = 0.65 mV/V.

Answer: Vo ≈ 6.50 mV (sign depends on which diagonal you call positive; magnitude is what matters).

Example 3 — using a datasheet. A 1000 N load cell with rated output 2 mV/V is excited at 10 V and carries 500 N. Vo = 2 × 10⁻³ × 10 × 500/1000 = 10 mV.

Common mistakes

  • Dropping the gauge factor: ΔR = GF·ε·R, not ε·R.
  • Treating rated output (mV/V) as mV per newton; it applies at full rated capacity.
  • Putting two tension gauges in adjacent arms — their effects cancel and the output vanishes.
  • Expecting a quarter bridge to be temperature compensated without a dummy gauge.
  • Forgetting the (1 + ν) factor when transverse gauges are used instead of true compression gauges.
  • Ignoring lead resistance in long cable runs with two-wire connection.

For GATE IN

Typical NATs: bridge output for quarter/half/full configurations, the exact (non-linear) quarter-bridge output, load-cell output from force, area and modulus, and finding which arms to place gauges in to cancel bending or temperature. MCQs test the adjacent-subtract/opposite-add rule. Practise writing the general bridge expression and simplifying it for each case.

Quick check

  1. Half bridge, GF = 2, ε = 500 µε (±), Vex = 5 V. Find Vo.
  2. Which arms must two gauges with equal strain occupy so that their outputs add?
  3. What is the role of a dummy gauge?
  4. A 2 mV/V, 50 kg load cell at 10 V excitation carries 20 kg. Output? Answers: 1. 5 × 2 × 500 × 10⁻⁶/2 = 2.5 mV; 2. opposite arms; 3. it sits in an adjacent arm and cancels temperature-induced resistance change; 4. 8 mV.

Interview questions

All Sensors and Transducers interview questions

Try answering each one aloud before you open it.

  1. 1.Explain the working principle of a Wheatstone bridge in the context of strain measurement.Concept

    A Wheatstone bridge is an electrical circuit used to measure unknown electrical resistances by balancing two legs of a bridge circuit. In strain measurement, it is used to measure the small changes in resistance of strain gauges. The bridge consists of four resistors, with one or more being strain gauges. When the strain gauge experiences strain, its resistance changes, causing an imbalance in the bridge. This imbalance results in a voltage difference across the bridge, which can be measured and related to the strain.

  2. 2.What is a load cell and how is it related to strain gauges?Concept

    A load cell is a transducer that converts force into an electrical signal. It typically consists of one or more strain gauges attached to a structural member. When a force is applied to the load cell, it causes deformation in the structural member, which in turn changes the resistance of the strain gauges. This change in resistance is measured using a Wheatstone bridge circuit, and the resulting electrical signal is proportional to the applied force.

  3. 3.Why do load cells use a bridge of several strain gauges instead of a single gauge?Application

    A single gauge's ΔR/R is about 10⁻³ and is swamped by temperature-induced resistance changes. Arranging several gauges in a Wheatstone bridge raises the output (up to four times a quarter bridge), makes it linear, cancels temperature effects because equal changes in adjacent arms subtract, and, with gauges on opposite faces in opposite arms, cancels bending from off-axis loads so only the wanted axial force is measured.

  4. 4.What happens if one of the strain gauges in a load-cell bridge fails?Application

    An open-circuit gauge drives the output to a large offset near half the excitation, which is easy to detect as an over-range reading. A partially failed or debonded gauge is worse: it gives a zero shift, reduced or non-linear span and loss of temperature compensation, so readings are wrong but plausible. Checking the bridge arm resistances and zero balance, and monitoring the insulation resistance to ground, are standard diagnostics, followed by recalibration.

  5. 5.How does temperature affect the performance of strain gauges and load cells?Application

    Temperature can affect the performance of strain gauges and load cells by causing changes in the resistance of the strain gauges and the material properties of the load cell. These changes can lead to measurement errors if not properly compensated. To mitigate these effects, temperature compensation techniques are used, such as using temperature-compensated strain gauges or designing the Wheatstone bridge to cancel out temperature-induced changes. Proper calibration at different temperatures can also help maintain accuracy.

  6. 6.Calculate the strain experienced by a strain gauge with a gauge factor of 2.0 and a change in resistance of 0.1 Ω, if the original resistance is 120 Ω.Numerical

    The strain (ε) can be calculated using the formula: ε = ΔR / (GF × R₀), where ΔR is the change in resistance, GF is the gauge factor, and R₀ is the original resistance. Substituting the given values: ε = 0.1 Ω / (2.0 × 120 Ω) = 0.1 / 240 = 0.0004167 or 416.7 microstrain.

  7. 7.A load cell of 1000 N capacity has a rated output of 2 mV/V and is excited at 10 V. What is the output at a load of 500 N?Numerical

    Rated output in mV/V applies at full rated capacity, so full-scale output = 2 mV/V × 10 V = 20 mV at 1000 N. The output is proportional to load, so at 500 N it is 20 mV × 500/1000 = 10 mV. A common error is to multiply 2 mV/V by the load in newtons, which ignores the capacity.

  8. 8.Explain how a four-arm (full) Wheatstone bridge enhances the sensitivity of a load cell.Application

    In a full bridge all four arms are active gauges, two strained positively and two negatively, placed so that every change adds at the output: opposite arms add and adjacent arms subtract. The output is Vex·GF·ε, four times a quarter bridge, and exactly linear. Because all four gauges are on the same element at the same temperature, temperature-induced resistance changes appear equally in adjacent arms and cancel. On a column load cell the negative gauges are transverse Poisson gauges, giving Vex·GF·ε(1+ν)/2.

  9. 9.What are the advantages of using a full-bridge strain gauge configuration over a half-bridge configuration?Application

    A full-bridge strain gauge configuration offers several advantages over a half-bridge configuration. It provides higher sensitivity and output signal, as all four arms of the bridge are active and contribute to the measurement. This configuration also offers better temperature compensation and reduces the effects of bending and other non-axial forces. Additionally, it can provide more accurate and stable measurements, making it suitable for precise applications.

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