Strain gauges · Measurement

Strain Gauge Bridge Equation Calculator

Calculate the bridge output signal from the strains in all four arms and the gauge factor – linearised and with the exact equation 4.6.

DMSKeil 2017

Inputs

Gauge factor kStrain ε₁ [µm/m]Strain ε₂ [µm/m]Strain ε₃ [µm/m]Strain ε₄ [µm/m]Excitation U_B [V]

Equal nominal resistances and gauge factors in all arms, constant-voltage excitation, unloaded output.

Results

Enter values and run the calculation.

Method

What is calculated?

With equal nominal resistances in all arms, eq. 4.1 yields the basic equation 4.6. Its numerator shows the sign rule: strains in arms 1 and 3 count positive, in 2 and 4 negative. The denominator carries the sum of all resistance changes – if non-zero the bridge becomes non-linear, which matters most with a single active arm.

Equations

U_M/U_B = (ΔR₁−ΔR₂+ΔR₃−ΔR₄) / (2(2R₀+ΔR₁+ΔR₂+ΔR₃+ΔR₄)) (Keil Gl. 4.6)

U_M/U_B ≈ k/4·(ε₁ − ε₂ + ε₃ − ε₄) (linearisiert, Keil Gl. 4.10 ff.)

ΔRᵢ/R₀ = k·εᵢ

Limits

Assumptions and typical mistake

Equal nominal resistances and gauge factors in all arms, constant-voltage excitation, unloaded output.

Entering compensation gauges with their temperature apparent strain although it is equal in all arms and cancels – or forgetting the sign of compression.

Gauge layout

Where the gauges sit and what the bridge makes of them

Where the grids sit and how they are wired, plus which load components remain in the signal and which the circuit compensates.

What does the bridge factor B mean?

The general bridge equation is U_M/U_B = k/4·(ε₁ − ε₂ + ε₃ − ε₄): strains in opposite arms count positive, in adjacent arms negative. Expressing every active strain as a multiple of one reference strain ε, B is exactly that signed sum of multiples – e.g. B = 2 for two adding contributions, B = 4 for four. U_M/U_B = k·B·ε/4 summarises this.

To assess it: B is a pure gain of the circuit, not a statement about temperature compensation or which load type is in the signal – that is covered by “Use” and the response chips below. A larger B gives a larger, more easily measured signal U_M for the same mechanical strain; this calculator inserts B into the calculation automatically once you pick a circuit, so you never enter it yourself.

Wheatstone bridge: arms 1–4 U_M/U_B = k/4·(ε₁ − ε₂ + ε₃ − ε₄)

Layout: U_M/U_B = k/4·(ε₁ − ε₂ + ε₃ − ε₄): strains in opposite arms (1 and 3, 2 and 4) add, strains in adjacent arms subtract.

ε₁ + ε₃ in the signal
ε₂ + ε₄ compensated

Use: Basic rule behind every circuit I–IX: disturbances that strain two adjacent gauges alike cancel.

Signal-adding or compensating – depending on whether the signal-producing resistors sit opposite or adjacent.

Valid for small resistance changes; the bridge equation becomes non-linear at large strains.

Keil Abschn. 4.5, Gl. 4.9

Bridge wiring diagrams after Keil, Dehnungsmessstreifen, 2nd ed. 2017 (secs. 4.5, 9.7, 10.2, 10.5) · bridge arms numbered R1–R4 with U_M/U_B = k/4·(ε₁ − ε₂ + ε₃ − ε₄).

Inputs

What you enter – and where the values come from

Gauge factor k: The gauge factor is the sensitivity of the strain gauge: it states how much the electrical resistance changes when the gauge is stretched (ΔR/R = k·ε). Without it no strain can be computed from an electrical signal. Source: printed on every gauge package or in the manufacturer datasheet, typically 2.0 to 2.1 for constantan foil gauges; valid at room temperature. The same value must be set in the amplifier.
Strain ε₁ [µm/m]: Strain under the gauge in bridge arm 1. Enter 0 for a fixed resistor or an unloaded compensation gauge. Strains in arms 1 and 3 enter the signal positively. Source: expected strain from calculation or measurement; mind the sign (compression negative).
Strain ε₂ [µm/m]: Strain in arm 2, adjacent to arm 1; strains here are subtracted from the signal. That is why a compensation gauge (temperature only) belongs here, and for bending the gauge on the opposite side so the magnitudes add.
Strain ε₃ [µm/m]: Strain under the gauge in arm 3, diagonally opposite arm 1. Strains here add to those of arm 1 – which is why in full bridges the two equally loaded gauges sit in arms 1 and 3. Enter 0 for a fixed resistor or compensation gauge.
Strain ε₄ [µm/m]: Strain under the gauge in arm 4, adjacent to arms 1 and 3; like arm 2 it is subtracted from the signal. In a bending full bridge the second compression-side gauge sits here so the magnitudes add. Enter 0 for a fixed resistor or compensation gauge.
Excitation U_B [V]: The bridge excitation is the DC or AC voltage with which the amplifier supplies the bridge circuit. The signal is proportional to it – which is why signals are usually stated as a ratio in mV/V and the excitation is only needed for the absolute signal voltage. Source: amplifier setting, usually 2.5 V, 5 V or 10 V; the upper limit is set by gauge self-heating (see the excitation calculator).

Context

What this calculator is for

Designing transducer circuits, checking temperature compensation (equal strain in all arms gives zero) and assessing quarter-bridge non-linearity on plastics or in the elasto-plastic range.

What comes next

Assign each arm its strain (fixed resistors and compensation gauges: 0).
Enter gauge factor and excitation.
Compare linearised and exact signals; for large strains evaluate with the exact value.

Results

How to read the results

Sign rule: arms 1 and 3 add, arms 2 and 4 subtract. A compensation gauge belongs in arm 2 or 4 next to the active gauge.
If the non-linearity stays below 0.1 %, the linear value (amplifier reading) can be used directly; otherwise work with the exact value or the correction in the signal calculator.

Result quantities

What each value means

Linearised signal ratio [mV/V]: Bridge signal per the linearised basic equation (what amplifiers compute internally): strains in arms 1 and 3 count positive, in 2 and 4 negative.
Exact signal ratio [mV/V]: Physically exact bridge signal that also accounts for the changed total resistances of both voltage dividers. Use this instead of the linear approximation for large strains or accuracy estimates.
Output voltage U_M (exact) [mV]: Actual bridge output voltage in millivolts at the entered excitation. Compare it with the amplifier's smallest input range and resolution.
Non-linearity [%]: Deviation of the linear approximation from the exact value in percent. Zero for symmetric full bridges (±ε), roughly equal to the strain itself with only one active arm.

Source

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 4.2, eqs. (4.2)–(4.8); sec. 4.5, eqs. (4.10)–(4.18).

The source supports the equation structure and worked examples; this calculator does not replace calibration of the measuring chain.

FAQ

Frequently asked questions

How large is the quarter-bridge linearity error?

Per Keil about equal to the strain: around 0.1 % at 1000 µm/m.

Why is the full bridge with ±ε linear?

Because the resistance changes in the denominator sum to zero.

Can I account for unequal gauge factors?

Approximately, by scaling an arm's strain with kᵢ/k.

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