Wheatstone Bridge and Measurement Circuits
A Wheatstone bridge converts a small resistance change into a measurable differential voltage. It is the core circuit behind load cells, strain gauges, pressure sensors, RTD measurement, and many precision resistance instruments.
Learning Objectives
By the end of this lesson, you should be able to derive the bridge balance condition, estimate bridge output for small sensor changes, choose quarter-, half-, and full-bridge arrangements, and identify the signal-conditioning problems that make bridge measurements difficult in real products.
Bridge Structure
A bridge is two voltage dividers powered by the same excitation voltage. The output is the difference between their midpoints.
$$
V_A = V_{EX}\frac{R_3}{R_1 + R_3}
$$
$$
V_B = V_{EX}\frac{R_4}{R_2 + R_4}
$$
$$
V_{OUT}=V_A-V_B
$$
$$
V_{OUT}=V_{EX}\left(\frac{R_3}{R_1+R_3}-\frac{R_4}{R_2+R_4}\right)
$$
The sign of VOUT depends on which midpoint is treated as positive. Instrumentation amplifier inputs should match the desired sign convention.
Balance Condition
The bridge is balanced when VOUT = 0:
$$
\frac{R_1}{R_3}=\frac{R_2}{R_4}
$$
or equivalently:
$$
R_1R_4=R_2R_3
$$
For a classical unknown-resistance measurement:
$$
R_3=R_1\frac{R_4}{R_2}
$$
At balance, the null detector reads zero, so the result depends mainly on resistor ratios rather than on the exact excitation voltage.
Worked Example: Unknown Resistance
A bridge uses R1 = 1 kOhm, R2 = 1 kOhm, and an adjustable R4. The bridge balances when R4 = 100.2 Ohm.
$$
R_3=1000\Omega\times\frac{100.2\Omega}{1000\Omega}=100.2\Omega
$$
The unknown resistance is 100.2 Ohm. The measurement accuracy now depends on ratio resistors, contact resistance, thermal EMF, and null detector sensitivity.
Sensor Bridges
Modern bridge circuits usually measure resistance change rather than absolute resistance.
Quarter Bridge
One active resistor changes. It is simple, but it has the lowest sensitivity and is most exposed to temperature drift.
For a small single-arm change:
$$
V_{OUT}\approx \frac{V_{EX}}{4}\frac{\Delta R}{R}
$$
Half Bridge
Two active elements change in opposite directions or one active element is paired with a dummy element for temperature compensation. Sensitivity is roughly twice a quarter bridge when both active gauges contribute.
Full Bridge
All four arms are active. This gives the highest sensitivity, best temperature compensation, and is common in load cells.
Worked Example: Strain Gauge Output
A 350 Ohm strain gauge has gauge factor GF = 2.1. It sees 1000 microstrain with VEX = 10 V.
$$
\frac{\Delta R}{R}=GF\epsilon
$$
$$
\frac{\Delta R}{R}=2.1\times0.001=0.0021
$$
$$
\Delta R=350\Omega\times0.0021=0.735\Omega
$$
For a quarter bridge:
$$
V_{OUT}\approx\frac{10V}{4}\times0.0021=5.25mV
$$
This is a very small signal. It needs a low-noise instrumentation amplifier, stable excitation, filtering, and careful wiring.
Load Cell Measurement Chain
A typical load cell is a full bridge rated in mV/V. A 2 mV/V load cell excited by 10 V produces 20 mV at full scale.
For a 0 V to 5 V ADC span:
$$
G=\frac{5V}{20mV}=250
$$
The ADC reference, bridge excitation, amplifier offset, and calibration weights all affect final accuracy.
RTD Bridges and Lead Resistance
RTDs such as Pt100 and Pt1000 change resistance with temperature. Around 0 degC, platinum RTDs are often approximated by:
$$
R_T \approx R_0(1+\alpha T)
$$
$$
\alpha\approx0.00385/degC
$$
A Pt100 at 100 degC is approximately:
$$
R_T=100\Omega(1+0.385)
$$
$$
R_T=138.5\Omega
$$
Long leads add resistance and create error. Use 3-wire compensation where lead resistance is matched, or 4-wire Kelvin sensing for precision work.
Practical Design Checks
- Use precision, low-TCR resistors in inactive bridge arms.
- Keep bridge excitation stable or measure it ratiometrically with the ADC reference.
- Choose an instrumentation amplifier with enough CMRR, input common-mode range, offset performance, and gain accuracy.
- Filter before the ADC to limit noise and aliasing.
- Shield long sensor cables and provide ESD protection at connectors.
- Calibrate zero and span in the final mechanical assembly.
Common Mistakes
- Treating a millivolt bridge output like a normal logic signal.
- Forgetting that self-heating changes sensor resistance.
- Using a single-ended ADC input for a differential bridge without checking common-mode range.
- Ignoring cable resistance, connector thermocouples, and mechanical mounting stress.
- Calibrating the electronics but not the full sensor structure.
Summary
A Wheatstone bridge compares two voltage dividers. At balance, resistor ratios define the result; away from balance, small resistance changes become small differential voltages. Useful bridge systems need stable excitation, low-noise differential amplification, filtering, wiring discipline, and calibration.
Further Reading
- Analog Devices, "Bridge Circuits and Instrumentation Amplifiers" application notes.
- Texas Instruments, "RTD Ratiometric Measurements and Bridge Sensors" design guides.
- Omega Engineering, "Strain Gauge and Load Cell Measurement" references.