Wheatstone Bridge Calculator
Compute unknown resistance R_x = R3·(R2/R1) at bridge balance, plus galvanometer current and percent imbalance for off-balance measurement.
The Wheatstone bridge has been the gold standard for precision resistance measurement since 1833, when Samuel Hunter Christie first published the circuit. Charles Wheatstone made it famous a decade later, and his name stuck. Despite 190 years of progress in electronics, the same four-resistor diamond is still the front end of strain gauges, RTD temperature sensors, load cells, and laboratory resistance standards. The reason: at balance, the measurement is independent of the supply voltage, independent of galvanometer calibration, and depends only on the ratio of two known resistors.
The circuit:
Four resistors arranged in a diamond. A voltage source drives one diagonal (A to C). A sensitive current detector (historically a galvanometer, today usually a high-impedance differential amplifier) bridges the other diagonal (B to D). R1 and R2 sit on the top arms; R3 and Rx sit on the bottom arms. The galvanometer hangs between the midpoint of (R1, R3) and the midpoint of (R2, Rx).
The balance condition:
The bridge is balanced when no current flows through the galvanometer. Algebraically:
V_B − V_D = 0
Which simplifies to the famous ratio:
R1 / R3 = R2 / Rx, so Rx = R3 · (R2 / R1)
To measure an unknown Rx, set R1 and R2 to known fixed values, then adjust R3 (typically a precision decade box) until the galvanometer reads zero. The balance ratio gives Rx directly.
Why null-balance is so accurate:
At balance, the galvanometer carries no current and the formula has no supply voltage in it. A noisy battery doesn’t matter. A non-linear galvanometer doesn’t matter. Only the ratio R2/R1 matters, and precision resistors at that level of accuracy are easy to make. The technique can routinely deliver 5-6 significant figures of resistance accuracy with ordinary lab equipment, and 8-9 with care.
Off-balance galvanometer current:
When the bridge is not balanced, the galvanometer current depends on the supply voltage V, all four resistances, and the galvanometer’s own resistance R_g. For small imbalances, the current is approximately:
I_g ≈ (V · ΔR) / [(R1 + R2)·(R3 + Rx) + R_g·(R1 + R2 + R3 + Rx)]
where ΔR is the deviation from balance. For strain gauges, this off-balance current (or voltage, in op-amp implementations) is the signal proportional to the strain.
Worked example, simple measurement:
A bridge has R1 = 100 Ω, R2 = 400 Ω, and R3 (calibrated potentiometer) is adjusted until the galvanometer reads zero, settling at R3 = 50 Ω.
Rx = R3 · (R2 / R1) = 50 · (400 / 100) = 200 Ω
The 1:4 ratio of R1:R2 amplifies the precision of the R3 reading by 4× — a 0.1 Ω increment in R3 translates to a 0.4 Ω resolution in Rx.
Strain gauge implementation:
A strain gauge is a thin foil resistor whose resistance changes with mechanical strain. Bonded to a structural member, it converts force or displacement into a tiny resistance change (typically 1 part in 10⁴ to 10⁶). On its own, this signal is too small to read easily. In a Wheatstone bridge, the imbalance produces a voltage that op-amps can amplify and digitize.
| Strain gauge configuration | Output |
|---|---|
| Quarter bridge (1 active gauge) | 1× sensitivity |
| Half bridge (2 active, opposite arms) | 2× sensitivity, cancels temperature drift |
| Full bridge (all 4 active) | 4× sensitivity, full temperature compensation |
Industrial load cells use full bridges for both sensitivity and the temperature stability that comes from having all four resistors at the same temperature.
Variations:
- Kelvin bridge (also called Thomson bridge): two extra resistors for very-low-resistance measurement, ohm to milliohm range. Removes contact-resistance error.
- Wien bridge: capacitor and resistor arms for AC measurements; used as a sine-wave oscillator (the first HP product, 1939).
- Maxwell bridge and Hay bridge: for measuring inductance and Q-factor of coils at audio frequencies.
- Anderson bridge: another inductance variant, less common today.
Sources of error:
- Lead resistance: at low Rx, the resistance of the wires connecting Rx becomes significant. Four-wire (Kelvin) connections solve this.
- Thermal EMFs: dissimilar metal junctions generate small voltages (the same effect as a thermocouple). At very high precision, these can dominate.
- Self-heating: I²R dissipation in Rx changes its temperature and thus its resistance. Low-current measurement protocols mitigate this.
- Galvanometer sensitivity: a poorly-sensitive null detector misses small imbalances; modern op-amps and lock-in amplifiers solve this.
Modern replacements that don’t replace it:
You might think digital multimeters and 4-wire ohmmeters would have killed the Wheatstone bridge. They have not, because:
- Strain gauge front-ends still use bridges.
- Precision thermometry (Pt-100 RTDs, thermistors) still uses bridges.
- Calibration laboratories use bridges to transfer the resistance standard.
- Bridge methods remain the easiest way to compare two nearly-equal resistors precisely.
The technique is older than electric power distribution, and it has outlived every “this will replace it” innovation since.