Wheatstone Bridge
Part of Basic Electrical Circuits
The precision measurement circuit that allows accurate resistance measurement with only a galvanometer and a set of known resistors.
Why This Matters
The Wheatstone bridge is one of the most elegant and useful circuits in electrical measurement. It allows the precise measurement of an unknown resistance using only a sensitive current detector (galvanometer) and a set of known resistors. It is immune to variations in battery voltage, requires no calibrated voltmeter or ammeter, and can measure resistance to a precision of 0.1% or better with carefully made components.
For a rebuilding community, the Wheatstone bridge provides the ability to calibrate the resistance boxes and resistors that are used in every other electrical instrument. It can measure the resistance of motor windings (allowing temperature monitoring), the resistance of wire (verifying conductor quality), the resistance of any unknown material, and the calibration of galvanometers being converted to voltmeters and ammeters.
Samuel Hunter Christie invented the circuit in 1833; Charles Wheatstone popularized it in 1843. For nearly a century before electronic instruments, it was the standard method for precision resistance measurement in laboratories worldwide.
The Circuit
The Wheatstone bridge consists of four resistors arranged in a diamond (or square) pattern:
A
/ \
R1 R2
/ \
B----G----D
\ /
R3 Rx
\ /
C
- B is the positive battery terminal; D is the negative
- A and C are the junction points
- G is the galvanometer, connected across the diagonal A-C
- R1, R2, R3 are known resistors; Rx is the unknown
When the bridge is balanced (no current through G), the following relationship holds:
R1/R2 = R3/Rx
Therefore: Rx = R3 × (R2/R1)
The galvanometer detects zero current—the null condition—without needing to be calibrated for absolute current values. It needs only to be sensitive enough to detect small imbalances.
Operating Principle
At balance, the potential at node A equals the potential at node C. No current flows through the galvanometer.
The potential at A relative to B: V_A = V_supply × R1/(R1+R2) The potential at C relative to B: V_C = V_supply × R3/(R3+Rx)
For balance: V_A = V_C R1/(R1+R2) = R3/(R3+Rx)
Cross-multiplying: R1(R3+Rx) = R3(R1+R2) R1×R3 + R1×Rx = R3×R1 + R3×R2 R1×Rx = R3×R2 Rx = R3 × R2/R1
The supply voltage cancels completely. The measurement is independent of battery voltage—a major advantage over direct measurement methods.
Building a Practical Wheatstone Bridge
Components needed:
- A galvanometer (see Galvanometer)
- R1 and R2: a “ratio arm” pair—two resistors with a known, precise ratio. Common choices: both equal (ratio 1:1), R1=10×R2 (ratio 10:1), or R1=100×R2 (ratio 100:1). Multiple ratio settings expand the measurement range.
- R3: a “rheostat arm”—a continuously variable precision resistor whose value can be set accurately and read off a dial
- Battery (any convenient voltage; 3–12V)
- Connecting wires
- Mounting board
Ratio arm construction: Wind two identical wire-wound resistors simultaneously by drawing two strands of the same resistance wire through the same coil form. This ensures R1 = R2 with high precision—far better than winding them separately.
For the 10:1 ratio: wind 10 turns of one value per 1 turn of another. For practical simplicity, use switchable decades: a set of resistors (10×100Ω, 1×100Ω) selectable by connection.
Variable arm construction: A sliding wire rheostat with a graduated scale. Stretch approximately 1 meter of manganin or constantan wire (preferred for low temperature coefficient) or nichrome wire on a wooden rule. Mark the scale in millimeters. Measure the total resistance and calculate resistance per millimeter. The position of the slider times this constant gives R3 at any setting.
Higher precision: a decade resistance box with individual decade switches (1Ω, 2Ω, 3Ω… 10Ω on one switch; 10Ω, 20Ω… 100Ω on the next; etc.). This allows setting any resistance to the nearest 1Ω.
Operating procedure:
- Connect the unknown resistor Rx in the appropriate bridge arm
- Set ratio arms R1:R2 appropriate for the expected magnitude of Rx
- Adjust R3 until the galvanometer shows zero deflection
- Read R3 from the scale or switch settings
- Calculate: Rx = R3 × (R2/R1)
If the galvanometer deflects strongly, the bridge is far from balanced—adjust R3 rapidly toward balance, then approach more slowly for precise nulling. Start with the galvanometer sensitivity reduced (a series resistor in the galvanometer circuit) to protect it from large imbalance currents.
Measurement Ranges
By changing the R1:R2 ratio, the same variable arm R3 covers different resistance ranges:
| R1:R2 ratio | Rx measurement range (if R3 covers 0–10,000Ω) |
|---|---|
| 1:1 | 0–10,000Ω |
| 10:1 | 0–1,000Ω |
| 100:1 | 0–100Ω |
| 1:10 | 0–100,000Ω |
| 1:100 | 0–1,000,000Ω |
This flexibility means one instrument can measure anything from milliohm contact resistances (with a suitable null detector) to megaohm insulation resistances.
Practical Applications
Calibrating resistors: Compare an unknown resistor against a standard resistor (one measured against a known-good reference). After calibration, the measured resistors become secondary standards.
Measuring wire conductivity: Sample a wire of known length and cross-section. Measure its resistance with the bridge. Calculate resistivity: ρ = R × A / L. Compare against copper reference to determine purity or identify the metal.
Motor winding temperature: Measure winding resistance when cold (note the exact temperature). Measure again when the motor has been running. Calculate temperature from the resistance increase: T₂ = T₁ + (R₂/R₁ - 1) / α Where α is the temperature coefficient (0.00393 for copper).
Insulation testing: Reverse the ratio: use very high known resistors in the bridge arms to measure the insulation resistance of cables and windings. Insulation resistance should be in the megaohm range; lower values indicate moisture ingress or degraded insulation.
Strain gauges: A metal wire changes resistance when stretched or compressed. Bonding fine resistance wire to a structural member creates a strain gauge—a sensor that measures mechanical stress. The Wheatstone bridge’s sensitivity to small resistance changes makes it ideal for reading these sensors.
Accuracy Limitations
- Contact resistance at the bridge terminals adds error. Use firm connections and clean contact surfaces.
- Lead resistance in the connecting wires from bridge to unknown resistor appears as part of Rx. For very low resistance measurements (below 1Ω), use the Kelvin double-bridge variant which eliminates lead resistance.
- Temperature effects on the known resistors cause drift. Use constantan or manganin wire for the precision elements; operate at stable temperature.
- Galvanometer sensitivity limits the smallest detectable imbalance. A more sensitive galvanometer allows finer balance adjustment and greater measurement precision.
With careful construction and operation, a homemade Wheatstone bridge achieves precision of 0.1–0.5%—far better than any direct measurement with a commercial analog meter, and adequate for all practical calibration and characterization work.