The Wheatstone Bridge: Turning a Null Reading Into a Precise Resistance

The circuit and the one equation that matters

A Wheatstone bridge is four resistors wired as a diamond. A source voltage V_s drives the two opposite corners (A and C); a detector — classically a galvanometer — bridges the other two corners (B and D). Label the arms so that going around one side you pass R₁ then Rₓ (the unknown), and around the other side R₂ then R₃.

Each side of the diamond is just a voltage divider. If the two dividers split V_s in the same ratio, corners B and D sit at exactly the same potential, so no current flows through the detector. That is the balance condition:

• Left divider voltage at B: V_B = V_s · Rₓ/(R₁ + Rₓ)
• Right divider voltage at D: V_D = V_s · R₃/(R₂ + R₃)
• Balance requires V_B = V_D, which reduces to R₁/Rₓ = R₂/R₃, i.e. Rₓ = R₃ · (R₁/R₂)

Read that last line carefully: the unknown Rₓ is fixed entirely by three known resistances and a ratio. V_s dropped out. So do the detector's calibration, the wire resistance to the source, and any drift in the supply voltage — none of them shift the null. You do not measure how much; you measure when it equals.

Why a null beats a reading

The genius of the bridge is that it trades an absolute measurement for a comparison, and comparisons dodge nearly every systematic error. A deflection instrument answers "how many volts?" — a question whose answer depends on the meter's gain, its zero offset, its temperature coefficient, and the exact supply voltage. A null instrument answers only "is it zero yet?", and zero is zero on any scale.

Concretely: suppose your galvanometer is a mediocre device with ±5% gain accuracy but excellent sensitivity — it can resolve 1 nA. In deflection mode your resistance is good to maybe 5%. In null mode you adjust R₃ until the needle stops moving; the gain error multiplies zero and vanishes. The precision is now set by how finely you can trim R₃ and how small a current the galvanometer can still detect — routinely giving 0.01–0.1% from ordinary parts.

This is the same philosophy behind a chemical balance (compare against known masses) and an interferometer (compare against a reference path). Whenever nature gives you a clean zero to hunt for, chase the zero.

Deriving balance from Kirchhoff's laws

The voltage-divider shortcut only strictly holds when the detector draws no current — which is exactly the balanced state — so it is worth confirming with the full Kirchhoff analysis. Call the detector current I_g through a galvanometer of resistance R_g, and let I₁ flow through R₁ and I₂ through R₂ from node A.

• At balance we demand I_g = 0, so all of I₁ continues through Rₓ and all of I₂ through R₃.
• Kirchhoff's voltage law around loop A→B→D: the drop across R₁ equals the drop across R₂, giving I₁R₁ = I₂R₂.
• Around loop B→C→D: the drop across Rₓ equals the drop across R₃, giving I₁Rₓ = I₂R₃.
• Divide the second equation by the first: Rₓ/R₁ = R₃/R₂ ⟹ Rₓ = R₃·(R₁/R₂). Identical to the divider result, with no assumption about V_s or R_g.

Notice R_g never appears in the balance condition. The detector's resistance affects only the sensitivity near balance — how big a current a given imbalance produces — not the balance point itself. That decoupling is why a bridge can be trimmed with a crude meter and still land on a precise ratio.

The unbalanced bridge: sensing with ΔR

Most modern bridges never sit at balance. Instead they start balanced and let a sensor's resistance change by a tiny ΔR, then read the resulting output voltage. This is the deflection mode, and it turns the bridge into an exquisitely linear transducer.

Take the common quarter bridge: three fixed resistors all equal to R, and one active element R + ΔR. The two divider outputs no longer match, and the open-circuit output is:

• V_out = V_s · [ (R+ΔR)/(2R+ΔR) − R/(2R) ]
• For ΔR ≪ R this linearizes to V_out ≈ (V_s/4)·(ΔR/R)

The factor of ¼ is the price of using only one active arm. Put it in numbers: a metal-foil strain gauge has gauge factor GF ≈ 2, so a strain ε = 1000 µε (0.1%, a big but non-destructive strain in steel) gives ΔR/R = GF·ε = 0.002. With V_s = 10 V, V_out ≈ (10/4)·0.002 = 5 mV — small, but perfectly resolvable with a modern instrumentation amplifier and 24-bit ADC. Using a full bridge (all four arms active, two increasing and two decreasing) removes the ¼ and gives V_out ≈ V_s·(ΔR/R), four times the signal, while automatically cancelling temperature drift because all four gauges shift together.

Real hardware: gauges, load cells, and thermometers

The bridge is the beating heart of the sensor industry:

• Load cells and scales: A bathroom scale or a truck weighbridge deforms a metal element by micrometers; foil strain gauges on it form a full bridge whose millivolt output is proportional to force. The best load cells are linear to 0.02% of full scale over the whole range precisely because the bridge geometry cancels nonlinearities.
• Pressure sensors: Silicon piezoresistive gauges diffused onto a flexing diaphragm form a bridge on a single chip; these are the MEMS pressure sensors in cars, phones, and altimeters.
• Platinum resistance thermometers (RTDs): A Pt100 element is 100.00 Ω at 0 °C and rises about 0.385 Ω per °C. Read in a bridge (often a 3- or 4-wire arrangement to cancel lead resistance), it delivers ±0.01 °C accuracy — the backbone of the ITS-90 temperature scale.
• Precision resistance metrology: The classic Kelvin double bridge is a bridge variant built specifically to measure very low resistances (below ~1 Ω, down to microohms) where the resistance of the connecting leads and contacts would otherwise swamp the reading.

Sensitivity, lead resistance, and the AC cousins

How small an imbalance can you see? Near balance, the detector current for a small fractional imbalance δ = ΔR/R in one arm of an equal-arm bridge is roughly I_g ≈ V_s·δ / [4(R + R_g)] (exact form depends on the four arm values). This says three sensible things: raise V_s for more signal (limited by self-heating of the arms — a strain gauge dissipates I²R and heats up, corrupting the reading), keep the arm resistances comparable to R_g for best power transfer to the detector, and match the source impedance to the bridge.

Two subtleties bite in practice:

• Lead resistance: Wires from the bridge to a remote sensor add series resistance in a bridge arm and mimic a real ΔR. The fix is the 3-wire and 4-wire (Kelvin) connections, which route the leads so their resistance appears symmetrically and cancels, or is measured separately.
• AC and reactance: Replace resistors with impedances and drive with AC, and the same topology measures capacitance and inductance. The Maxwell bridge finds an unknown inductance by balancing it against a known capacitance; the Wien bridge balances at one frequency and became the frequency-selective network in the classic Wien-bridge oscillator. Balance now requires both magnitude and phase to null — you tune two knobs to zero the same detector.

Common misconceptions

"The bridge measures the voltage across the unknown." No — at balance there is no current through the detector and the interesting quantity is the ratio of resistances, not any single voltage. The detector reads zero on purpose.

"You need a precise, high-quality meter." Exactly backwards for null mode. You need a sensitive detector (small currents visible) but its calibration is irrelevant, because you only ask it whether the current is zero. The precision lives in the known resistors R₂, R₃ and your ability to trim.

"Supply voltage accuracy matters." Not for the balance point — V_s cancels out of Rₓ = R₃·(R₁/R₂). It matters only in deflection mode, where V_out ∝ V_s, which is why load cells specify output per volt of excitation (e.g. 2 mV/V) and why precision systems ratio-measure V_out against V_s to cancel supply drift.

"It's obsolete." The galvanometer-and-dial box is a museum piece, but the four-arm topology is everywhere — every digital scale, every automotive pressure sensor, every lab-grade RTD channel is a Wheatstone bridge feeding a modern amplifier.

Two ways to run the same four-resistor bridge: null (balanced) vs. deflection (unbalanced).

Property	Null / balanced mode	Deflection / unbalanced mode
What you read	A resistor dial at zero current	The bridge output voltage V_out
Detector role	Null indicator (I → 0)	Calibrated voltmeter / ADC
Meter accuracy needed	Only sensitivity, not calibration	Full absolute accuracy
Governing relation	Rₓ = R₃·(R₁/R₂)	V_out ≈ (V_s/4)(ΔR/R)
Best for	Precise static resistance (0.01%)	Fast, continuous sensing (strain, ~kHz)
Typical use	Lab resistance standards, bridge box	Strain gauges, load cells, RTDs