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The Wheatstone Bridge

Last guide turned the world into a resistance — but a strain gauge's signal is a whisker of change riding on a huge fixed value. The Wheatstone bridge is the elegant trick that subtracts away that baseline before you ever amplify, so the tiny wiggle is all that's left.

The haystack problem: a whisker on a mountain

The previous guide handed you a family of resistive sensors — the strain gauge, the RTD, the thermistor — each a transducer that turns some real-world quantity into a change in resistance. The trouble is the change is tiny next to the resistance it sits on. A typical foil strain gauge rests at 350 ohm; squeeze it to 1000 microstrain (a strain of 0.001, near the limit for many metals) and with a gauge factor of 2 its resistance changes by only dR/R = 2 times 0.001 = 0.002. That is 0.7 ohm of change on top of 350 ohm — a whisker on a mountain.

Drop that gauge into a plain voltage divider against a matching 350 ohm resistor across 5 V, and the midpoint sits at 2.5 V. The strain nudges it by only about (5/4) times 0.002 = 2.5 mV — a 0.1% wiggle perched on a 2.5 V cliff. To read it you would need a converter that resolves millivolts on top of volts, and any drift in that 2.5 V baseline would swamp the signal entirely. The honest answer is not to amplify harder, because amplifying the wiggle amplifies the cliff with it. You must first cancel the baseline.

Four arms, two dividers, one difference

The Wheatstone bridge is nothing more than two voltage dividers fed from the same excitation supply, with the output taken as the difference between their two midpoints. Call the four resistors R1, R2 (top arms) and R3, R4 (bottom arms). The left divider's midpoint is node A at Va = Vex times R3/(R1+R3); the right divider's midpoint is node B at Vb = Vex times R4/(R2+R4). The bridge output is the difference Vout = Va - Vb. One divider holds a steady reference; the other carries the sensor.

            +Vex
              |
       +------+------+
       |             |
      [R1]          [R2]
       |             |
   A o-+             +-o B      Vout = Va - Vb   (a small differential)
       |             |
      [R3]          [R4]
       |             |
       +------+------+
              |
             GND

   Va = Vex x R3/(R1+R3)        Vb = Vex x R4/(R2+R4)
   BALANCED  (Vout = 0)  when   R1/R3 = R2/R4
   put the sensor in one arm: at rest it balances; a change dR tips it.
Two dividers side by side. The bridge reads the difference between their midpoints, A and B — so the big voltage both share simply cancels.

Here is the magic. When the two dividers have the same ratio — R1/R3 = R2/R4, equivalently R1 times R4 = R2 times R3 — the two midpoints sit at exactly the same voltage and Vout = 0, no matter what the excitation is. The bridge is balanced. Now place your sensor in one arm and pick the other three so it balances at rest: the 2.5 V cliff from the haystack problem is gone, subtracted by the matching arm, and the output reads a clean zero. When the sensor's resistance shifts by dR, the balance tips and a small difference appears. Originally people dialed a variable resistor to null it back to zero and read the dial; today we just amplify the imbalance directly.

How big is the wiggle? Quarter, half, full

With one active arm — a quarter bridge — the output works out to Vout = (Vex/4) times (dR/R). Plug in our gauge: dR/R = 0.002, Vex = 5 V, so Vout = (5/4) times 0.002 = 2.5 mV. Yes, that is the same tiny number as the divider gave — but now it is a difference centered on zero, not a wiggle stranded on a 2.5 V cliff, which is exactly what makes it amplifiable. Be honest about one wrinkle: the active arm also changes its own divider's denominator, so the exact output is (Vex/4) times (dR/R) divided by (1 + dR/2R). For dR/R = 0.002 that bend is about 0.1% — negligible here, but it grows with strain, so a quarter bridge is slightly nonlinear.

  1. Quarter bridge — one active gauge, three fixed resistors. Sensitivity (Vex/4) times (dR/R). Simple and cheap, but slightly nonlinear, and the gauge's resistance also drifts with temperature, which masquerades as fake strain.
  2. Half bridge — two gauges in opposite arms, one stretched (+dR) and one compressed (-dR) by the same load. The output doubles to (Vex/2) times (dR/R), the nonlinearity cancels exactly, AND a common temperature change moves both gauges the same way, so it cancels too.
  3. Full bridge — four active gauges, two going +dR and two -dR. Output reaches Vex times (dR/R), four times the quarter, perfectly linear, with the best temperature rejection. This is exactly how a real load cell is built: four gauges sealed onto one flexure, shipped as a finished full bridge.

Why does the half and full bridge shrug off temperature? Because the gauges all sit on the same metal at the same temperature, a heat change nudges every arm by the same fraction — a common shift that moves both nodes together, and the difference Va - Vb politely ignores it. That single insight is why a load cell is a sealed four-gauge bridge rather than a lone sensor: the configuration cancels the things you don't want (drift, temperature, nonlinearity) while doubling or quadrupling the thing you do.

Two gifts and a debt: differential, common-mode, ratiometric

The bridge hands the next stage a differential signal: the information lives in Va - Vb, not in either node alone. But notice what else rides on those two wires — both nodes sit near Vex/2, roughly 2.5 V, a large common-mode voltage that carries no information whatsoever, plus any mains hum that couples equally onto both leads. The job of throwing away that shared 2.5 V (and the hum) while faithfully amplifying the millivolt difference is precisely what a instrumentation amplifier with a high common-mode rejection ratio does — the whole reason the next guide exists.

Now the gift. Look again at every formula: the output scales with Vex. Double the excitation and you double the signal — which sounds like a curse, because any drift or ripple in the excitation looks exactly like a change in the measurement. The elegant cure is ratiometric measurement: derive the reference of your analog-to-digital converter from the same Vex that powers the bridge. Then the converter reports Vout/Vex, a pure ratio, and any wander in the excitation cancels top and bottom. You no longer need a laboratory-grade excitation — only a stable ratio, which is far easier and cheaper to guarantee.

Practical realities: where the real error lives

Matching sets the zero. The bridge only nulls if the fixed arms truly equal the sensor's rest value, and real resistors carry tolerance. A mere 0.1% mismatch in one arm shifts that node by about Vex times 0.00025 = 1.25 mV at 5 V — as big as our entire 2.5 mV signal! So a bridge almost always needs either a physical zero trim or a measure-and-subtract offset step during calibration. The bridge gives you a difference; it does not, by itself, give you accuracy.

Two more gotchas haunt resistive bridges. First, lead resistance: long wires out to a remote RTD add their own ohms in series with the sensor arm, faking a temperature reading — cured by 3-wire or 4-wire (Kelvin) connections that place the lead resistance where it cancels or is sensed separately. Second, self-heating: the excitation current flows through the sensor itself, and a thermistor or RTD warmed by its own current reports its warmth, not the world's. Excitation is therefore a compromise — enough current for a usable signal, little enough to keep self-heating below your error budget.

Step back for the honest framing of this whole rung. The bridge is a beautiful subtractor, but it is not magic: it does not amplify, it does not straighten a nonlinear sensor (that is linearity correction, later), and it can never beat the matching of its own arms. The real accuracy of the entire signal chain is decided right here — in the sensor, its bridge, and its conditioning — long before the bits of any converter. A glittering 16-bit ADC reading a bridge whose arms match to only 0.5% is still a 0.5% measurement dressed in very fine-looking numbers. For a sensor far from its electronics you would often hand the conditioned signal off as a 4-20 mA current loop to survive the long, noisy wire — the closing topic of this rung.