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Measurement Engineering Simulator

Wheatstone Bridge Simulator — Balance Condition & Sensitivity

Visualize the open-circuit output voltage Vout and balance condition of a bridge with four resistors R1 to R4 and excitation Vin. Experience the principle behind strain gauges and precision resistance measurement.

Parameters
Resistor R1
Ω
Resistor R2
Ω
Resistor R3
Ω
Resistor R4
Ω

Excitation voltage Vin is fixed at 5 V. Default values R1=R2=R3=1000 Ω and R4=1010 Ω model a single strain gauge with about 1% resistance change.

Results
—
Output voltage Vout
—
Sensitivity Vout/Vin
—
R4 at balance (R2·R3/R1)
—
Imbalance ΔR4
Bridge Circuit Diagram

Diamond of four resistors / Vin across the L-R nodes, voltmeter Vout across the T-B nodes / green = balanced (|Vout|<1mV), red = unbalanced

Theory & Key Formulas

The two voltage dividers in a Wheatstone bridge split the excitation Vin in two different ratios. The output Vout appears as the difference between them.

Open-circuit output voltage (high-impedance load):

$$V_\text{out} = V_\text{in}\left(\frac{R_2}{R_1+R_2} - \frac{R_4}{R_3+R_4}\right) = V_\text{in}\,\frac{R_2 R_3 - R_1 R_4}{(R_1+R_2)(R_3+R_4)}$$

Balance condition (gives Vout = 0):

$$R_1 R_4 = R_2 R_3 \quad\Longleftrightarrow\quad R_4 = \frac{R_2 R_3}{R_1}$$

Strain-gauge relation (GF: gauge factor, epsilon: strain):

$$\frac{\Delta R}{R} = G_F\,\varepsilon, \quad \frac{V_\text{out}}{V_\text{in}} \approx \frac{G_F}{4}\,\varepsilon \;(\text{single-gauge})$$

Near balance, Vout responds linearly to resistance change, allowing tiny displacement, temperature or strain to be measured with high precision. This is why the bridge has remained in use for over a century.

What is the Wheatstone Bridge Simulator

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"Wheatstone bridge" comes up all the time in textbooks, but what is it actually good for? Why not just use a regular ohmmeter?
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Roughly speaking, a bridge lets you do "comparison against zero". A normal ohmmeter has to read 1000 Ω going to 1001 Ω as a 0.1% change against a large background. But if you start from a balanced bridge, you only need to read the deviation from zero volts. That gives you about 1000x more precision for tiny changes. Try setting R1=R2=R3=R4=1000 in the simulator — Vout drops to exactly 0.
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Oh, it really does! And by default only R4 is 1010 Ω — what does that represent?
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That's a strain-gauge model. Imagine a strain gauge bonded to R4. A metallic gauge changes by dR/R ≈ 2% for 1% strain (gauge factor GF ≈ 2). Here we simplify to dR/R = 1%, i.e. R4 going from 1000 to 1010. Look at Vout — it should be about −12.4 mV. That is the output you would read from your strain sensor.
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−12.4 mV is tiny! You can't read that without an amplifier, can you?
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Right — in a real instrument an instrumentation amplifier multiplies it by 100 to 1000. Commercial sensors like load cells are spec'd as "2 mV/V": with Vin = 10 V you get up to 20 mV at full scale. The "Sensitivity" card here uses that same mV/V unit. To get more sensitivity you put gauges on all four arms — a "full bridge" combining tension and compression sides — for 4x sensitivity. That is why almost every commercial sensor uses a full bridge.
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When I press "Balance R4", Vout drops to exactly zero. Is that the famous "balance condition"?
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Exactly. The balance condition is R1·R4 = R2·R3. The "R4 at balance" card shows R2·R3/R1; setting R4 to that value zeros out Vout. In the 19th century Wheatstone himself measured an unknown resistor by adjusting a variable arm until Vout was zero — the "null method". Today we usually digitize Vout directly, but the principle is the same.

Frequently Asked Questions

A two-resistor divider does turn a resistance change into a midpoint-voltage change, but that voltage is an absolute value. If the excitation Vin wanders, the output wanders with it, and any temperature drift goes straight into the signal. A bridge measures the difference between two dividers, so excitation drift is rejected by both halves equally and temperature compensation can be built in on adjacent arms. That is the essential benefit of the four-resistor configuration.
Vout is proportional to Vin, so doubling Vin doubles the sensitivity. But the power dissipated in each resistor (P = Vin^2/(R1+R2)) scales as Vin^2, and a strain gauge whose resistance changes with temperature will then suffer self-heating error. In practice gauge dissipation is kept under about 25 mW: roughly 5 V for 120 Ω gauges and 10 V for 350 Ω gauges. Cranking Vin too high actually worsens accuracy — a classic textbook trap.
Setting the parenthesis in Vout = Vin·(R2/(R1+R2) − R4/(R3+R4)) to zero gives R2·(R3+R4) = R4·(R1+R2). Expanding to R2·R3 + R2·R4 = R4·R1 + R4·R2 and cancelling R2·R4 on both sides leaves R2·R3 = R1·R4. It is the elegant symmetric statement that "the products of opposite resistors are equal" — and that beautiful form is exactly what Wheatstone made famous in 1843.
Strictly, Vout = Vin·(R2·R3 − R1·R4)/((R1+R2)(R3+R4)) has R4 in the denominator, so for large dR/R there is a nonlinearity error. With one gauge at dR/R = 1% the error is about 0.5%, and at 5% about 2.5%. In a full-bridge (all four arms strained symmetrically), the denominator is symmetric and cancels, giving perfectly linear output. This is another reason commercial load cells use a full bridge.

Real-World Applications

Strain gauges and load cells: From strain gauges bonded to aircraft wings to truck scales, electronic kitchen scales and grip-strength meters, almost every device that converts force, load or torque into an electrical signal contains a bridge circuit. Full-bridge load cells combine temperature compensation and high linearity, and have become the standard for industrial weighing.

Platinum RTDs and temperature measurement: A Pt100 sensor (100 Ω at 0 °C) measures temperature through resistance change, and standard 3- and 4-wire bridge connections cancel the resistance error of the lead wires. This is the technology at the heart of precision thermometers with 0.01 °C accuracy, indispensable for semiconductor process control and reference instrumentation.

Pressure sensors and pressure transducers: A MEMS pressure sensor with piezoresistors diffused into a silicon diaphragm reads diaphragm strain via a full bridge. Smartphone barometers, automotive tire-pressure monitors (TPMS) and medical blood-pressure cuffs all rely on this principle for modern pressure measurement.

Gas detection and chemical sensors: A catalytic combustible-gas detector uses the fact that combustible gas burning on a catalyst heats the sensing element and raises its resistance; paired with a dummy element in a bridge, it can detect trace methane or hydrogen. Such bridges are widely deployed as safety devices in mines, chemical plants and gas distribution facilities.

Common Misconceptions and Cautions

The most common misconception is to think that "the bridge is old technology and modern ADCs make it unnecessary". There is research on connecting 24-bit delta-sigma ADCs directly to gauges, but in industrial measurement the bridge is still the dominant choice. The reason is simple: physically cancelling excitation drift, temperature drift and common-mode noise is still cheaper and more reliable with a bridge circuit. Try changing R1 to R4 by the same temperature coefficient in the same direction in the simulator — Vout barely moves. That is the bridge's "self-compensation".

The next common error is to assume that "the higher the excitation Vin, the higher the sensitivity". Vout is indeed proportional to Vin, but power dissipation in the gauge grows as Vin^2 and the gauge's own temperature rise generates a spurious dR. For example, 10 V on a 120 Ω gauge dissipates 80 mW and self-heats by several degrees, mixing temperature drift into the signal. In practice "excitation optimization" matters and the gauge datasheet's maximum excitation must be respected. The simulator's sensitivity card uses mV/V precisely so you can read off the sensor's intrinsic sensitivity independent of Vin.

Finally, it is dangerous to over-rely on the idea that "balancing the bridge solves everything". In a real instrument lead-wire resistance puts a spurious dR in series with one arm, temperature gradients leave the four arms at different temperatures so cancellation is incomplete, adhesive creep drifts the offset over time, and so on. The simulator is an idealized four-resistor model, but in real designs 3- and 4-wire connections, shielding, sense leads and calibration resistors (CAL) — the auxiliary technology around the bridge — are what really decide the measurement accuracy.

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