The Wheatstone bridge calculator above solves a four-arm resistance bridge in both directions. In balance mode it returns the resistance in the fourth arm that nulls the output, which is the classical way an unknown resistor is measured against three known ones. In output mode it returns the voltage that appears across the detector when the bridge is deliberately off balance, which is how every resistive sensor built on this topology actually works: strain gauges, load cells, platinum resistance thermometers, pressure transducers and torque sensors all present themselves to the world as a small differential voltage from a bridge.
Arb Digital publishes two adjacent circuit pages, and it is worth being clear about where each stops. The voltage divider calculator computes the output of a two-resistor divider and models what a load does to it; it deals with one divider, not two, and does not address bridges. The resistor combination calculator reduces resistors in series, in parallel, or in two parallel series branches, and reports equivalent resistance and per-resistor dissipation — but a bridge is precisely the case that cannot be reduced that way, because the detector arm cross-connects the two branches. This page is the one that handles that cross-connection.
What This Wheatstone Bridge Calculator Does
Picture two voltage dividers hung across the same supply. The left one is R1 above R2; the right one is R3 above Rx. Each divider produces a mid-point voltage, and the bridge output is the difference between them. When the two dividers happen to divide in the same ratio, the mid-points sit at the same potential, the difference is zero, and the bridge is balanced. When Rx moves, the right mid-point moves with it and a voltage appears.
The tool reports that output voltage, the value of Rx that would return the bridge to null, the sensitivity expressed as millivolts of output per volt of excitation, the total current the bridge draws from the supply, and the Thevenin resistance looking back into the output terminals. It also converts the output ratio into an implied quarter-bridge strain when you give it a gauge factor, and it lets you add lead resistance into the Rx arm so you can see what a long cable run costs you.
How to Use It
- Choose the mode. Balance mode is for a null measurement of an unknown resistor. Output mode is for a sensor bridge sitting slightly off null.
- Enter the excitation voltage. This scales the output directly and has no effect on the balance point at all, which is the first thing worth internalising about the topology.
- Enter the three known arms. R1 and R2 form one divider, R3 sits above the arm that changes. In a laboratory bridge R1/R2 is the fixed ratio arm and R3 is the adjustable standard.
- Enter Rx. In output mode this is the sensor's current resistance. In balance mode it is computed for you and the input is hidden.
- Add lead resistance if the sensor is remote. A two-wire run of any length puts real resistance in series with Rx, and the note tells you what it did to the apparent reading.
The Formula: Balance and Off-Balance
Take the negative supply rail as zero. The left mid-point sits at Vex R2 / (R1 + R2) and the right mid-point at Vex Rx / (R3 + Rx), so the output voltage is the difference:
Vout = Vex [ R2 / (R1 + R2) − Rx / (R3 + Rx) ]
Setting that to zero gives the balance condition R1 / R2 = R3 / Rx, and therefore Rx = R2 R3 / R1. Notice what has vanished: the excitation voltage. The balance point depends only on ratios of resistances, which is why a null measurement is so accurate. A supply that drifts by five per cent does not move the null by anything at all, it only changes how loud the imbalance sounds on the way to it.
Work the default through by hand. With Vex = 5 V and R1 = R2 = R3 = 1000 Ω, an Rx of 1010 Ω gives a left mid-point at exactly 2.5 V and a right mid-point at 5 × 1010/2010 = 2.512438 V. The difference is −12.4378 mV, and the balance value is 1000 Ω. The Thevenin resistance at the output is the two source impedances in series, R1 ∥ R2 plus R3 ∥ Rx, which is 500 + 502.49 = 1002.49 Ω. Total supply current is 5/2000 + 5/2010 = 4.9876 mA. The underlying loop and junction analysis is the standard treatment in OpenStax's University Physics chapter on Kirchhoff's rules.
Excitation Voltage: More Signal, More Self-Heating
Output voltage is directly proportional to excitation, so the temptation with a small signal is always to turn the supply up. The limit is thermal. Every arm dissipates V²/R, and in a strain gauge that heat has nowhere to go except into the specimen the gauge is bonded to. A warmed specimen expands, the gauge reads that expansion as strain, and you have manufactured a signal that has nothing to do with the load.
The severity depends on what the gauge is stuck to. A large aluminium part is a heat sink and tolerates a lot; a thin plastic coupon or a small composite sample does not, and the same excitation that is comfortable on metal will produce a slow drifting output on plastic that looks exactly like creep. This is why gauge manufacturers publish excitation guidance in terms of power density on the gauge grid rather than a single voltage, and why a bridge that reads steadily on the bench can drift on a real part.
The second reason to think about excitation is stability rather than magnitude. Because output is proportional to it, a one per cent supply error is a one per cent reading error. Precision bridges are therefore either run from a very stable reference or read ratiometrically, with the same reference feeding both the bridge and the converter that digitises its output, so the supply term cancels in the division. Working out the dissipation in any one arm is what the resistor power rating calculator is for, and Ohm's law calculator covers the arithmetic underneath it.
Lead Resistance Is the Classic Field Error
A gauge on a test article is rarely next to its instrument. Run two wires out to it and the resistance of those wires sits in series with the gauge, inside the bridge arm, indistinguishable from the gauge itself. Thirty metres of thin copper can easily add a couple of ohms, and on a 350 Ω gauge that is a substantial offset before any load has been applied.
The offset itself is survivable because it can be nulled out. The real damage is twofold. First, the added resistance desensitises the arm: the fractional change the gauge produces is now diluted across gauge plus leads, so every subsequent reading is low by roughly the ratio of gauge resistance to gauge-plus-lead resistance. Second, and worse, copper's resistance changes by about 0.4 per cent per degree Celsius, so the leads themselves become a temperature sensor wired in series with your strain sensor. A cable run through changing ambient conditions produces a wandering baseline that no amount of care at the gauge will fix.
The standard answer is a three-wire connection, which splits the lead resistance between two opposite arms of the bridge so that the temperature drift largely cancels, or a four-wire Kelvin arrangement where the sense wires carry no current. Both are worth the extra conductor. To see the size of the problem for a particular run, the wire resistance calculator gives the resistance from gauge and length, and the voltage drop calculator covers the supply side of the same cable.
The Output Is Not Quite Linear
A quarter bridge — one active gauge, three fixed arms — has an output ratio of Vout/Vex = (GF ε / 4) ÷ (1 + GF ε/2). The denominator is the part people drop, and dropping it is fine for small strains and wrong for large ones. At 1000 microstrain with a gauge factor of 2, the nonlinearity error is about 0.1 per cent. At 10,000 microstrain it is about one per cent, which is no longer negligible in a calibrated measurement.
Half and full bridges reduce or eliminate this. A full bridge with four active gauges arranged so that opposite arms move in opposite senses is exactly linear in strain and produces four times the output of a quarter bridge, which is why commercial load cells are built that way and why their data sheets quote sensitivity in millivolts per volt. The relationship between the strain a gauge reports and the stress it implies is a separate step, handled by the stress strain calculator.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Expecting the excitation voltage to affect the balance point — it does not. Balance depends only on the ratio of resistances, which is why a null measurement is so much more accurate than a deflection one.
- Reading the output with a low-impedance meter — the bridge has a Thevenin resistance of the same order as its arms, so a loading instrument pulls the reading down. The tool reports that resistance for exactly this reason.
- Running two-wire leads to a remote gauge — the cable resistance sits inside the arm, desensitises it, and drifts with ambient temperature at about 0.4 per cent per degree.
- Turning the excitation up to chase signal — the extra heat goes into the specimen, and a warmed specimen expands, producing apparent strain that no filtering can remove.
- Using the linear quarter-bridge approximation at large strain — the exact expression has a denominator, and ignoring it costs about one per cent at 10,000 microstrain.
Related Free Tools From Arb Digital
For a single divider rather than two, use the voltage divider calculator, and for series and parallel networks the resistor combination calculator. The Ohm's law calculator and the resistor power rating calculator cover currents and dissipation in each arm, the resistor color code calculator decodes the bands on the parts you are fitting, and the resistor noise calculator gives the Johnson noise floor that limits how small a bridge output can usefully be resolved. For the cable, use the wire resistance calculator. Everything Arb Digital publishes is indexed on the free online tools hub. The resistivity and temperature-coefficient data behind the lead-wire discussion is tabulated in OpenStax's chapter on resistivity and resistance, and the wider circuit theory is developed in MIT OpenCourseWare's 6.002 Circuits and Electronics.
Frequently Asked Questions
The bridge is balanced when the two dividers divide in the same ratio, which means R1 divided by R2 equals R3 divided by Rx. Rearranged, the unknown arm is Rx equals R2 times R3 divided by R1. The excitation voltage does not appear anywhere in that condition, so the balance point is completely independent of the supply, and only the ratios of the four resistances matter.
That page handles one two-resistor divider and models what connecting a load does to its output. A bridge is two dividers across the same supply with the measurement taken between their mid-points, and the cross-connection is exactly what a single divider cannot describe. Use the divider page for a potential divider or a sensor pull-up, and this page whenever there are four arms and a differential output.
Because balance is the condition that two ratios are equal, and multiplying both dividers by the same supply voltage leaves those ratios untouched. Raising the excitation makes an unbalanced bridge produce a proportionally larger output, so it makes the imbalance easier to see, but it moves the null by nothing. That independence is the reason a null measurement can be far more accurate than a deflection measurement.
In a two-wire connection it sits in series with the gauge inside the arm, so it produces an initial offset, it desensitises the arm because the gauge's fractional change is now diluted across gauge plus leads, and it drifts with temperature at roughly 0.4 per cent per degree Celsius for copper. A three-wire connection splits the leads between opposite arms so the drift largely cancels, and a four-wire arrangement removes it from the measurement entirely.
Enough to lift the signal above the instrument's noise, and not enough to heat the specimen. Every arm dissipates power, and in a bonded gauge that heat goes into the part, which expands and reads as strain. Manufacturers publish guidance as power density on the gauge grid rather than a single voltage, because a gauge on a large metal part tolerates far more than the same gauge on a thin plastic coupon.
Not exactly. A quarter bridge has an output ratio of the gauge factor times strain over four, divided by one plus half the gauge factor times strain. The denominator is small at small strains, so the linear approximation costs about 0.1 per cent at 1000 microstrain, but about one per cent at 10,000 microstrain. A full bridge with four active arms arranged in opposition is exactly linear and gives four times the output.
Because it tells you what the detector or amplifier is connected to. Looking back into the output terminals, the bridge appears as the two arm pairs in parallel and then in series, which for four equal arms is the arm value itself. An instrument with an input impedance comparable to that will load the bridge and read low, which is why bridge outputs are read with high-impedance instrumentation amplifiers rather than ordinary meters.
Yes, the topology is the same. A platinum element is simply the Rx arm and its resistance rises with temperature along a published characteristic. This tool computes what the bridge does electrically, in output voltage, sensitivity and source resistance. Turning that resistance into a temperature needs the element's own standardised resistance-temperature relation, which is not part of this page.
This page models an ideal four-arm resistive bridge and omits amplifier input current, thermocouple effects at dissimilar-metal junctions and gauge self-heating. Instrumentation for a measurement that carries consequences should be specified and calibrated by a qualified engineer.
