Wheatstone Bridge Formulas: Balance and Output

Wheatstone bridge formulas: the balance condition, unknown resistance, output voltage of an unbalanced bridge and a quarter-bridge strain gauge signal.

The formulas

The left arm is R1 on top and R2 below it. The right arm is R3 on top and Rx below it, and the output is the difference between the two midpoints.

Vs is the excitation voltage. At balance the output is zero whatever the supply voltage, which is why the bridge measures resistance accurately with a null meter. The quarter-bridge formula applies when one arm changes by a small amount and the other three equal R. GF is the gauge factor, about 2 for a metal-foil gauge.

Worked example: balance

R1 = 1 kΩ, R2 = 2 kΩ and R3 = 330 Ω.

1. Rx = 2000 × 330 / 1000 = 660 Ω gives zero output

Worked example: an unbalanced bridge

The same bridge on 10 V with Rx = 700 Ω.

1. Left midpoint: 10 × 2000 / 3000 = 6.667 V
2. Right midpoint: 10 × 700 / (330 + 700) = 6.796 V
3. Vout = 6.796 − 6.667 = 0.1294 V

Worked example: a strain gauge

A 350 Ω gauge with GF = 2 at 1,000 microstrain on a 5 V bridge.

1. ΔR = 350 × 2 × 0.001 = 0.7 Ω
2. Vout ≈ 5 / 4 × 0.7 / 350 = 2.5 mV (exactly 2.498 mV)

Balancing resistance with R1 = 1 kΩ and R3 = 330 Ω

Frequently asked questions

Why is the bridge used for strain gauges?

It turns a tiny resistance change into a voltage, and the three fixed arms cancel changes in the supply voltage and, with a dummy gauge, in temperature.

Does the supply voltage affect the balance?

No. At balance the two dividers give the same ratio, so the output is zero for any supply voltage.

How small a change can I detect?

It depends on the amplifier. The signal is a few millivolts for a gauge, so a bridge amplifier with low noise is used.

What about the resistance of the lead wires?

It adds to the gauge arm. A three-wire connection cancels most of it for a remote sensor such as an RTD.

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