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Provide any three of \( R_1 \)–\( R_4 \) to solve for the fourth using the balance condition. Provide all four plus \( V_s \) (\(A\) to \(C\)) to calculate the output voltage \( V_{out} \) of an unbalanced bridge.
A Wheatstone bridge consists of four resistors arranged in a diamond configuration with a supply voltage \( V_s \) across one diagonal and a measurement point across the other. It is used to measure an unknown resistance precisely, or to detect small resistance changes in sensor applications such as strain gauges and RTDs.
The bridge is balanced when no current flows through the galvanometer — i.e. when both branches have the same voltage ratio: \[ \frac{R_1}{R_2} = \frac{R_3}{R_4} \] Which is equivalently written as: \[ R_1 \cdot R_4 = R_2 \cdot R_3 \] At balance, the output voltage \( V_{out} = 0 \). Solving for any one resistor given the other three: \[ \begin{align} R_1 &= \frac{R_2 \cdot R_3}{R_4} \\ R_2 &= \frac{R_1 \cdot R_4}{R_3} \\ R_3 &= \frac{R_1 \cdot R_4}{R_2} \\ R_4 &= \frac{R_2 \cdot R_3}{R_1} \end{align} \]
When the bridge is not balanced, a voltage appears across the output. For a high-impedance load (no current drawn from the output): \[ V_{out} = V_s \cdot \left(\frac{R_3}{R_1 + R_3} - \frac{R_4}{R_2 + R_4}\right) \] The sign of \( V_{out} \) indicates which branch has the higher voltage. This formula assumes no loading. If a low-impedance load is connected, the output voltage will be lower due to the bridge's output impedance.
In practice one or more of the resistors is a sensor — a strain gauge, thermistor, or RTD — whose resistance changes with a physical quantity. The bridge is first balanced at a reference condition. Any deviation from balance then produces a \( V_{out} \) proportional to the resistance change, which can be amplified and measured. This is the principle behind load cells, pressure sensors, and precision temperature measurement.
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