5.7 Wheatstone bridge

Wheatstone bridge basics

A Wheatstone bridge is a circuit that measures an unknown resistance by comparing it to known resistances. It's one of the most precise tools for resistance measurement because, at its core, it relies on detecting zero current rather than measuring an absolute value. That zero-detection approach eliminates many sources of error that plague other methods.

Definition and purpose

The Wheatstone bridge measures unknown electrical resistance with high precision by comparing it to a known reference resistance. It can detect very small changes in resistance, which makes it ideal for sensitive measurements in lab settings.

The circuit was invented by Samuel Hunter Christie in 1833 and later popularized by Charles Wheatstone in 1843 (so the name is a bit misleading).

Circuit configuration

The bridge consists of four resistors arranged in a diamond (or "bridge") shape:

• Two parallel branches, each containing two resistors in series
• A voltage source connected across one diagonal of the diamond
• A galvanometer (sensitive current meter) connected across the other diagonal
• The unknown resistance sits in one arm of the bridge

Think of it as two voltage dividers side by side, with the galvanometer measuring the voltage difference between their midpoints.

Bridge components

• Four resistors: $R_1$, $R_2$, $R_3$, and $R_x$ (the unknown)
• Voltage source (battery or power supply) providing excitation voltage
• Galvanometer or sensitive voltage meter to detect the null condition
• Variable resistor (rheostat) for fine-tuning the balance
• Connection wires and terminals

Bridge operation principles

The Wheatstone bridge works by balancing electrical potentials across two branches of the circuit. When the two midpoints of the diamond sit at the same voltage, no current flows through the galvanometer, and you've found your balance point.

Balanced vs. unbalanced states

In the balanced state, no current flows through the galvanometer. This happens when the ratio of resistances in one branch equals the ratio in the other branch:

$\frac{R_1}{R_2} = \frac{R_3}{R_x}$

In an unbalanced state, current does flow through the galvanometer. The direction and magnitude of that current tell you which way the resistance ratio is off.

Null detection method

The null detection method is what gives the Wheatstone bridge its precision:

1. Connect the unknown resistance $R_x$ into one arm of the bridge.
2. Adjust the variable resistor until the galvanometer reads zero current.
3. At zero current, the bridge is balanced, and you can calculate $R_x$ from the known resistor values.

Because you're detecting zero current rather than measuring a specific current value, the internal resistance of the galvanometer doesn't affect the result. That's a big deal for accuracy.

Sensitivity and accuracy

• Sensitivity depends on how well the galvanometer can detect small currents. A more sensitive galvanometer lets you find the null point more precisely.
• Accuracy depends on how precisely you know the values of $R_1$, $R_2$, and $R_3$.
• Higher excitation voltage increases sensitivity but can cause resistors to heat up, which changes their resistance.
• Proper shielding and grounding reduce electromagnetic interference.

Mathematical analysis

Bridge equation

The fundamental relationship for a balanced Wheatstone bridge comes from setting the voltage at both midpoints equal. At balance:

$\frac{R_1}{R_2} = \frac{R_3}{R_x}$

Solving for the unknown resistance:

$R_x = R_3 \cdot \frac{R_2}{R_1}$

This equation holds when the bridge is perfectly balanced (zero galvanometer current). It also assumes ideal components with no significant parasitic resistance or temperature drift.

Resistance calculations

Here's how to actually solve for $R_x$ step by step:

1. Balance the bridge by adjusting the variable resistor until the galvanometer reads zero.
2. Record the values of $R_1$, $R_2$, and $R_3$ at that balance point.
3. Plug into $R_x = R_3 \cdot \frac{R_2}{R_1}$.

Example: If $R_1 = 100 \, \Omega$, $R_2 = 400 \, \Omega$, and $R_3 = 50 \, \Omega$, then:

$R_x = 50 \cdot \frac{400}{100} = 200 \, \Omega$

For high-precision work, you also need to account for lead resistances (the resistance of the wires connecting to $R_x$) and contact resistances at the terminals.

Error analysis

Common sources of error include:

• Component tolerances: If your "known" resistors are only accurate to $\pm 1\%$, your final answer inherits that uncertainty.
• Temperature coefficients: Resistor values drift as temperature changes.
• Lead and contact resistance: Especially problematic when $R_x$ is small.

Use standard error propagation to estimate uncertainty in $R_x$. Since $R_x = R_3 \cdot R_2 / R_1$, the relative uncertainty is:

$\frac{\delta R_x}{R_x} = \sqrt{\left(\frac{\delta R_1}{R_1}\right)^2 + \left(\frac{\delta R_2}{R_2}\right)^2 + \left(\frac{\delta R_3}{R_3}\right)^2}$

Applications in physics

Resistance measurement

The most direct application: measuring unknown resistances in electronic components or materials. You can characterize a material's resistivity, determine wire lengths from their resistance, or perform quality control on resistive elements.

Strain gauge systems

A strain gauge is a thin conductor whose resistance changes when it's stretched or compressed. By placing a strain gauge in one arm of a Wheatstone bridge, you can detect tiny mechanical deformations. Full bridge configurations (all four arms are strain gauges) provide temperature compensation because temperature changes affect all arms equally and cancel out.

These systems are used in structural analysis, load cells, and material testing.

Temperature sensing

Resistance Temperature Detectors (RTDs) exploit the fact that a metal's resistance changes predictably with temperature. A platinum RTD (like the Pt100, which has $100 \, \Omega$ at $0°C$) placed in a Wheatstone bridge allows precise temperature measurement. The bridge detects the small resistance change caused by temperature shifts.

You need to keep the excitation voltage low enough to avoid self-heating of the RTD, which would throw off the reading.

Variations and extensions

Kelvin bridge

For measuring very low resistances (below $1 \, \Omega$), a standard Wheatstone bridge struggles because lead and contact resistances become significant compared to $R_x$. The Kelvin bridge (also called a double bridge) solves this by using separate current and voltage connections (four-wire measurement), which eliminates lead resistance from the measurement.

AC bridges

Replacing the DC source with an AC source lets you measure complex impedances, not just resistance. This opens the door to measuring capacitance and inductance. Specific configurations include:

• Maxwell bridge: measures inductance
• Wien bridge: measures capacitance and frequency
• Schering bridge: measures capacitance and dielectric loss

These go beyond the scope of a DC circuits unit but are worth knowing about.

Anderson loop

A modified bridge that uses a current source instead of a voltage source. This provides better linearity and reduced sensitivity to lead resistance changes, making it useful for high-precision strain gauge measurements where long cable runs are involved.

Experimental techniques

Bridge balancing procedure

1. Start with a rough balance using coarse adjustment of the variable resistor.
2. Switch to fine adjustment, watching the galvanometer closely.
3. Alternate between coarse and fine adjustments until the galvanometer reads as close to zero as possible.
4. Allow time for thermal equilibrium before recording your final reading. Current flowing through the resistors generates heat, and you want the system to stabilize.

Calibration methods

Before measuring an unknown, verify your bridge works correctly:

• Substitute a known standard resistor for $R_x$ and confirm the bridge gives the correct value.
• Calibrate at several points across your expected measurement range.
• Note the ambient temperature and account for temperature coefficients if needed.

Data interpretation

• Record all relevant parameters: resistance values, excitation voltage, and ambient temperature.
• Take multiple measurements and apply basic statistical analysis (mean, standard deviation).
• Propagate uncertainties through the bridge equation to report a final result with error bounds.
• Compare your results with expected or theoretical values.

Limitations and considerations

Temperature effects

Resistance changes with temperature, and this can introduce errors. Self-heating from the measurement current itself is a subtle but real problem: the act of measuring changes the thing you're measuring. Temperature compensation techniques (like using matched resistors in adjacent arms) help mitigate this.

Contact resistance issues

Poor or dirty connections add extra resistance that gets lumped in with your measurement. This is especially problematic when $R_x$ is small, because the contact resistance becomes a significant fraction of the total. Four-wire (Kelvin) connections minimize this effect. Always ensure connections are clean and secure.

Measurement range constraints

Standard Wheatstone bridges work best for resistances roughly between $1 \, \Omega$ and $1 \, M\Omega$. Below $1 \, \Omega$, lead and contact resistances dominate. Above $1 \, M\Omega$, leakage currents and insulation resistance become problematic. Specialized configurations exist for measurements outside this range, but they add complexity.

Modern implementations

Digital Wheatstone bridges

Modern bridges use analog-to-digital converters (ADCs) to measure the bridge voltage precisely. Microcontrollers automate the balancing process and calculate $R_x$ directly, displaying the result on a digital readout. Many include data logging and computer interfaces for analysis.

Integrated circuit applications

Wheatstone bridge configurations are built into single-chip sensor interfaces for pressure sensors, accelerometers, and load cells. On-chip amplification and signal conditioning improve measurement resolution while keeping the system compact.

Automated measurement systems

Computer-controlled systems can sweep through excitation voltages, automatically balance the bridge, and log data in real time. These integrate with other lab instruments for comprehensive experimental setups, removing much of the manual effort from bridge measurements.