21.5 Null Measurements

Null Measurements

Null measurements determine unknown quantities by balancing them against known references rather than reading a value off a meter scale. This approach draws essentially zero current from the circuit being measured, which avoids the "loading effect" that makes ordinary meters less accurate. Two classic null measurement devices are the Wheatstone bridge (for resistance) and the potentiometer (for voltage).

Accuracy of Null Measurement Devices

Standard voltmeters and ammeters have a fundamental problem: they draw current from the circuit they're measuring. That current changes the very thing you're trying to measure, especially in high-impedance circuits. The accuracy of these meters also depends on how well their scales are calibrated, which introduces reading errors.

Null measurement devices sidestep both issues:

• They work by balancing an unknown quantity against a known reference. At balance, the detector reads zero, so virtually no current flows through the measurement branch.
• Accuracy depends on the sensitivity of the null detector (a galvanometer that can detect currents as small as a few microamperes), not on the precision of a meter scale.
• Because the detector only needs to identify zero, even a roughly calibrated detector gives highly accurate results. The precision comes from the known reference standards, not the detector itself.

Resistance Calculation with the Wheatstone Bridge

A Wheatstone bridge measures an unknown resistance by comparing it to known resistances in a balanced circuit. It consists of four resistors arranged in a diamond (or bridge) configuration:

• $R_x$ is the unknown resistance.
• $R_1$, $R_2$, and $R_3$ are known, adjustable resistors (often decade resistance boxes).
• A voltage source (battery) connects across one diagonal of the diamond.
• A null detector (galvanometer) connects across the other diagonal.

How to find the unknown resistance:

1. Connect the circuit with $R_x$ in one arm of the bridge.
2. Adjust the known resistors until the galvanometer reads zero current.
3. At this balance point, the voltage across the top pair of resistors equals the voltage across the bottom pair, so no current flows through the galvanometer.
4. Calculate the unknown resistance:

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

This works because at balance, the ratio of resistances in one branch equals the ratio in the other branch: $\frac{R_x}{R_3} = \frac{R_2}{R_1}$. Since $R_1$, $R_2$, and $R_3$ are all precisely known, $R_x$ can be determined with high accuracy.

Components and Principle of Potentiometers

A potentiometer (in the null-measurement sense) measures an unknown voltage by comparing it to a known voltage along a uniform wire. Don't confuse this with the simple three-terminal "pot" used as a variable resistor.

Key components:

• A long, uniform resistance wire (slide wire) with constant resistance per unit length
• A sliding contact (wiper) that moves along the wire
• A standard cell with a precisely known voltage (historically a Weston cell, about 1.0183 V)
• A null detector (galvanometer)
• A driving battery that sends current through the full length of wire

How it works:

1. The driving battery pushes current through the entire slide wire, creating a uniform voltage gradient along its length.
2. The unknown voltage source is connected so that it opposes the voltage drop along a portion of the wire, with the galvanometer in the loop.
3. Slide the wiper along the wire until the galvanometer reads zero current.
4. At balance, the voltage drop across the wire segment exactly equals the unknown voltage. Calculate it using:

$V_x = \frac{l_x}{l_{\text{total}}} \times V_{\text{standard}}$

where $l_x$ is the length of wire between the wiper and one end, and $l_{\text{total}}$ is the total wire length.

Because zero current flows through the unknown source at balance, the potentiometer measures the true EMF without any drop across internal resistance.

Error Reduction and Calibration in Null Measurements

Null methods reduce measurement errors in several ways:

• Zero-current condition: At balance, no current is drawn from the unknown source, eliminating errors from internal resistance and loading effects.
• Reference standards: Accuracy traces back to well-characterized standards (precision resistors, standard cells) rather than meter calibration.
• Sensitivity over accuracy: The null detector only needs to identify zero, so even small detector imperfections don't affect the final result.

To maintain this precision, the reference standards themselves need regular calibration against primary standards. Any drift in a standard cell's voltage or degradation of precision resistors will directly affect measurement accuracy.