IN10 · A null reveals a resistance

Wheatstone & Kelvin Resistance Bridges

Why can zero detector current reveal an unknown resistance? At a bridge null, the detector connects two points at equal potential. Known arm ratios then determine the unknown resistance. The detector need only resolve which way the imbalance points; the standards and actual contact locations still determine measurement accuracy.

Balance a finite-input bridge, move sense contacts and reverse excitation to measure resistance and temperature-corrected strain.

Compare, reverse, find null

Drag the gold ratio dial, or move the sense clips between inner and outer posts. The saved pair below stays unchanged.

Instrument topologyRatio arm dial

1.03

Bridge excitation

0.2\,\mathrm V

A valid reversed pair is loaded.

Leads, detector and calibration

Each force lead / Kelvin link (ohms)Auxiliary ratio mismatchBridge detector resistance (ohms)Thermal voltage (microvolts)Bounded detector noise (microvolts)Excitation reversal asymmetrySeedSynthetic specimen reference resistance (ohms)

Voltage, source monitors and the gauge zero/span/temperature fixture have separate saved observations. Four-terminal inputs are finite ten-megaohm loads.

Model assumptions

Five explicit resistor networks distinguish Wheatstone null balance, Kelvin double bridge, two-wire/four-terminal sensing and a quarter-bridge gauge. All node voltages, detector currents and branch powers share the same linear DC solution. The detector has finite input resistance; its electronic pointer range changes without changing that input. Stable thermoelectric voltage cancels through measured unequal-polarity excitation. Gauge factor and temperature coefficient come from an independent maintained-current fixture. Resistors remain at their prescribed temperature; self-heating, contact nonlinearity, thermal drift during a pair and leakage to earth are omitted. Layout, current-marker speed and component bodies are schematic. ADC observations and independent standards are synthetic; no confidence interval, total uncertainty or manufacturer certification is claimed.

Simulated ADC observations

What the reversed pair tells us

Inspect a saved pair

Resistance from the pair

126\,\Omega

Half-difference voltage

950\,\mu\mathrm V

Even voltage intercept

2\,\mu\mathrm V

Raw polarity observations and frozen calibration
Polarity	Detector code	Source monitor code

Known standards and measurement limits

V=aD+b,\qquad a=\frac{V_+-V_-}{D_+-D_-}

The two measured source magnitudes need not match. Their slope removes a thermal voltage stable over the pair. Bridge inversion uses known arms and the finite detector; wire measurements report the segment between the actual clips, including any lead inside it. The even intercept is a circuit-loaded offset, not an independent thermometer. Noise, ADC quantization and standard errors remain; no total uncertainty or confidence interval is claimed.

The gauge fixture uses an ideal maintained current of five milliamperes in each direction. Its known input value, ampere unit and fixture version are saved. Fixture raw current codes are monitoring diagnostics; the gauge fit uses the saved known current. Main measurement pairs instead use their measured excitation difference.

Linear, ohmic, isothermal teaching components. Gauge resistance is multiplicative in strain and measured temperature, with factor and temperature coefficient fitted from independent fixture observations. Self-heating, lead/contact drift during reversal, leakage to earth and nonlinear contacts are omitted. A current marker is schematic; the board layout identifies the actual solved nodes.

Start with the essentials

Focus question: Which circuit and contacts make a null or voltage pair identify the intended resistance?
One-sentence intuition: The node graph decides what is compared. Null balance, a double bridge and four-terminal sensing remove different error paths.

Core mathematical model

Wheatstone null

R_x=R_2R_3/R_1

The left arms run from A through R1 and R2 to D; the right arms use R3 and Rx. At a true null their midpoint voltages agree, independently of finite detector resistance.

Kelvin double bridge in this board orientation

R_x=\frac P Q S+\frac{qr}{p+q+r}\left(\frac P Q-\frac p q\right)

Main path A–B–C–D contains Rx, link r and standard S. The main ratio path runs A–E–D; the auxiliary path runs B–F–C. The detector compares E to F. Matching ratios cancels the link term.

A finite voltage input loads the chosen wire segment

a=\frac{V_+-V_-}{I_+-I_-},\qquad R_{\mathrm{segment}}=\frac{a}{1-a/R_{\mathrm{in}}}

The slope uses measured source currents, not assumed equal commands. A finite sense-input current is corrected. Leads within the chosen segment remain part of that measured resistance.

Known strain and temperature standards

R=R_0(1+G_F\varepsilon)[1+\alpha(T-T_0)]

A known-current fixture measures zero strain, known strain and a second temperature. These observations identify the reference resistance, factor and temperature coefficient used by the inverse.

Common difficulties

A null is not an unpowered circuit

Typical misconceptionIf the detector current is zero, no current flows anywhere.

Better mental modelThe arms carry current. Only the comparison branch has equal potentials at its endpoints.

Run the experiment

01
Move through null
Drag the gold ratio dial, then save a reversed pair.
What to observe: The detector changes sign. Its input current vanishes at balance, while current still flows in the bridge arms.
02
Move the actual sense clips
Choose Sense at the specimen, then drag each clip to an outer post.
What to observe: Every included lead adds its resistance to the measured segment. The force current and voltage contacts have distinct paths.
03
Break the paired ratio
Choose Two ratios disagree, then change the auxiliary mismatch in advanced settings.
What to observe: The complete double bridge changes its detector response; a main-ratio-only interpretation would miss the link term.
04
Separate temperature from strain
Choose Stretch a warm gauge. Change temperature, acquire independent standards and save a new pair.
What to observe: Known strain/current/temperature fixture observations set the gauge conversion. Old data retain their original calibration.

Wheatstone & Kelvin Resistance Bridges

What the reversed pair tells us

Follow the comparison current

Core mathematical model

Wheatstone null

Kelvin double bridge in this board orientation

A finite voltage input loads the chosen wire segment

Known strain and temperature standards

Common difficulties

A null is not an unpowered circuit

Run the experiment

Move through null

Move the actual sense clips

Break the paired ratio

Separate temperature from strain

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