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Keithley Instruments is an instrument manufacturer associated with precision electrical measurement. Its Low Level Measurements Handbook, Seventh Edition by Keithley Instruments explains how current sources, voltmeters and connection arrangements determine what a low-level reading means, with concrete treatments of lead resistance and thermoelectric offset.^[ref]

A useful example is the handbook's comparison of two-wire resistance measurement with four-terminal sensing. In two-wire measurement, current and voltage readout share the leads, so their resistance contributes to the measured drop. Separate voltage-sensing connections can avoid including that lead drop when sensing current is negligible.^[ref]

For an original ideal calculation, a $1\Omega$ resistor has two current leads of $0.1\Omega$ each. At $10\mathrm{mA}$, the total drop is $12\mathrm{mV}$, giving $1.2\Omega$ if all of it is attributed to the specimen. A separate voltmeter connected directly across the specimen, with negligible sensing current, instead measures $10\mathrm{mV}$ and gives $1\Omega$.

That result follows from the connection arrangement, rather than from the manufacturer's name. Finite input impedance, contact placement, thermoelectric offset and source uncertainty can still matter. The manufacturer's technical handbook is useful as an apparatus-level account of such limits; its example methods do not establish the accuracy of an unspecified instrument or a particular laboratory setup.

Lead-resistance subtraction in two-wire measurements corrects a specimen reading using a separate shorted-lead reading intended to estimate the unwanted series resistance. The correction is useful only to the extent that the lead and contact contributions remain known between the two arrangements. Keithley identifies lead resistance as an added term in the two-wire resistance measurement circuit and distinguishes it from additive thermoelectric offsets.^[ref]

An original measurement model makes the residual explicit. Let the specimen reading bewhere $R_x$ is the desired electrical resistance, $R_L$ and $R_C$ are included lead and contact resistances, and $E$ is an additive potential-difference offset. If the shorting connection has resistance $R_s$, its reading isSubtraction givesThe short is an actual electrical connection, not necessarily an exact zero. Replacing the specimen also changes contacts, so their contributions need not be identical.

For a numerical example, take $R_x=0.200\Omega$, stable total lead resistance $0.300\Omega$, negligible other contacts and offsets, and $R_s=0.005\Omega$. The specimen reading is $0.500\Omega$, the short reading $0.305\Omega$, and subtraction gives $0.195\Omega$, a $2.5\%$ negative error. If the leads subsequently rise to $0.312\Omega$, retaining that old short reading instead gives $0.207\Omega$, a $3.5\%$ positive error. Accurate arithmetic cannot establish that the two series contributions were equal.

A simple temperature model predicts one drift mechanism. If lead resistance follows $R_L(T)\simeq R_L(T_0)[1+\alpha (T-T_0)]$, then $\Delta R_L\simeq \alpha R_L\Delta T$. With stipulated $R_L=0.300\Omega$, $\alpha =0.0039{\mathrm{K}}^{-1}$ and $\Delta T=10\mathrm{K}$, the residual is $0.0117\Omega$, or $5.85\%$ of the example specimen. The actual temperature coefficient of resistance and temperature history must be established for the leads used.

Uncertainty in a difference includes covariance:This is the difference specialization of the primary uncertainty-propagation law.^[ref] With $u_t=u_0=1\mathrm{m\Omega }$ and correlation $0.9$, the difference uncertainty is about $0.447\mathrm{m\Omega }$; assuming independent readings would give $1.414\mathrm{m\Omega }$. Positive correlation can reduce a shared additive contribution, but a common gain error still scales the corrected specimen resistance.

An unchanged thermal offset also fails to cancel if the test currents differ: for $E_t=E_0=20\mu \mathrm{V}$, $I_t=10\mathrm{mA}$ and $I_0=1\mathrm{mA}$, its residual is $-0.018\Omega$. Current reversal within each arrangement can address a stable additive offset, but does not remove series resistance drift. Four-terminal sensing addresses the lead-drop geometry by measuring potential difference separately at the chosen specimen terminals.

A proposed check alternates short and specimen measurements while recording current and lead temperature, repeats after reconnecting the clips, and compares against a four-terminal result. The spread after reconnection tests a different contribution from rapid repeatability without disturbing the circuit. Report the defined specimen boundary, short construction and delay between readings rather than treating a successful zero display as a complete accuracy assessment.

Polarity reversal measurement separates a signal that changes sign on reversal from an additive contribution that does not. If readings in two polarities obeythen the half-difference estimates the signal, $\hat{s}=(y_{+}-y_{-})/2$, while the half-sum estimates the unchanged offset, $\hat{b}=(y_{+}+y_{-})/2$. The procedure depends on reversing the intended signal while preserving the relevant offset.

A primary electrical application is cancellation of thermoelectric offset in low-level measurements. The Keithley handbook describes reversing sources when comparing small electric potential differences, and reversing test current for low-electrical resistance measurements. A thermal offset can arise at dissimilar-metal junctions and remain approximately unchanged during a sufficiently short cycle.^[ref]

For an original resistance example, a bipolar current source applies $+I$ and $-I$ to a resistor while a voltmeter measuresLet $I=10\mathrm{mA}$ and readings be $103$ and $-97\mu \mathrm{V}$. Their half-difference is $100\mu \mathrm{V}$, giving $R=10\mathrm{m\Omega }$; their half-sum reveals a $3\mu \mathrm{V}$ offset. Using only the positive reading would instead give $10.3\mathrm{m\Omega }$.

An offset changing from $b_{+}$ to $b_{-}$ leaves resistance error $(b_{+}-b_{-})/(2I)$. A $2\mu \mathrm{V}$ change in this example therefore leaves $0.1\mathrm{m\Omega }$ error. Allow electrical settling after reversal while keeping the cycle short relative to material temperature changes. The handbook explicitly notes this time-scale requirement.^[ref] Reversal does not automatically remove lead resistance, current-magnitude errors, self-heating or every offset in the circuit. Record the actual reversal operation and timing with the measurement procedure.

A thermoelectric offset is an unwanted electric potential difference associated with Seebeck effect responses in a measurement circuit. Dissimilar conductors at different junction temperatures can generate a signal even when the intended electrical input is absent. Keithley's handbook discusses these offsets as a major limitation in low-voltage and low-resistance measurements.^[ref]

For a simple two-material circuit, a useful model iswhere the Seebeck coefficients depend on conductor material and temperature. With a stipulated constant difference $S_A-S_B=20\mu \mathrm{V}{\mathrm{K}}^{-1}$ and junction-temperature difference $0.5\mathrm{K}$, the offset is $10\mu \mathrm{V}$. These are teaching inputs, not coefficients assigned to a particular real connector pair.

That small offset can dominate an intended signal. For an original $1\mathrm{m\Omega }$ resistor tested at $10\mathrm{mA}$, the wanted drop is also $10\mu \mathrm{V}$. An additive offset of the same sign doubles the indication to $20\mu \mathrm{V}$ and gives an apparent $2\mathrm{m\Omega }$ if it is ignored. Reversing the current produces readings $20$ and $0\mu \mathrm{V}$ in this model; their half-difference recovers the wanted $10\mu \mathrm{V}$.

Polarity reversal measurement cancels the unchanged offset, not every temperature effect. Thermal changes during the cycle can leave a residual, while current-induced heating can alter the resistor itself. The handbook therefore connects reversal timing with thermal stability.^[ref]

An offset investigation records connector materials, junction temperatures, warm-up and handling, and compares readings before and after reversal. Reducing temperature gradients and avoiding unnecessary dissimilar-metal connections can reduce the generated signal. Keep residual correction uncertainty in the measurement result. A thermoelectric offset and random thermal noise have different roles in the measurement model; they are not interchangeable merely because both involve temperature.

A two-wire resistance measurement sends a known electric current through a specimen and measures the resulting electric potential difference using the same two connecting leads. The instrument estimates electrical resistance from their ratio. Because the leads carry the test current, their voltage drops enter the reading along with the specimen’s. Keithley’s primary handbook illustrates the two-wire circuit and identifies added lead resistance as its principal low-resistance limitation.^[ref]

For an original linear series-circuit model, let $R_x$ be the specimen resistance, $r_1+r_2$ the total lead resistance, and $r_c$ an included contact resistance. With actual test current $I\ne 0$ and additive offset potential difference $E$, the terminal reading isThe division assumes the actual current is known and the specified resistive elements are sufficiently linear at that current. Non-ohmic contacts require a different response model.

For $R_x=0.100\Omega$, $r_1=r_2=0.150\Omega$, $r_c=0$ and $I=10\mathrm{mA}$, the true terminal potential difference is $4.00\mathrm{mV}$. The indicated resistance is $0.400\Omega$: a $300\%$ positive error relative to the specimen value. Adding $E=20\mu \mathrm{V}$ raises the reading to $0.402\Omega$. The same $0.300\Omega$ lead contribution is only $0.003\%$ of a $10\mathrm{k\Omega }$ specimen, illustrating why the acceptable arrangement depends on the measurement scale.

Lead-resistance subtraction in two-wire measurements estimates that series contribution using a shorted-lead reading. It can remove a stable offset resistance, but reconnecting clips, moving contacts or changing lead temperature can alter the quantity subtracted. If residual lead resistance changes by $0.020\Omega$ after zeroing, the error is already $20\%$ for the $0.100\Omega$ specimen above. Display resolution alone cannot establish that subtraction’s accuracy.

Reversing the test current cancels a different effect. If $E$ and the resistances remain unchanged between equal-magnitude positive and negative currents,The thermoelectric offset cancels, but the lead resistance remains. In the example, $V_{+}=4.02\mathrm{mV}$ and $V_{-}=-3.98\mathrm{mV}$ yield $0.400\Omega$, not $0.100\Omega$. The handbook treats current reversal separately from the four-wire connection.^[ref]

Four-terminal sensing instead separates the current-carrying connections from potential-sensing connections placed at the desired specimen boundaries. A high-input-impedance sense circuit draws sufficiently little current that its lead drops can be much smaller; the exact contact placement still defines what is measured. Raising the two-wire test current to suppress $E/I$ also raises specimen heating, $I^2{R}_x$. Increasing this example’s current from $10$ to $100\mathrm{mA}$ raises specimen dissipation from $10\mu \mathrm{W}$ to $1\mathrm{mW}$.

A proposed comparison measures the same stable reference specimen with short and long leads, repeats with current reversal, then uses a four-wire connection at fixed specimen terminals. Record current, temperature and connection geometry. Lead-length dependence and reversal dependence distinguish series resistance from additive thermal offsets, while repeating at lower current helps test whether self-heating changed the specimen during the measurement.