Two-wire resistance measurement

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.

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.