= Lead-resistance subtraction in two-wire measurements
<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.<Low Level Measurements Handbook, Seventh Edition by Keithley Instruments>[§3.3.1, Lead Resistance and Four-Wire Method, Figure 3-16; §3.3.2, current reversal]{ref}
An original measurement model makes the residual explicit. Let the specimen reading be
$$
R_t=R_x+R_{L,t}+R_{C,t}+E_t/I_t,
$$
where $R_x$ 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 is
$$
R_0=R_{L,0}+R_{C,0}+R_s+E_0/I_0.
$$
Subtraction gives
$$
\boxed{\widehat R_x=R_t-R_0
=R_x+\Delta R_L+\Delta R_C-R_s+E_t/I_t-E_0/I_0.}
$$
The 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:
$$
u^2(\widehat R_x)=u_t^2+u_0^2-2\operatorname{Cov}(R_t,R_0).
$$
This is the difference specialization of the primary uncertainty-propagation law.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A.3, equation A-3]{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 $E_t=E_0=20\,\mathrm{\mu V}$, $I_t=10\,\mathrm{mA}$ 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.
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