= Branch-current mismatch in a resistance ratio measurement
{title2=$I_T/I_{\rm ref}$}
<Branch-current mismatch in a resistance ratio measurement> is a residual scale error when the sensor and <reference resistor> carry unequal <electric currents>. A common supply or common nominal current setting does not establish the shared excitation required by a <shared-excitation resistance ratio measurement>.
Suppose separate branches carry positive currents $I_T$ and $I_{\rm ref}$. Ideal potential sensing gives
$$
r=\frac{I_TR_T}{I_{\rm ref}R_{\rm ref}},\qquad
\widehat R_T=rR_{\rm ref}=R_T\frac{I_T}{I_{\rm ref}}.
$$
Thus the fractional resistance error is exactly $I_T/I_{\rm ref}-1$. If $I_T=I_0(1+\epsilon_T)$ and $I_{\rm ref}=I_0(1+\epsilon_{\rm ref})$, the common scale $I_0$ cancels, while
$$
\frac{\widehat R_T-R_T}{R_T}
=\frac{\epsilon_T-\epsilon_{\rm ref}}{1+\epsilon_{\rm ref}}
\simeq\epsilon_T-\epsilon_{\rm ref}.
$$
In an unbranched series circuit the same <electric current> passes through both components; this particular error instead concerns separate paths, leakage, or unequal effective sampling times.
For an original example, use $R_T=200\,\Omega$, $R_{\rm ref}=2000\,\Omega$, $I_T=1.002\,\mathrm{mA}$ and $I_{\rm ref}=0.998\,\mathrm{mA}$. Their potentials are $0.2004\,\mathrm V$ and $1.996\,\mathrm V$. The inferred resistance is $200.801603\ldots\,\Omega$, high by approximately $0.400802\%$. Each current is only $0.2\%$ from its nominal setting, but the opposed errors add to first order.
A useful ideal comparison is <current-source swapping in a resistance ratio measurement>. In two otherwise unchanged acquisitions, assign source A to the sensor and source B to the reference, then exchange them. If $q=R_T/R_{\rm ref}$, the two ratios obey
$$
r_A=q\frac{I_A}{I_B},\qquad r_B=q\frac{I_B}{I_A},\qquad
\sqrt{r_A r_B}=q.
$$
This exact <geometric mean> cancellation assumes unchanged source outputs and resistances between acquisitions; switching transients and <sensor self-heating in a ratiometric resistance measurement> can violate that assumption. The primary circuit guide independently treats current-source mismatch and source interchange in its three-wire lead-cancellation circuit; that topology's correction must follow its own equations rather than be replaced by the two-branch model above.<A Basic Guide to RTD Measurements by Joseph Wu>[§1.4, printed pp. 5–6, Figures 1-5–1-6 and excitation-source mismatch]{ref}
A proposed test measures both branch currents through characterized sensing <resistors>, then changes their assignments while holding the sensor environment fixed. Agreement after swapping tests the mismatch model; agreement from merely changing the common nominal current tests a different cancellation.
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