= Resistance pressure gauge
{title2=$R=R(p,T,\text{history})$}
A <resistance pressure gauge> is a <pressure sensor> whose <electrical resistance> is calibrated against <pressure>. Even a simple local response needs both <pressure> and <temperature>. For an original illustrative model define
$$
y=\frac{R}{R_0}-1=\alpha p+\beta(T-T_0)+h,\qquad
p=\frac{y-\beta(T-T_0)-h}{\alpha},
$$
where $R_0$ is the reference <electrical resistance> at $p=0$, $T=T_0$, and zero history offset. The <derivatives> $\alpha$ and $\beta$ are local sensitivities with units $\mathrm{Pa^{-1}}$ and $\mathrm{K^{-1}}$, respectively; $h$ is a dimensionless history offset. Constant sensitivities are an approximation over a chosen range, not material constants valid at every <pressure>.
Choose synthetic coefficients $\alpha=0.020\,\mathrm{GPa^{-1}}$ and $\beta=2.0\times10^{-4}\,\mathrm{K^{-1}}$, with $T-T_0=10\,\mathrm K$ and $h=0$. A measured ratio $R/R_0=1.022$ then gives $p=(0.022-0.002)/0.020=1.00\,\mathrm{GPa}$. Ignoring the <temperature> change gives $1.10\,\mathrm{GPa}$, a $0.10\,\mathrm{GPa}$ error. This model is deliberately stipulated; these numbers are not measured <manganin> coefficients or Bridgman's data.
\Image[/pressure-resistance-calibration-example.png]
{title=<Pressure calibration>: a synthetic <electrical resistance> response with <temperature> and <hysteresis> offsets}
{description=Original calculated example. The blue reference line has $T=T_0$, $h=0$; the orange line has $T-T_0=10\,\mathrm K$, $h=0$. Dashed lines illustrate two histories with $h=\pm0.0002$, not observed data. A ratio 1.022 maps to 1.00 GPa after correcting <temperature>, or 1.10 GPa if read against the blue line. The lower chain distinguishes the specimen, the internal sensor, the electrical readout and the calibrated <pressure>.}
If two histories differ by $\Delta h=0.0004$, their outputs at the same <pressure> differ by $0.0004R_0$; interpreting both with a single history-free curve gives an apparent <pressure> separation $\Delta h/\alpha=0.020\,\mathrm{GPa}$. The half separation is $0.010\,\mathrm{GPa}$. Reversing the drive can additionally cause mechanical <friction> <hysteresis>, independently of this sensor model. Record both the electrical calibration history and the loading direction.
For independent small standard <measurement uncertainties> $u_y,u_T,u_h$ and exact stipulated $\alpha,\beta$, first-order propagation gives
$$
u_p^2=\frac{u_y^2+\beta^2u_T^2+u_h^2}{\alpha^2}.
$$
For $u_y=10^{-4}$, $u_T=0.5\,\mathrm K$, and $u_h=10^{-4}$, this is $u_p=0.00866\,\mathrm{GPa}$. Real calibration also includes coefficient <measurement uncertainties>, correlations, reference-point <measurement uncertainty>, and any unresolved nonhydrostatic loading. A sensor calibrates its local environment; it cannot by itself certify the absence of <deviatoric stress> throughout the sample.
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