= Capacitance uncertainty in heterodyne current recovery
<Capacitance uncertainty in heterodyne current recovery> is the contribution of uncertain shunt <capacitance> to the inferred generated beat <electric current>. In the <capacitive current division in heterodyne receivers> model, a measured load-current amplitude $A$ is corrected to
$$
F=A\sqrt{1+x^2},\qquad x=\omega RC,
$$
with load <electrical resistance> $R$, angular <frequency> $\omega$ and shunt <capacitance> $C$. These amplitudes must refer to the stated <current-plane convention in heterodyne calibration> and use the same peak or <root mean square> convention.
Differentiation gives the logarithmic <sensitivity coefficient>
$$
g=\frac{\partial\log F}{\partial\log C}=\frac{x^2}{1+x^2},\qquad
d\log F=d\log A+g(d\log\omega+d\log R+d\log C).
$$
Thus <capacitance> contributes little when $x\ll1$, contributes half its fractional change at $x=1$, and contributes nearly its full fractional change when $x\gg1$. High-frequency correction increases the required amplitude estimate; it does not make uncertainty in the correction disappear.
Let $y=(\log A,\log\omega,\log R,\log C)^{\mathsf T}$ have <covariance matrix> $\Sigma_y$. First-order <uncertainty propagation> gives
$$
u^2(\log F)\simeq b^{\mathsf T}\Sigma_yb,\qquad b=(1,g,g,g)^{\mathsf T}.
$$
This form retains correlations rather than silently assigning four independent inputs. It is the GUM's sensitivity-and-covariance law applied to this particular <measurement model>.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1.2–5.1.3 and 5.2.2, equations 10 and 13]{ref}
For an original small-error example, set $x=1$, neglect <frequency> uncertainty, and assign fractional <standard uncertainties> of $2\%$ to $A$, $1\%$ to $R$ and $5\%$ to $C$. With independent inputs,
$$
\frac{u(F)}F\simeq\sqrt{0.02^2+\tfrac14(0.01^2+0.05^2)}\approx3.24\%.
$$
If only the $R,C$ errors have <correlation coefficient> $+0.8$, the result becomes about $3.54\%$; correlation $-0.8$ instead gives about $2.92\%$. The same marginal uncertainties can therefore produce different output uncertainty.
Nonlinearity also shifts the mean. Holding $A,\omega,R$ fixed and writing $C=C_0+\epsilon$ with centered small error, a second-order <Taylor series> gives
$$
\frac{\mathbb E[F(C)]-F(C_0)}{F(C_0)}
\simeq\frac{x_0^2}{2(1+x_0^2)^2}\frac{u(C)^2}{C_0^2}.
$$
At $x_0=1$ and $u(C)/C_0=0.05$, this is about $0.0313\%$, distinct from the first-order spread. Broad or asymmetric inputs require a fuller distribution model.
A proposed diagnostic characterizes the load and shunt response at the actual bias and relevant beat frequencies, then propagates the joint calibration record. NIST's primary apparatus separately calibrates electrical networks and mismatches as well as controlling optical beating.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF pp.1–2, Heterodyne measurements and equations 1–4]{ref} A fitted capacitance is only one part of that response: a single-pole model should not absorb untested cable, transit-time or amplifier effects.
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