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 towith 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 coefficientThus capacitance contributes little when $x\ll 1$, contributes half its fractional change at $x=1$, and contributes nearly its full fractional change when $x\gg 1$. 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 givesThis 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.^[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,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 $C=C_0+ϵ$ with centered small error, a second-order Taylor series givesAt $x_0=1$ 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.^[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.