User:Sufficient statistics/Calculable capacitor

Calculable capacitor

A calculable capacitor (often the Thompson–Lampard capacitor) is a precision instrument that realizes electrical capacitance based on one mechanical dimension. Since the 2019 revision of the SI, realizations are linked to fixed values of fundamental constants via the metre and the second.

Principle of operation

The calculable capacitor is based on the Thompson–Lampard theorem, which states that the value of the cross capacitance satisfies a simple relation for a system with uniform cross section. This implies that the capacitance per unit length only depends on permittivity of free space. In practice, this is realized with 4 cylindrical rods whose center is placed on a square. 2 central grounded rods are placed within the space within the 4 rods, which are significantly pointed inside the extend of the rods. The quantity of interest is the change in capacitance when the central rods are displaced by a known amount, measured by laser interferometry.

The cross section has 4 electrically isolated segments labeled 1,2,3,4. In SI units, the cross capacitor satisfies relationship

$\exp {\bigg (}-\pi C_{13}/\epsilon _{0}{\bigg )}+\exp {\bigg (}-\pi C_{24}/\epsilon _{0}{\bigg )}=1$ where $C_{13},C_{24}$ are the cross-capacitance per unit length, $\epsilon _{0}$ is the permittivity of free space.

In a symmetric system where $C_{13}=C_{24}=C$, the cross-capacitance is $C=\epsilon _{0}{\frac {\ln 2}{\pi }}=1.953549043...\times 10^{-12}Fm^{-1}$

In a practical realization, the average of the 2 cross capacitance is equal to the above within second order of the asymmetry. I.e. if the difference is within $10^{-4}$, then the average is within $10^{-8}$ with the above value.

A practical realization uses four vertical cylindrical electrodes at the corners of a square and two guard electrodes introduced in the inter-electrode region. The measurand is the change in mean cross-capacitance when one guard electrode is displaced by a calibrated distance $\Delta L$ measured interferometrically; the generated capacitance increment is $\Delta C=\left({\frac {\varepsilon _{0}\ln 2}{\pi }}\right)\,\Delta L$ for an ideal symmetric geometry, which links the farad to the metre and second via the laser standard used in the interferometer.

Mathematical formulation and sensitivity

Complementary cross-capacitances

Let $C_{m}\equiv {\tfrac {1}{2}}(C_{13}+C_{24}),\qquad \Delta \equiv {\tfrac {1}{2}}(C_{13}-C_{24}).$ From the Thompson–Lampard identity one finds the exact relation $C_{m}\;=\;{\frac {\varepsilon _{0}}{\pi }}\,{\Big [}\ln 2+\ln \!\cosh \!{\Big (}{\frac {\pi \Delta }{\varepsilon _{0}}}{\Big )}{\Big ]}.$ Thus the mean cross-capacitance equals the ideal value $C_{0}=\varepsilon _{0}\ln 2/\pi$ up to a second-order correction in the asymmetry $\Delta$. Expanding for small $|\Delta |\ll \varepsilon _{0}$, ${\frac {C_{m}-C_{0}}{C_{0}}}\;=\;{\frac {\ln 2}{2}}\,\eta ^{2}\;+\;O(\eta ^{4}),\qquad \eta \equiv {\frac {\Delta }{C_{0}}}.$ Numerically, ${\frac {\ln 2}{2}}\approx 0.3466$. Hence if the fractional half-difference $|\eta |\lesssim 10^{-4}$, the bias of the average is $\lesssim 3.5\times 10^{-9}$. (This follows algebraically from the theorem; see also the expositions in van der Pauw and Jackson.)

Misalignment, tilt, and local imperfections

Practical realizations must control: (i) lateral eccentricity of the movable guard; (ii) tilt; (iii) cylindricity and taper of the main bars; (iv) localized surface defects. Model tests at NIST with a full-scale mock-up showed that changes in cross-capacitance due to small eccentricity are second order in the lateral offset, and that spike- or cone-tipped guards reduce sensitivity to bar taper and local defects; empirical sensitivities were quantified to guide tolerances and alignment budgets. Modern BIPM/NMIA designs specify $\sim$100 nm cylindricity and parallelism, use semi-conical nosepieces carrying semi-transparent mirrors for the interferometer, and operate in vacuum to suppress air-permittivity and refractive-index corrections. Finite-element studies indicate sub-micrometre tolerances on intra-electrode geometry and guard positioning are needed for uncertainties below $10^{-7}$.

Fringing fields and outer space

If the exterior region is free space, there is also an “outer” contribution equal to $C_{0}$ per unit length. In practice this is reduced below $10^{-8}$ by a grounded cylindrical screen; the dominant contribution then comes from the interior cross-capacitances. Guard electrodes further suppress end effects, so the measurand is the difference between two positions of the guard, for which the fringing contribution cancels to first order.

Accuracy and uncertainty attained

Early realizations achieved about $10^{-7}$ relative accuracy (in e.s.u. at the time). The NIST calculable capacitor links the U.S. capacitance unit to SI with a quoted relative standard uncertainty of $2\times 10^{-8}$. The BIPM/NMIA program targets a relative uncertainty at the level of a few parts in $10^{9}$ for the generated capacitance increment (to enable $10^{-8}$ determinations of the von Klitzing constant $R_{\mathrm {K} }$). International key comparisons of 10 pF and 100 pF standards show participating NMIs maintaining uncertainties at or below the $10^{-7}$ level using calculable capacitors and quantum traceability. Regional (APMP) comparisons report similar performance for 0.5 pF and 1 pF standards, with axial length determination often dominating the uncertainty budget.

Frequency behavior and bridge measurements

Realizations are compared at audio frequencies (≈ 100–1000 Hz) using transformer-ratio and quadrature bridges. Frequency dependence is modeled via the loss tangent (dissipation factor) $\tan \delta$, which in the simple series model gives $\mathrm {ESR} ={\frac {\tan \delta }{\omega C}},\qquad \omega =2\pi f,$ with complex impedance $Z\approx {\big (}1/j\omega C{\big )}{\big (}1+j\tan \delta {\big )}$. The calculable capacitor anchors the farad; traceability chains transfer the value to 10 pF and 100 pF fused-silica standards, and then to resistors via quadrature (resistance–capacitance) bridges.

Nonlinearity of bridges and standards

Because the calculable capacitor provides a continuously variable, low-loss reference, it is used to characterize the nonlinearity of precision capacitance bridges. Recent work at LNE used a Thompson–Lampard standard spanning ~0.8 pF with sub-ppm resolution to diagnose a saw-tooth nonlinearity pattern in the AH 2700A bridge; internal calibration restored linearity to the specified level. Residuals from a linear fit of $C_{m}(z)$ versus interferometric displacement $z$ quantified nonlinearity at or below $5\times 10^{-7}$ over the full range, with planned checks at $10^{-8}$. Earlier NIST investigations also used a continuously variable calculable capacitor to study bridge behavior and stability.

Proof sketch (conformal mapping)

The theorem is two-dimensional. Let $D$ be a simply connected cross-section bounded by four conductor arcs $E_{1},E_{2},E_{3},E_{4}$ (in cyclic order). Solve the Dirichlet problem for the potential $\phi$ with boundary data $\phi =0$ on $E_{1}\cup E_{3}$ and $\phi =V$ on $E_{2}\cup E_{4}$. The electrostatic energy per unit length is ${\mathcal {E}}={\frac {\varepsilon _{0}}{2}}\!\int _{D}\!|\nabla \phi |^{2}\,dA={\tfrac {1}{2}}C_{13}V^{2}.$ By the Riemann mapping theorem there exists a conformal map $w=f(z)$ from $D$ onto a rectangle $R=\{0<\operatorname {Re} w<\Lambda ,\ 0<\operatorname {Im} w<\pi \}$ that sends $E_{1},E_{3}$ to the vertical sides and $E_{2},E_{4}$ to the horizontal sides; $\Lambda >0$ is the (conformal) modulus of the quadrilateral $(D;E_{1},E_{2},E_{3},E_{4})$. Harmonic functions compose with conformal maps, so the potential becomes a linear function of $\operatorname {Re} w$ in $R$. Direct integration in $R$ yields ${\frac {\pi C_{13}}{\varepsilon _{0}}}=\Lambda ,\qquad {\frac {\pi C_{24}}{\varepsilon _{0}}}=\ln \!\left({\frac {1}{1-e^{-\Lambda }}}\right).$ Eliminating $\Lambda$ gives $\exp \!\left(-{\frac {\pi C_{13}}{\varepsilon _{0}}}\right)+\exp \!\left(-{\frac {\pi C_{24}}{\varepsilon _{0}}}\right)=1,$ which is the Thompson–Lampard relation. (Alternate proofs proceed via harmonic measure or the Schwarz–Christoffel mapping; see Lampard’s original paper and Jackson’s exposition for rigorous details and the generalizations to arbitrary cross-sections.)

History

The theorem and concept were introduced by A. M. Thompson and D. G. Lampard in 1956–1957 at CSIRO (Australia). Practical instruments followed at CSIRO/NML and other NMIs, enabling absolute ohm determinations via impedance chains and revealing long-term drifts of resistance standards at the $10^{-7}$–$10^{-6}$/year level. Later work generalized and clarified the mathematics (van der Pauw; Jackson) and integrated calculable capacitors into modern metrology chains alongside quantum standards.

Five-electrode variant

An extension of the Thompson–Lampard calculable capacitor (TLCC) uses five static electrodes arranged as a regular pentagon, together with a movable guard electrode. This design, pioneered at the Laboratoire national de métrologie et d’essais (LNE), offers several advantages over the original four-bar square geometry.

In the ideal symmetric case the mean cross-capacitance per unit length is $\gamma _{0}={\frac {\varepsilon _{0}}{\pi }}\ln \varphi \;\approx \;1.35623\ {\text{pF·m}}^{-1},$ where $\varphi =(1+{\sqrt {5}})/2$ is the golden ratio. The measurable capacitance increment is $C=2\,r\,\gamma \,\Delta L,$ where $\Delta L$ is the interferometrically determined guard displacement, $r$ the bridge ratio (e.g. $r=8/3$ at LNE), and $\gamma$ the mean of the five cross-capacitances.

Internal consistency checks

With five electrodes the Thompson–Lampard theorem yields five exponential relations, such as $\exp \!\left[-{\tfrac {\pi }{\varepsilon _{0}}}(\gamma _{1,3}+\gamma _{1,4})\right]+\exp \!\left[-{\tfrac {\pi }{\varepsilon _{0}}}\gamma _{2,5}\right]=1,$ and analogous expressions for cyclic permutations. These provide an over-determined system, allowing residuals $e_{i}$ to be formed. In the symmetric case all $e_{i}=0$, while in practice their mean residual $e=-{\tfrac {1}{\kappa }}\,{\tfrac {u(\gamma )}{\gamma _{0}}},\qquad \kappa \approx 1.5,$ links directly to the relative uncertainty. Thus the instrument possesses an internal self-check of its geometry.

Reduced sensitivity to imperfections

In the pentagonal geometry, first-order sensitivity to static asymmetries such as small bar radius differences or position errors cancels in the average $\gamma$. By contrast, the four-bar capacitor achieves such cancellation only to second order. The 5-bar version is therefore more robust against machining tolerances.

Parallelism and trajectory error analysis

The main mechanical error sources are (i) tilt of a static electrode by angle $\alpha$ and (ii) lateral trajectory error $\rho$ of the movable guard. The relative uncertainty can be written $\left[{\tfrac {u(\gamma )}{\gamma _{0}}}\right]^{2}=\left(-\kappa \,{\tfrac {\partial e}{\partial \alpha }}\,u(\alpha )\right)^{2}+\left(-\kappa \,{\tfrac {\partial e}{\partial \rho }}\,u(\rho )\right)^{2},$ so experimental determination of the sensitivity coefficients $\partial e/\partial \alpha$ and $\partial e/\partial \rho$ allows direct uncertainty budgeting. LNE reports at least a four-fold reduction of the mechanical-imperfection contribution compared with earlier four-bar standards, reaching a target relative uncertainty of order $10^{-8}$.

Redundancy and alignment

Because any four of the five electrodes define a valid Thompson–Lampard quadrilateral, multiple independent realizations of the farad can be made with one instrument. These redundant data sets support averaging and A/B checks. In practice, alignment is achieved by using one electrode as reference and aligning the others with capacitive probes to within ≈50 nm, then confirming via the TL residuals.

Trade-offs

The per-unit-length slope of the 5-bar capacitor ($\gamma _{0}\approx 1.356\ {\text{pF·m}}^{-1}$) is lower than that of the 4-bar square ($\approx 1.954\ {\text{pF·m}}^{-1}$). Thus the capacitance change per unit travel is smaller, requiring more sensitive bridge electronics or slightly greater guard displacement. Moreover, mechanical construction and alignment are more complex. Nevertheless, the built-in consistency checks and reduced sensitivity to asymmetry make the five-electrode variant attractive for the most demanding realizations of the farad.

See also

• Quantum Hall effect
• Josephson effect