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Arbeitsgruppe 2.62 Quanten-Hall-Effekt, Widerstand

The Capacitance Unit Farad

The unit of electrical capacitance is the farad (abbreviated F), named after the English physicist and chemist Michael Faraday. The capacitance C of a capacitor is the ratio of the charge Q stored in the capacitor to the applied dc voltage U:  

C = Q/U.

In the case of alternating current (ac), the capacitance is defined by the alternating current I which flows when an ac voltage U is applied to the impedance Z of the capacitor:

Z = U/I    with    Z = 1/(jωC)      ⇒      C = I/( jωU)

with j as the imaginary unit (j2= -1) and ω the angular frequency.

Hence it holds both for direct and alternating current:

1 F = 1 As/V = 1 s/Ω

The realisation and dissemination of the farad is accomplished world-wide with alternating current. Therefore, in the following only the ac capacitance is considered. Often used capacitance standards are commercial parallel-plate capacitors made of Invar and thermostated fused-silica standards because they, amongst other features, have a very small dissipation factor.

1 nF capacitor of the type "General Radio 1404 A"

1 nF capacitor of the type "General Radio 1404 A", for didactic purposes with a cut case to make the stack of parallel capacitor plates visible.

The realisation of the capacitance unit at PTB by coaxial measuring bridges

The capacitance unit is realised at PTB by means of a so-called quadrature bridge which links a 10 nF capacitance standard under calibration to the known quantum Hall resistance. The following figure shows the scheme of such a quadrature bridge:


Scheme of a quadrature bridge          

Scheme of a quadrature bridge.


Note that the same alternating current I flows through both standards. Using Ohm’s law I = U/RH for the left arm of the bridge and I = ωCU for the right arm of the bridge (which by the way is the definition of resistance and capacitance, respectively), the capacitance of the standard under calibration can be expressed in terms of the known quantum Hall resistance RH:

C = (1 + Δ)/(ωRH)         

with ω = 2πf the angular frequency and f = 1233,147 Hz the frequency, traced to the frequency standard of PTB (Department 4.4). Δ is the (usually very small) relative deviation of the 10 nF capacitance standard from nominal and is determined from a bridge balance system which, for the sake of simplicity, is not shown in the figure above.

It is important to make sure that the ac value of the quantum Hall resistance agrees with the quantised dc value and in particular does not deviate due to parasitic ac dissipation. To avoid such unwanted effects, PTB developed a special shielding technique.

According to the recommendation of the CIPM, the quantum Hall resistance is referred to RK-90, to ensure the best possible agreement with the SI farad. The relative difference between RK-90 and the current SI value of RK amounts to less than 2.10-8, which is practically not relevant and will fall away with the new SI.

The accuracy of the quadrature bridge shown above is limited by an inaccuracy of technical origin in creating the quadrature voltage jU. Expanding the quadrature bridge to a mirror-symmetrical double bridge allows to eliminating this effect and to achieving the desired accuracy. Indeed, this increases the measurement effort. In particular, two ac quantum Hall resistances are needed. They are operated in one cryostat with a superconducting solenoid and are provided with coaxial leads and shields.

Scheme of a double quadrature bridge

Scheme of a double quadrature bridge.




 Main part of the quadrature brigde

Photo of the main part of the quadrature brigde. The width of the photo corresponds to about 2.5 m.


In this way, the 10 nF capacitance standards are calibrated. Capacitance standards with nominal values of 10 pF and 100 pF (1 pF = 10-12 F) exhibit the best time stability and transportability. Therefore, they are most suitable for medium-term preservation, at PTB as well as for its customers. They are thus the "working horses" of capacitance metrology. To calibrate such a 10 pF or 100 pF capacitance standard, a sequence of 10:1 steps – starting with the already calibrated 10 nF standards – is carried out by means of a coaxial ratio bridge.


Measuring chain from the quantum Hall resistance

Measuring chain from the quantum Hall resistance to a 10 pF capacitance standard and to a dc resistance standard.

The quantum Hall resistance is thus a fixed point not only for the resistance scale, but also for the capacitance scale. This is an advantage for the consistency of the system of units. The uncertainty achievable for a 10 pF standard amounts to 1.10-8 (k = 2), which is clearly smaller than the uncertainty of the world-beating calculable capacitance artefacts. The reasons for this low uncertainty are not only the special properties of the quantum Hall resistance, but also the special coaxial measurement technique which allows very precise measurements at a low noise level.

Coaxial double holder for two GaAs quantum Hall resistances with overlain measured curves

Bottom end of a coaxial double holder for two GaAs quantum Hall resistances for application at low temperatures and strong magnetic fields. The overlain measured curves show the plateaus of the quantum Hall resistance. 

For the preservation at PTB, 10 pF and 100 pF working standards of known drift behaviour are traced in this way to the quantum Hall resistance, about two times a year as and when required. These capacitance standards are then used in Working Group 2.13 for the calibration of customer standards. There, also the capacitance scale with larger nominal values up to 10 mF is built up.

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