kilogram, the definition of the unit of mass from 1889 to 2019. As such, both methods would assign a number to a transfer standard, for example a national prototype, which would then be used to work-up and work-down to larger and smaller masses. It was always part of traditional mass metrology to produce multiples and submultiples of the kilogram. This task is achieved by substitution-weighing of combinations of masses that add up to the known standards together with comparisons of the individual masses within that combination. For example, a mass set comprised of a 500 g, two 200 g and two 100 g masses can be used to work-down from 1 kg to 100 g. A minimum of five substitution comparisons are required to solve a system of equations of the five unknown masses. In practice, more comparisons are performed, and the system of equations becomes overdetermined. It can be solved with a least-squares procedure which will result in the mass values and their uncertainties. The example above shows how the unit of mass is divided down by one order of magnitude. This division must occur for many orders of magnitudes. Some laboratories calibrate masses as small as 100 μg, a total of seven orders of magnitudes below 1 kg. Clearly, subdividing the unit of mass is an involved process that requires several weighings. Hence, it is an appealing option to build Kibble balances capable of measuring small mass directly via quantum electrical standards instead of subdividing a mass standard. For smaller masses, smaller forces must be created with the coil and since the force is proportional to the current, smaller current is necessary. The last statement is true if the geometric factor remained the same. However, when designing a Kibble balance for smaller mass values, one could decide on a magnet system with a smaller geometric factor and, hence, maintain a large current. The problem with this approach is that the induced voltage in the generator mode will also go down and therefore be more difficult to measure. Increasing the velocity is often not a practical solution, either. Hence, it becomes clear that for smaller and smaller masses and forces, the Kibble balance may not be the ideal tool. An alternative solution is to use an electrostatic balance. In an electrostatic force balance (EFB) [15], [16], the actuator is a capacitor with one fixed capacitor plate and a second moveable capacitor plate mounted to a balancing mechanism. The electrostatic energy of the capacitor is given by E = ½CV2, where C is the capacitance and V the potential difference between the capacitor plates. The force acting on each capacitance plate is given by Fz = dE 1 2 dC (9) = V dz 2 dz Like the measurement with the Kibble balance, the measurement with the EFB is performed in two modes. In the weighing mode, the force or weight that has to be measured is balanced against the electrostatic force. In this mode, the potential difference between the capacitor plates is measured against a voltage standard that is ultimately derived from a Josephson Voltage standard. In the second mode, the capacitance of the capacitor is obtained as a function of position C(z). From 14 this measurement the capacitance gradient dC can be obtained. dz The capacitance is ultimately traceable to either the calculable capacitor or the ac quantum Hall effect. The capacitance per unit length of a calculable capacitor is given by C 0 = ln 2 (10) L π where the electric constant is given by 0 = e2 (11) 2α hc0 Here, e, h, and c 0 are defining constants in the SI and, hence, have no uncertainty. In contrast, the unitless fine structure constant, α has to be measured and, thus, carries a (negligible) uncertainty. The finite uncertainty in ϵ0 is collateral damage that was incurred with the introduction of the present SI last year. The other traceability chain starts with the ac quantum Hall effect, see below, where the capacitance is given by C= ne 2 (12) 4π hf Since the fine structure constant is a dimensionless number, no matter which of the two traceability chains are used the capacitance gradient is given by dC β 2 e 2 = (13) dz hc0 where β2 is a known numerical factor. Combining this with the Josephson voltage measurements in the force mode, V= β3h 2e f (14) yields a force given by F = β4 f 2h (15) c0 with β4 = β 2 β 32 4 . (16) Again, the force is realized as the product of two frequencies and the Planck constant divided by the speed of light. Note, this assertion also holds for the Kibble balance, since the velocity of the coil can be seen as a tiny fraction of the speed of light. While the final force equations are identical for the Kibble balance and the electrostatic balance, the differences occur in the steps needed to reach this result. The main advantage of the electrostatic balance over the Kibble balance is that calibration mode is carried out in a static fashion, whereas in the Kibble balance, the velocity mode is a dynamic measurement. For the Kibble balance, the induced voltage is measured simultaneously with the coil's velocity while the coil is moving. This requires synchronization between the voltage measurement and the velocity measurement (usually an interferometer). For a highly precise measurement the synchronization is not IEEE Instrumentation & Measurement Magazine May 2020

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