# Instrumentation & Measurement Magazine 23-3 - 14

```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

```

# Instrumentation & Measurement Magazine 23-3

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