**Watt Balance

Introduction

The kilogram is the unique unit among the International System (SI) that is still defined from a single artefact, in accordance with the definition that was adopted by the General Conference on Weights and Measures (CGPM) in 1889. It is equal to the mass of the international prototype kilogram, platinum-iridium material standard, kept by the International Bureau of Weights and Measures (BIPM), and named .

However, even if represents the mass unit by definition, the international prototype is not perfectly stable, owing to its interaction with the surrounding medium.
Since 1880, three comparisons of the international prototype to test specimen and national prototypes of similar nature (the French national prototype appears under the number 35) were carried out by the BIPM. They have shown a non negligible drift between these elements, which mean relative value over some one hundred years is in the order of 30 µg with dispersion in the order of 100 µg.

Copyright figure : M. Girard - BIPM
Copyright figure : M. Girard - BIPM

Even though the comparison measurements were carried out with high accuracy, they do not provide any information regarding the absolute variations in the international prototype, which knowledge is conditioned by the comparison of the latter to a quantity invariant over time.

Because of the dependence of a certain number of other units (the ampere, the mole and the candela are defined from the mass unit), characterizing the behaviour of the kilogram and putting forward a new definition constitute a major challenge for today’s metrology.

Over these last decades, many NMIs and the BIPM have proposed potential routes or methods to answer this question.
The most promising routes consist in establishing a relationship between the kilogram and the Planck’s constant, h, or the Avogadro’s number NA.

Encouraged by the 20th and 21th CGPM (1995-resolution 5 and 1999-resolution 7) and to contribute to the international effort, France has made the decision to realize an experiment called "Watt Balance" based on a principle devised in 1976 by B.P. Kibble (NPL).

It has a twofold objective :

If this determination can be carried out with a sufficiently low uncertainty, satisfying both objectives could allow obtaining an evolution in the definition of the mass unit based on a fundamental constant, as was the case for the metre in 1983 by setting the value of the light speed conventionally, and hence relate the metre to the second.

Watt balances are already in existence (NPL – United Kingdom, NIST - United States of America, METAS - Switzerland) which let to believe that it will be possible to determine the Planck’s constant with an uncertainty in the order of 1.10-8, and hence confirm the method relevance.

Experiment principle

The experiment consists in making a comparison between mechanical force and electromagnetic force (hence its name). It is the result of a two-step measurement: a static phase, in which electromagnetic force (Laplace force) exerted on a coil of current-carrying wire in a strong magnetic field is compared with the weight of a standard mass; and a dynamic phase in which one measures the voltage produced at the coil terminals by moving the coil in the flux at a known speed.

Static phase

Static phase
Static phase

A wire with length l carrying current I is placed in a magnetic field B so that the electromagnetic force Fz (Laplace force) generated is vertical. Force exerted on the wire, hanged to a mass comparator, is compensated by the weight P of a mass m subjected to the acceleration due to gravity g.

Fz = B.l.I = P = m.g

Dynamic phase

Dynamic phase
Dynamic phase

In the dynamic phase, the same coil is moved in the same strong magnetic field with vertical speed V.
A voltage induced ε (equal to the variation in the flux cut during the displacement) then appears at the coil terminals :

ε = - dΦ/dt = - B.l. (dz/dt) = - B.l.V

Induction B and of the length l of the coil remain constant during the measuring time, the dynamic phase consists in an indirect determination of the product Bl.

The combination of the relationships describing the static and dynamic phases then leads to a relationship that expresses the equivalence between mechanical force and magnetic force (which gives its name to the experiment)

mgV= ε.I

In practice, current I is determined by the Ohm‘s law measuring the potential drop U that it causes at the terminals of a resistance R.

mgv = ε.U/R

How to establish a relationship between mass and Planck’s constant (h) :

Two quantum macroscopic effects, coming from solid-state physics, are currently commonly used by the NMIs to ensure the volt and ohm maintenance within relative uncertainties in the order of 10-9.

The Josephson effects allows generating voltages quantified at the superconductor-insulator-superconductor junctions exposed to a high-frequency radiation of frequency f. A Josephson junction is a frequency voltage converter (U = f/KJ) which proportional factor KJ (Josephson constant) is expressed according to the electron charge e and the Planck’s constant h (KJ=2e/h).

Similarly, the quantum Hall effect (quantization of the Hall resistance of a two-dimensional electron gas, often obtained at low temperatures in gallium arsenide heterostructures) sets resistance values proportional to the resistance quantum RK (von Klitzing constant, RK=h/e²).

Determining the values of voltages U and the resistance R by comparison to the Josephson effects and the quantum Hall effect allows expressing the value of mass m according to a combination of the KJ and RK constants independent from the electron charge.


The determination of the ratio h/m relates directly to determining the acceleration due to gravity g, to the coil displacement V and to the one of the term A in relation to the measurement of electrical quantities.

In order to realize the relative uncertainty within the order of 1.10-8, it is necessary that each of the quantities mentioned above be determined with a relative uncertainty in the order of 1 to about 10-9.

Prototype of beam for force comparator
Prototype of beam for force comparator

Dimensioning the experiment and outlook

The experimental device that is under development includes a balance beam (force comparator) to which is hanged a moving armature carrying on the one hand a coil (600 revolutions, 270 mm in diameter) and a standard mass (500 g), and on the other side a tare mass. During the dynamic phase, this set-up is jointly moved (2 mm/s) vertically by a guiding device actuated by a translation stage. A interferometer allows to measure at any time the coil position and control its speed in the air gap of a magnetic circuit generating radial induction in the order of 1T.

Configuration of measurement device
Configuration of measurement device.

The project, which first prototype should see the light of day by late 2007, was launched in 2002. It requires different competencies in mass metrology (realization of transfer mass standards), in mechanics (balance beam of force comparator, guidance system; vacuum chamber, magnetic circuit), in optical interferometry (control of coil speed, alignment of different components), atomic interferometry (construction of absolute cold atom gravimeter), and electrical and magnetic metrology (construction of measuring benches for voltage, resistors, currents and magnetic circuit characterization).

It is therefore the fruit of combined efforts made by several national metrology laboratories: the LNE, the LNE-INM and the LNE-SYRTE. Many collaborations have been initiated with universities, engineer schools, laboratories (Universités de Versailles - Saint Quentin en Yvelines, Ecole nationale supérieure des mines de Paris, the physics laboratories of the Université de Bourgogne and the CNAM, the Ecole nationale supérieure d'arts et métiers du Centre of Lille, the Laboratoire du génie électrique of Paris, the Ecole Normale Supérieure of Cachan), as well as some foreign NMIs more particularly in the United Kingdom (NPL), Germany (PTB) and of Sweden (SP).

Magnetic circuit with coil
Magnetic circuit with coil

A few references