# How can a Lego version of a Kibble balance measure the Planck constant?

As the picture shows below

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in a Kibble balance, one can drop out the measurement uncertainty of $$B$$ (magnetic flux intensity) and $$L$$ (length of coil) by the use of two modes, force mode and velocity mode.

And the voltage and electric current (resistor) value is determined by the Josephson effect and the quantum Hall effect; these two phenomena are what permits the measurement of electrical quantities in terms of the Planck constant to the precision required for the watt balance and redefinition.

So before the 2018 Nov 16 General Conference on Weights and Measures (CGPM) in Versailles, France, we measure the Planck constant using a known reference mass, such as Le Grand K. And, at the 2018 Nov 16 BIPM,the Planck constant, as defined by the ISO standard, was set to $$6.626070150\times 10^{-34}\:\rm J \cdot s$$ exactly.

That is, before the 2018 redefinition of units, the equation in the red rectangle is used to measure $$h$$ from a mass traceable to the IPK. After the redefinition, the equation will be used to realize the definition of the kilogram from a fixed value of $$h$$ in joule-seconds.

And I saw a Lego version kibble balance, designed by NIST people, reported in the paper

• A LEGO Watt balance: An apparatus to determine a mass based on the new SI. LS Chao et al. Am. J. Phys. 83, 913 (2015); L. S. Chao, S. Schlamminger, D. B. Newell, and J. R. Pratt

which states

Before the 2018 redefinition of units, the equation [...] is used to measure $$h$$ from a mass traceable to the IPK. After redefinition, the equation [...] will be used to realize the definition of the kilogram from a fixed value of $$h$$ in joule-seconds.

In a classroom setting, quantum electrical standards are typically unavailable. However, it is still possible to measure the Planck constant due to the way the present unit system is structured. While the SI is used for most measurements, a different system of units has been used worldwide for almost all electrical measurements since 1990. For these so-called conventional units, the Josephson and von Klitzing constants were fixed at values adjusted to the best knowledge in 1989. These fixed values are named “conventional Josephson” and “conventional von Klitzing” constants and are abbreviated $$K_{J–90}$$ and $$R_{K–90}$$, respectively. Since 1990, almost all electrical measurements are calibrated in conventional units. By comparing electrical power in conventional units to mechanical power in SI units, $$h$$ can be determined.

How to understand this?

• I do not really understand what kind of classroom setting the people at NIST had in mind. It is unfortunate that the new definition of the kg is unexplainable in school.
– user137289
Dec 12, 2018 at 9:10

## 2 Answers

Basically, since 1990, the bulk of all electrical measurement equipment has been calibrated to measurements that are traceable to fixed values of $$h$$ and $$e$$. When we talk about redefining the SI units, on the electrical side, this is best understood as simply elevating those previously non-SI units to full SI status. (For more details see this larger thread.)

When you're doing the measurement in the Chao et al. paper, you're making a comparison between two measurements of power which are traceable to two different standards:

• On one side, you're measuring mechanical power using a test weight whose mass is traceable to the IPK.
• On the other side, you're measuring electrical power using electrical measurement equipment whose calibration is traceable to the conventional electrical units, i.e. to $$K_{J\text{-}90}$$ and $$R_{K\text{-}90}$$.

The connection to $$h$$ is via the latter, since your electrical equipment has been calibrated to a fixed (though non-SI) value of $$h$$. By comparing this value to a measurement of an identical quantity made exclusively using SI-traceable measurements of mass, speed and time.

Ultimately, of course, in a classroom setting, you've lost so much precision in the calibration chain on both sides that you're unlikely to get more than two significant figures out of the process (their result, $$h/h_{90}=0.998$$, looks to me to be rather more precise than what you'll be able to do in a regular classroom). But the traceability principles look sound to me.

The measurement provided by the LEGO Kibble balance does NOT constitute a primary mass standard. This is because the measurement is NOT in any way directly related to Plancks constant $$h$$.

Rather, the authors claim, the volt and ammeters used to measure the voltages and currents in the lego machine are $$h$$-traceable in that they were calibrated to standards that were at point calibrated to standards... that were calibrated in a way related to the Josephson effect.

What's impressive about the LEOG Kibble balance is that it can pedagogically demonstrate how the complex Kibble balances used at advanced metrological labs work and that it is surprising accurate.