Kibble balance

Prior to 2019, the definition of the kilogram was based on a physical object known as the International Prototype of the Kilogram (IPK).

A conducting wire of length ${\displaystyle L}$ that carries an electric current ${\displaystyle I}$ perpendicular to a magnetic field of strength ${\displaystyle B}$ experiences a Lorentz force equal to the product of these variables. In the Kibble balance, the current is varied so that this force counteracts the weight ${\displaystyle w}$ of a mass ${\displaystyle m}$ to be measured. This principle is derived from the ampere balance. ${\displaystyle w}$ is given by the mass ${\displaystyle m}$ multiplied by the local gravitational acceleration ${\displaystyle g}$. Thus,

${\displaystyle w=mg=BLI.}$

The Kibble balance avoids the problems of measuring ${\displaystyle B}$ and ${\displaystyle L}$ in a second calibration step. The same wire (in practice, a coil) is moved through the same magnetic field at a known speed ${\displaystyle v}$. By

${\displaystyle U}$

${\displaystyle BLv}$

${\displaystyle U=BLv.}$

The unknown product ${\displaystyle BL}$ can be eliminated from the equations to give

${\displaystyle UI=mgv}$

${\displaystyle m=UI/gv.}$

With ${\displaystyle U}$, ${\displaystyle I}$, ${\displaystyle g}$, and ${\displaystyle v}$ accurately measured, this gives an accurate value for ${\displaystyle m}$. Both sides of the equation have the dimensions of power, measured in watts in the International System of Units; hence the original name "watt balance". The product ${\displaystyle BL}$, also called the geometric factor, is not trivially equal in both calibration steps. The geometric factor is only constant under certain stability conditions on the coil.^[1]

Implementation

The Kibble balance is constructed so that the mass to be measured and the wire coil are suspended from one side of a balance scale, with a counterbalance mass on the other side. The system operates by alternating between two modes: "weighing" and "moving". The entire mechanical subsystem operates in a vacuum chamber to remove the effects of air buoyancy.^[20]

While "weighing", the system measures both ${\displaystyle I}$ and ${\displaystyle v}$. The system controls the current in the coil to pull the coil through a magnetic field at a constant velocity ${\displaystyle v}$. Coil position and velocity measurement circuitry uses an

While "moving", the system measures ${\displaystyle U}$. The system ceases to provide current to the coil. This allows the counterbalance to pull the coil (and mass) upward through the magnetic field, which causes a voltage difference across the coil. The velocity measurement circuitry measures the speed of movement of the coil. This voltage is measured, using the same voltage standard and integrating voltmeter.

A typical Kibble balance measures ${\displaystyle U}$, ${\displaystyle I}$, and ${\displaystyle v}$, but does not measure the local gravitational acceleration ${\displaystyle g}$, because ${\displaystyle g}$ does not vary rapidly with time. Instead, ${\displaystyle g}$ is measured in the same laboratory using a highly accurate and precise

Like the early atomic clocks, the early Kibble balances were one-of-a-kind experimental devices and were large, expensive, and delicate. As of 2019, work is underway to produce standardized devices at prices that permit use in any metrology laboratory that requires high-precision measurement of mass.^[21]

As well as large Kibble balances, microfabricated or

Measurements

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Accurate measurements of electric current and potential difference are made in

von Klitzing constant

, ${\displaystyle K_{\text{J-90}}}$ and ${\displaystyle R_{\text{K-90}}}$ respectively. The current Kibble balance experiments are equivalent to measuring the value of the conventional watt in SI units. From the definition of the conventional watt, this is equivalent to measuring the value of the product ${\displaystyle K_{\text{J}}^{2}R_{\text{K}}}$ in SI units instead of its fixed value in conventional electrical units:

${\displaystyle {\frac {1}{K_{\text{J}}^{2}R_{\text{K}}}}={\frac {1}{K_{\text{J-90}}^{2}R_{\text{K-90}}}}{\frac {\{mgv\}_{\text{W}}}{\{UI\}_{W_{90}}}}.}$

The importance of such measurements is that they are also a direct measurement of the Planck constant ${\displaystyle h}$:

${\displaystyle h={\frac {4}{K_{\text{J}}^{2}R_{\text{K}}}}.}$

The principle of the electronic kilogram relies on the value of the Planck constant, which is as of 2019 an exact value. This is similar to the metre being defined by the speed of light. With the constant defined exactly, the Kibble balance is not an instrument to measure the Planck constant, but is instead an instrument to measure mass:

${\displaystyle m={\frac {UI}{gv}}.}$

Effect of gravity

Gravity and the nature of the Kibble balance, which oscillates test masses up and down against the local gravitational acceleration g, are exploited so that mechanical power is compared against electrical power, which is the square of voltage divided by electrical resistance. However, g varies significantly—by nearly 1%—depending on where on the Earth's surface the measurement is made (see