## How Watt Balances Work

A watt balance is an exquisitely accurate weighing machine. Like any balance, it is designed to equalize one force with another: In this case, the weight of a test mass is exactly offset by a force produced when an electrical current is run through a coil of wire immersed in a surrounding magnetic field.

The surrounding field is provided by a large permanent magnet system or an electromagnet. The moveable coil, once electrified, becomes an electromagnet with a field strength proportional to the amount of current it conducts. When the coil's field interacts with the surrounding magnetic field, an upward force is exerted on the coil. The magnitude of that force is controlled by adjusting the current.

The instrument is called a "watt" balance because it makes measurements of both current and voltage in the coil, the product of which is expressed in watts, the SI unit of power. That product equals the mechanical power of the test mass in motion.

Current and voltage are measured in two separate stages, or modes, of operation.

### Weighing/Force Mode

In "weighing" or "force" mode, a test mass is placed on a pan that is attached to the coil. It exerts a downward force -- its weight -- which is equal to its mass (*m*) times the local gravitational field (*g*). The current applied to the coil is then adjusted until the upward force on the coil precisely balances the downward force of the weight. When the system reaches equilibrium, the current is recorded.

At this point, it might seem that the job is finished. After all, the force (*F*) on the coil – which equals the weight of the mass – can be calculated with a simple equation that dates from the 19th century: *F = IBL*, where *I* is the current, *B* is the magnetic field strength, and* L* is the length of the wire in the coil. However, as a practical matter, the product BL is extremely hard to measure directly to the necessary accuracy.

Fortunately, physics provides a way around that problem via yet another relationship revealed in the mid-19th century: induction. Michael Faraday discovered that a voltage is induced in a conductor when it travels through a magnetic field, and that the voltage is exactly proportional to the field strength and the velocity. So if the velocity is constant, the induced voltage is a sensitive measure of the field strength.

### Velocity/Calibration Mode

That phenomenon is the basis for the watt balance's second stage, called "velocity" or "calibration" mode. For this operation, the test mass is removed and the applied current through the coil is shut off. The coil is then moved through the surrounding field at a carefully controlled constant velocity. The resulting induced voltage is measured.

Again, an uncomplicated formula governs the magnitude of the induced voltage: *V = νBL*, where *B* and *L* are the very same field strength and wire length as in weighing mode, and *ν* is the velocity. When this equation is combined with the one above for force on the coil, the problematic *B* and *L* cancel out, leaving *IV = mgν*. (That is, electrical power equals mechanical power,* both expressed in watts.) Or, solving for mass, m = IV/gν.

Everything on the right side of that equation can be determined to extraordinary precision: The current and voltage by using quantum-electrical effects that are measurable on laboratory instruments; the local gravitational field by using an ultra-sensitive, on-site device called an absolute gravimeter; and the velocity by tracking the coil's motion with laser interferometry, which operates at the scale of the wavelength of the laser light.

Where is the Planck constant,* h*, in all this? It comes into play in the way current and voltage are measured using two different quantum-electrical physical constants. Both constants are defined in terms of *h* and the charge of the electron, *e. *Those are very small quantities. Yet both are manifest in measurable macroscopic phenomena.

Current in a watt balance is measured by way of a resistor in the circuit. Resistance can be determined to about 1 part per billion using the von Klitzing constant, which describes how resistance is quantized in a phenomenon called the quantum Hall effect. Voltage is measured using the Josephson effect (and its associated Josephson constant), which relates voltage to frequency in a superconducting circuit, with measurement uncertainties in the range of 1 part in 10 billion. The Josephson effect is the world's de facto standard for quantifying voltage.

Because of these connections to the Planck constant, a watt balance can measure *h* when the mass is exactly known (as in the case of a 1 kg standard), or it can measure an unknown mass if *h* is exactly known. The impending redefinition of the kilogram will assign a specific fixed value to *h*, allowing watt balances to measure mass without recourse to the IPK or any physical object.

* Actually, neither* IV *nor *mgν *is measured *directly *in either operational mode. The watt balance thus makes a *virtual* comparison between electrical and mechanical power.