An Alternative Approach to Mass Metrology

In anticipation of the redefinition of the kilogram and to avoid a parallel non-SI dissemination system for mass, an alternative approach to mass measurement is being developed at NIST. Magnetic levitation is utilized to create a mass comparison system in which a test mass artifact in air can be directly compared to a standard mass artifact in vacuum using the same high precision comparator balance. Due to the extreme sensitivity of this comparator balance (in the order of 10 μg), any change in temperature near the balance may affect measurement results. Therefore, permanent magnets will be used in this project to achieve the levitation. A solenoid is added to the system to stabilize the magnetic levitation. To test the feasibility of this project, a levitation system has been constructed as shown in Figure 1.

This is a photograph of the proof-of-concept apparatus. For simplicity, the proof-of-concept apparatus was constructed with both masses in air at atmospheric pressure. Two attractive permanent magnets are used to make the levitation. A solenoid is added to the system to stabilize the magnetic levitation.

A close-up of the levitation parts in the proof-of-concept apparatus is shown in Figure 2.

The is a photograph of the close-up of the levitation parts in the proof-of-concept apparatus. The vertical position signal is provided by a "shadow sensor" photodiode that detects a laser beam that just skims the top of the magnetic pole that is attached to the levitated assembly.

A model of the system with permanent magnetic levitation and electromagnetic control was derived. Using the derived model, a feedback control was applied to our proof-of-concept apparatus. The purpose of modeling and controlling such a system is to have an automatic levitation system with a steady balance reading. That is, when the controller is initiated, the levitated body will automatically move from its resting position to a stable levitated position. Furthermore, this controlled system will oscillate at a higher frequency than the balance and with an amplitude low enough that the balance will filter the oscillation and will be able to produce a steady mass measurement result.

According to Earnshaw's theorem, stable magnetic levitation cannot be achieved by using only permanent magnets. It is necessary to add an active feedback control to regulate the perturbation affecting the system. When the controller is initiated, the vertical position signal V_actual is servo controlled about a desired equilibrium position corresponding to an output V_desired of the shadow sensor. That is, based on the difference of V_actual to V_desired, the controller provides an input V₀ for the current power supply to maintain stable levitation. The current power supply provides a current i to the solenoid based on the input voltage V₀. The magnetic field produced by the solenoid exerts an attractive force on the levitated body and raises it to position x. The shadow sensor detects this position and provides an output in the form of a voltage V_actual. The block diagram of this system in an open loop with input V₀ and output V_actual is shown in Figure

Block Diagram of the Open Loop System.
We modeled different components of this system separately. Using the first order Taylor series expansion of the equation of motion, plant model was derived to be a second order linear differential equation. Power supply and shadow sensor models are both determined experimentally to be linear. Using these models, a PID (proportional, integral, and derivative position loop) controller was established. This controller was then applied digitally to the levitation apparatus to produce an automatic stable levitation with a vertical position oscillation of 3.4 μm around the desired levitation distance. On this balance of 1 mg accuracy, steady measurement is achieved using the proposed controller.

For the construction of NIST's next generation magnetic levitation balance with bigger mass and stronger magnets, computer experiments were run on a surrogate-FEA computational black boxes MagNet so as to gain insight into dominant factors as well as deduce optimal settings. We utilized important functionalities like scripting, parameterization and interoperability of MagNet to conduct computer experiments efficiently. Verification and Validation were performed to checks that MagNet meets specifications and that it fulfills its intended purpose. We used factorial designs to study multiple factors simultaneous to optimize the magnetic force gradient and the distance between magnetic poles. The factors studied include coercivity of the magnets, the shape of the magnetic poles and the current for the solenoid.

The main limitation in our result comes from the noise level of the shadow sensor. An improvement of this sensor will give us the accuracy needed to test other control theory to determine the optimal control for our application. We will apply this work to bigger mass and stronger magnets in vacuum and air when the next generation magnetic levitation balance is ready.