Stabilization of an electromagnetically controlled oscillator

We discuss a stabilization problem that arises from the problem of forming an optical aperture by deforming a flexible membrane. The basic idea is that a planar circular membrane with uniform radial in-plane tension subjected to a uniform normal stress assumes a paraboloidal shape, which is very close to the ideal shape for a primary mirror. We can generate the uniform out-of-plane load by applying an electrostatic potential between the conducting membrane and a plate electrode that is parallel to the undeformed configuration of the membrane. The difficulty arises from the fact that if the membrane is deformed so that the gap between the membrane and electrode becomes sufficiently small, then the equilibrium position of the membrane is unstable. This equilibrium position is desirable for two reasons. First, the voltage needed to maintain this equilibrium is a relatively small value, in spite of the fact that the membrane deformations are large. And, second, the large curvature of the membrane produces a small f number and thus a small focal length, that is, a smaller distance to the subreflector. The drawback, of course, is the need for active control to achieve stabilization. In this paper we describe a related experiment designed to investigate the problem described above. This experiment, which involves a mass, spring, and electromagnet, is essentially a lumped-parameter idealization of the membrane problem. This idealized problem is classical in nature and has been considered in studies involving magnetic levitation. The interesting aspect of this problem is the fact that the equilibrium position of the mass is unstable if the gap between the mass and the electromagnet is less than two-thirds of the open-loop equilibrium gap