On Characterizing Microelectromechanical Processes Using Coupled Resonators

In this paper, a simple method of measuring process-induced variations is proposed using a chain of nominally identical microelectromechanical resonators. The method is based upon the fact that the n eigenfrequencies of a chain of mechanically coupled resonators can be determined from the response of only one resonator in the chain. However, these n values do not provide enough information to determine the 2n - 1 elements of the system matrix. The extra information needed to obtain the system matrix is therefore obtained by perturbing the characteristics of one of the resonators and measuring the resulting eigenfrequencies. A resonator whose effective spring constant can be perturbed by varying an applied voltage has therefore been used to validate the proposed method. The validity of the proposed method is then demonstrated in several different ways. First, the extracted system matrix is used to predict the effects of perturbations to one or more of the resonators. Second, the eigenvectors of the system matrix are shown to correspond to the measured eigenmodes of the system. Finally, it is shown that, as expected, a change to one resonator only changes the corresponding diagonal element of the system matrix. Most importantly, this test shows that the method can determine the critical diagonal elements of a system matrix to an accuracy of 0.1%.