## Peformance-On-Demand MEMS

Due to process variations, no two devices have been able to behave identically. The goal of PODMEMS (Performance On Demand MEMS) is to not only allow MEMS to correct for process variations and quadrature error, but also to increase the utility of MEMS by enabling MEMS to extend their behavior beyond conventional mechanical limits. This is done by imparting active electrical forces onto MEMS that mimic additional mechanical forces. In doing so, MEMS will be able to modify their apparent mass, damping, and stiffness, whereby changing their performance on demand.

[Top] PODMEMS circuit. To convert a MEMS device to a PODMEMS device, the real-time displacement sensing electronics of the device need to be processed and fed back to the driving electronics of the device. This displacement signal processing creates feedback forces that are proportional to displacement (i.e. stiffness), proportional to velocity (i.e. damping), and proportional to acceleration (i.e. mass).

That is, the conventional equation of motion for MEMS is  Mx'' + Dx' + Kx = F,  where M, D, and K are the effective mass, damping, and stiffness that are easily measured by EMM (Electro Micro Metrology).

However, the equation of motion for PODMEMS is  (M+Me)x'' + (D+De)x' + (K+Ke)x = F,  where Me, De, and Ke are electrically generated forces. They can be positive or negative, large or small, to counter process variations, quadrature error, harsh temperature variations, damping, etc.

In the above equations, F is the sum of all other forces, including those from inertial frames of reference. And x, x', and x'' are displacement, velocity, and acceleration of the MEMS.

[Bottom-Left] Illustration of a wafer consisting of exact copies of a 30kHz MEMS resonator. However, upon fabrication, each MEMS device has a slightly different resonance frequency due to process variation. They resonate near, but not, at 30kHz.

[Bottom-Right] Illustration of a multitude of PODMEMS achieving the desired resonance of 30kHz, in which they have corrected for process variations. In addition, the PODMEMS are able to resonate at other frequencies, such as 55kHz, if necessary.

Example of how PODMEMS can stabilize resonators in the face of fluctuating temperatures. The performance of MEMS is often subject to fluctuations in temperature, where MEMS geometry expands due increasing temperature. Likewise, operating electronics is also subject to changes in temperature. Semiconductor resistances decrease with temperature, and capacitances increase with temperature.

[Left] This is a plot of resonance frequency of a MEMS as a function of temperature. The 170K sweep in temperature if accompanied by a 0.49krad/s shift in resonance.

[Right] With the PODMEMS circuit turned on to counter the change in temperature, the conventional frequency shift is reduced by a factor of 5400. These simulations have taken into account thermal expansion of structural materials, thermal effect on electronic components, and thermally induced noise.

This plot shows the transient DDE (delay differential equation) simulation of PODMEMS subject to various feedback dynamics over 7ms. The plot shows the applied and feedback forces, as well as the displacement response. The effective mass, damping, and stiffness change every 1ms as the PODMEMS goes from being underdamped, to being over-damped, to changing its resonance frequency.

Throughout the simulation, a noise disturbance force is always active (see magnification inset near t = 0 ms). Its induced vibration on the trajectory is shown within magnified inset. Noise is modeled because it is always present in real systems. Also throughout the simulation, a 0.5ms step pulse repeats every  1ms for excitation.

At t=0ms, PODMEMS feedback is inactive, so the mass M, damping D, and stiffness K are 100% mechanical. At t=2ms, the apparent mass is reduced by 1/4, which doubles resonance. This is accomplished by feeding back an electrical mass that is Me=-3M/4. At t=3ms, the apparent stiffness is reduced by 1/4 by feeding back Ke=-3K/4, which halves resonance. At t=4ms, stiffness is increased by 4, where Ke=3K, which doubles resonance. At t=5ms, damping is set to critical. And at t=6ms, damping is set to a tenth of critical.

The MEMS structure used in this simulation is shown below, where the electronic circuits of the sense, bias, drive and feedback is the analog PODMEMS circuit shown at the top of the page.