The Effect of High Order Non-linearities on Sub-harmonic Excitation with Parallel Plate Capacitive Actuators

Electrostatic actuation of motion is commonly used in resonant MEMS. Drive signal feed-through is undesirable as it masks the detection signal. This paper reports analysis and demonstration of a resonant motion excitation scheme, which uses a combination of a DC bias with a sinusoidal AC voltage, frequency of which is twice of the mechanical resonant frequency. This configuration is experimentally demonstrated to excite a gyroscope with a parallel plate drive capacitor into high amplitude periodic vibrations. The feed-through has a frequency higher than the main motional harmonic and thus can be eliminated by simple low pass filtering. Full nonlinear dynamics along with several higher-order approximations are considered. Analysis of the effect of the approximation order on the frequency response accuracy shows that the complete nonlinear equation should be used for modeling of high amplitude actuation. Two distinct types of frequency responses are examined as functions of driving AC and DC voltages and damping. INTRODUCTION Several important classes of MEMS devices, such as resonators [1], gyroscopes [2], and chemical sensors [3], rely on resonance phenomenon in their operation. In these devices, resonant motion needs to be actuated, detected, and controlled. Capacitive phenomena are commonly used for transduction in vi∗Address all correspondence to this author. 1 bratory MEMS devices due to the ease of fabrication, low sensitivity to temperature changes, and other practical advantages [4], [5]. Conventional electrostatic actuation schemes use a combination of DC and AC voltages to excite mechanical motion at the resonant frequency of the structure. This is typically achieved by matching the frequency of the drive AC voltage to the resonant frequency of the device. In this case, the amplitude of motion is dictated by the amplitudes of the driving voltages and the quality factor of the mechanical resonator. For this type of actuation, the electrical drive feed-through from drive to sense electrodes caused by parasitic capacitance has the same frequency content as the mechanical motion. Thus, the motional signal is masked by the feed-through, and intricate detection schemes, such as electromechanical amplitude modulation [6], are required to extract and measure the mechanical response. These factors increase the overall system complexity and limit the performance. Nonlinearity of parallel plate capacitors presents design opportunities for introducing new electrostatic actuation schemes. Issues of large amplitude parallel plate actuation using a conventional combination of DC and AC voltage were extensively studied, for example in [7–9]. Parametric excitation phenomenon in MEMS was reported in [10]. In its simplest form, parametric excitation using electrostatic parallel plate force is described by a classical linear time variant (LTV) Mathieu equation. Dynamics of a clampedclamped beam under parametric excitation was studied in [11] Copyright c © 2007 by ASME using small amplitude assumption. The effect of an additional third order term was studied in [12]. An interesting parametrically enhanced harmonic actuation scheme using two separate sets of capacitors and AC voltages was proposed in [13]. Most of the relevant literature studies parallel plate parametric effects using approximate equations of motion. The focus of this paper is actuation of high amplitude vibrations in capacitive MEMS using a combination of a DC bias with a single AC sinusoidal voltage, frequency of which is twice of the mechanical resonant frequency. In Section 1 we introduce a general electro-mechanical model of a MEMS resonator with parallel plate actuation and derive the equations of motion. The phenomenon of high amplitude parametric excitation is demonstrated experimentally in Section 2 using a prototype of a capacitive MEMS gyroscope. In Section 3 we define two different norms of solutions used for frequency response analysis for the nonlinear system. Section 4 presents a comparative numerical study of system dynamics using the complete nonlinear model and its several finite order approximations. Section 5 summarized the obtained results. 1 PARAMETRIC EXCITATION Figure 1 shows a general schematic of a capacitive microresonator, a basic element of various micro-sensors. The electromechanical diagram includes the mechanical resonator and the electrostatic drive and sense electrodes. Mass m of the resonator is suspended by a spring k and is constrained to move only along the horizontal x-axis. The variable sense capacitance is defined as Cs(x), and the drive capacitance as Cd(x), where x is the displacement. Typically in MEMS devices, the drive and sense terminals are not completely insulated, but are electrically coupled by stray parasitic capacitors and resistors [6,14,15]. In this paper we assume, without loss of generality, that the parasitic circuit consists of a single lumped capacitor Cp. An AC driving voltage Vd(t) = vd sin(ωdt) is applied to the drive electrode, and a DC bias Vdc is applied to the mass (voltage values are referenced with respect to a common ground). The sense capacitor Cs is formed between the mobile mass and the fixed sense electrode. The sense electrode is connected to the inverting input of an operational amplifier which is configured as a trans-impedance amplifier, [16]. Due to motion the sense capacitance Cs(x) changes, causing a flow of motional current Is = d(CsVs) dt , where Vs is the sensing voltage across the sense capacitor. The total pick-up current I(t) = Is(t)+Ip(t) consists of both the motional and the parasitic feed-through currents and is converted to the final output voltage V (t) with trans-impedance gain −R. Parasitic current is induced by the drive voltage Vd and therefore has the same frequency ωd . The total sensing voltage is the DC bias Vdc. According to the laws of electrostatics, the total pick-up current that flows through the feed-back resistor of the trans-impedance amplifier is I(t) = d dt [Vd(t)Cp +VdcCs(t)]. Drive Sense