Analog Integrated Circuits and Signal Processing, 37, 17–33, 2003 c  2003 Kluwer Academic Publishers. Manufactured in The Netherlands. First Harmonic (2f ) Characterisation of Resonant Frequency and Q-Factor of Micromechanical Transducers V.J. LOGEESWARAN, 1, ∗ F.E.H. TAY, 1 M.L. CHAN, 1 F.S. CHAU 1 AND Y.C. LIANG 2 1 Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Tel.: 065-68744567, Fax: 065-67791459 2 Department of Electrical & Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 E-mail: mpelv@nus.edu.sg Received May 31, 2002; Revised September 13, 2002; Accepted December 31, 2002 Abstract. In this paper, the response to the first harmonic component (2 f ) of the electrostatic force in single terminal driven electrostatic comb-drive and parallel-plate drive was used as a signal to extract device parameters, namely, the Q-factor and resonant frequency instead of the fundamental (1 f ) resonance response. It is shown that the difficulty in motional measurement due to electrical cross-talk (parasitics) using 1 f measurement can be overcome with a higher signal-to-noise ratio of the 2 f signal. Both atmospheric (low-Q) and reduced pressure environment were investigated using off-chip electronics and lock-in amplifier. The measurements were done on the electrostatic comb-drive and capacitive parallel plate sensing plates that form the two core modules of a yaw rate sensor (dual-axis resonator). The effects of AC and DC bias voltages on the measured response have been investigated. Experimental amplitude and phase response data have been analysed using the Lorentzian curve-fit, Resonance Curve Area (RCA) method, the half-power bandwidth method (3 dB) and the Nyquist plot for data fitting and determination of the Q-factor and resonance frequency. Key Words: electrostatic drive, first harmonic (2 f ), Lorentzian curve-fitting, lock-in technique, resonance curve area (RCA), experimental modal analysis 1. Introduction A comprehensive characterisation of the mechanical and electrical parameters is required at the wafer level (die-level) for vibrating (not necessarily operating in resonance) micromechanical structures for design ver- ification and subsequently for integration (hybrid) with the discrete electronics or ASIC’s. Consequently wafer level testing proves to be critical in determining known- good-dies (KNG) prior to package level testing. For most vibrating microstructures like accelerometers and gyroscopes, either discrete or continuous, the system parameters, namely masses, m j , damping constants, c j , spring constants, k j and resonant frequencies, ω j need to be identified. Determination of these parameters at wafer level (die-level) is done using the frequency re- sponse function (FRF) and/or transient method. The input (actuation) and output (detection) of the device can be optical, e.g., laser and infrared source [1–5] ∗ Corresponding author. and/or electrical, e.g., electrostatic, electrothermal and piezoelectric. For wafer level probing, the input force has to be directly applied to the mass. Hence internal input actuation (e.g. electrostatic pulse) and detection (e.g. capacitive) has been the practice. Major drawbacks of this application especially in bulk micromachined devices are the interference of the system parameters (e.g. spring softening) and con- siderable cross-talk between the actuation and detec- tion terminal. Therefore accurate parameter determi- nation in the presence of electrical and mechanical noise is a challenge. Larmer et al. [6] have discussed the importance of this cross-talk in the electrical pick- up of piezoelectric, electrostatic-capacitive (1 f ) and electrothermal-piezoresistive sensors. W.C. Tang [7] suggested the use of a carrier modulation method and frequency doubling (2 f ) to detect the small motional current which was further expanded by C.T.C Nguyen [8] but neither presented any experimental data on the 2 f measurement. Yao et al. [9] presented experimental results on SCREAM resonators measured in vacuum