October 2006 By Florian Bousquet TFE 06 – ASICs for MEMs 3 rd order intermodulation in Micromechanical resonators

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 2 Outline I.Introduction II.Background III.1 st Order Formulation for IIP 3 IV.Complete Formulation for IIP 3 V.Experimental results VI.Conclusion

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 3 I. Introduction Nowadays: cellular and cordless phone applications MUST satisfy strict specifications for LINEARITY µmechanical resonator technology to RF communication circuits have been delayed so far due to this linearity µmechanical signal processors possess sufficient linearity for such applications? Need to find a complete analytical formulation for the IIP 3 Paper deals with a method which is computing IIP 3 of capacitively driven CC-beam µmechanical resonators

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 4 II. Background 1) Third Intermodulation distortion (IM3) IM 3 for a frequency filter occurs when system nonlinearities allow out-of- band signal components (tones) to generate an in-band component S IM3 back at w 0 (in-band frequency) Quantitatively (in & out) : Finally the transfer function:

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 5 II. Background Common case: interferers located at frequencies Δ w and 2 Δw from the fundamental the quantity (2w 1 -w 2 ) will be equal to w 0 Possibly masking fundamental w 0 Even if Interfering tones outside the filter passband: Still generate an in-band response: highly undesirable The 3rd-order nonlinear term MUST be constrained below a minimum acceptable value. Most useful metrics to gauge the ability of a system to suppress IM 3 distortion : 3 rd -order input intercept point IIP 3

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 6 III. 1 st Order Formulation for IIP 3 Principle: Electrode and beam : 2 plates of the transducer capacitor C(x) DC-bias V p + AC v i : drive the beam into vibration. not mechanical nonlinearity that governs the degree of IM 3 : nonlinearity in the capacitive transducer. Why? : vibrate with amplitudes much smaller than their lengths (e.g., 100 Angstrom amplitude for a 40pm-long beam)

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 7 III. 1 st Order Formulation for IIP 3 Total force acting on the suspended mass under an applied input V p -v i : And with: You finally get:

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 8 III. 1 st Order Formulation for IIP 3 And finally, with the fundamental force, we can find the input voltage magnitude at the IIP 3 : To be noticed: IIP 3 can be increased by reducing V p and A 0 and by increasing d 0 and k reff Actually all modifications that will increase the series motional resistance R x of the resonator Trade off between linearity and resistor size can be found (which for matching purposes often must be small)

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 9 IV. Complete formulation for IIP 3 Final equation is fairly close from the reality, but actually not exact: Neglect of the beam bending due to V p => Gap spacing is a function of y k reff should be distributed rather than lumped Fully modelisation of effects : d(y) and k reff (y) must be used to attain F IM3 and F fund

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 10 V. Experimental results Characteristics of resonator: IIP 3 measurement: using the previous test set-up with Δw =2π (200kHz).

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 11 V. Experimental results

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 12 V. Experimental results Strong dependence of IIP 3 on the initial gap spacing d 0 : it must be accurately known => sufficiently accurate theoretical prediction for comparison with measurement. Plot of frequency f, versus DC-bias V p is measured, from which the value of d0 is measured using results from others papers : curve fitting with a theorical curve

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 13 V. Experimental results Verify accuracy of results :very close agreement between measurement and theory There is an optimum Vp at which the P IIp3 is maximized –3 dBm): Vp increases, R x decreases, hence R Q decreases, leading to an increase in P IIp3. BUT as Vp becomes even larger, the IM3 force also steadily increases, due to both the direct increase in Vp and due to a decrease in d 0 caused by Vp-induced beam bending. This latter effect begins to dominate after some threshold voltage V p, beyond which the P IIp3 decreases with increasing V p

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 14 V. Experimental results Quantify the dependence of IIP 3 on the loaded Q: Obtained Q of the resonator was controlled by adding an R Q resistor in series with the resonator.

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 15 V. Experimental results

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 16 VI. Conclusion Analytical expression for the IIP 3 has been presented and verified experimentally This paper shows us IIP 3 ’s as high as -3 dBm for a 10 MHz CC-beam terminated via an impedance 3X its own series motional resistance. This measured value at 10 MHz easily satisfies GSM receive path requirements, it is still short of the +7.6 dBm needed for RF channel-selection in CDMA handsets

October 2006 Florian BOUSQUET – 3 rd order intermodulation in Micromechanical resonators 17 Questions ?