In this paper the effect of parametric amplification in MEMS-based air-flow hairsensors is studied. With an AC-voltage controlled torsional stiffness the rotation of the hair can be influenced. With the appropriate amplitude, phase and frequency, the rotation of the torsional hair system is increased with respect to the case without parametric amplification. Therefore, parametric amplification is identified as a method to improve the performance of MEMS-based hair air flow sensors

Droogendijk, H.

Krijnen, G.J.M.

English

Krijnen, Gijsbertus J.M.

NARCIS 

Simulation Studies of Parametric Amplification in Bio-inspired Flow Sensors

University of Twente Research Information

Back to the ProgrammeSIMULATION STUDIES OF PARAMETRIC AMPLIFICATION IN BIO-INSPIRED FLOW SENSORSH. Droogendijk1 and G. J. M. Krijnen11University of Twente, MESA+ Research Institute; P.O. Box 217, 7500 AE, Enschede, The NetherlandsAbstract — In this paper the effect of parametricamplification in MEMS-based air-flow hair-sensors is studied. With an AC-voltage controlledtorsional stiffness the rotation of the hair can beinfluenced. With the appropriate amplitude, phaseand frequency, the rotation of the torsional hairsystem is increased with respect to the case withoutparametric amplification. Therefore, parametricamplification is identified as a method to improvethe performance of MEMS-based hair air flowsensors.Keywords: Cricket hair, Bio-inspired, Flow sensor,Non-linear effects, Parametric amplificationI – IntroductionInspired by crickets (figure 1) and their exquisiteperception of flow phenomena down to thermal noiselevels [1], artificial hair-based flow sensors have beendeveloped successfully in our group [2]. The realiza-tion of array structures and improvement of fabricationmethodologies have led to better performance, makingit possible to measure (illustrated in figure 2) sub-mm/sflow velocities [3]. To further improve the performanceof these artificial hair flow sensors, we propose to makeuse of non-linear effects. In nature a wide range of sucheffects exist (filtering, parametric amplification, etc.)giving improvements in sensitivity, dynamic range andselectivity.Figure 1: Flow perception by crickets (SEM pictures courtesyof Jérôme Casas, Université de Tours).II – Parametric amplificationParametric amplification (PA) is based on a nonlinearresponse of a material or structure to excitation. Dueto this nonlinearity the simultaneous presence of morethan one excitation signal will in general lead to a com-Figure 2: Performing flow measurements with artificial hair-sensors.plex interaction where amplitude, frequency and phaseof each of the excitatory signals play important roles inthe entanglement of the signals and the overall response.In the case of our capacitive hair-based flow sensors,which are in fact electrostatic transducers, we have thefreedom to apply electrostatic torques in addition to theflow driven torques. The former causes the flow-sensingcapabilities of the sensor to be modulated and this canbe considered as adaptation of the sensor performance.By controlling the mechanical properties of the hair sen-sory system in time an interesting dynamical system canbe obtained. It is shown that with the appropriate choiceof parameters the input is amplified by the system com-pared to the case without parametric amplification [4].Generally, with a well-defined configuration one canachieve filtering and selective gain of the system.Air flowT (t)JR SyLθ(t)Figure 3: Model of a flow sensing hair [5].MME2010 Workshop 149Back to the ProgrammeTo investigate how this principle can be used inour bio-inspired hair sensor (figure 3), we consider thesecond-order differential equation describing its behav-ior, where J is the moment of inertia, R the torsionalresistance, S the torsional stiffness, T the drag torqueamplitude due to oscillating air flow with angular fre-quency ωa (=2π fa):Jd2θdt2+ Rdθdt+ S (t)θ = T0 cos(ωat) (1)Va cos(ωat)Up cos(ωpt + θp) Up cos(ωpt + θp)Figure 4: Modulating the torsional spring stiffness in time.Note that J and R may contain contributions due to airflow viscous and inertial effects. Normally the torsionalstiffness is given by a spring constant. But it may wellbe modulated in time applying an AC voltage on thetwo capacitor plates of the sensors. Assume this signalhas amplitude Up, frequency ωp (=2π fp) and phase φp(illustrated in figure 4):S (t) = S 0 − 14U2pd2Cdθ2− 14U2p cos(2ωpt+ 2φp)d2Cdθ2(2)θg2L wxFigure 5: Tilted rectangular capacitor.The capacitance C is calculated from the sensorgeometry, which is considered as a tilted rectangularplate (figure 5). For this configuration, 2L is the lengthof the membrane, w is the width of the membrane, gis the distance between the (rotational) center of themembrane and the substrate and θ is the rotation angle.The capacitance C(θ) as function of the rotation angleis:C(θ) =∫ L−Lε0w cos(θ)g − x sin(θ)dx (3)Approximation of its solution by a second orderTaylor expansion leads to:C(θ) ≈ ε0w2Lg(1 +16(2L2g2− 1)θ2)+ O(θ4) (4)where the first and third order in θ drop becauseof symmetry of the capacitor geometry. Using thisapproximation, the value for d2Cdθ2 in equation 2 can becalculated for small values of the rotation angle θ.An analytical solution for equation 1 with S (t) givenby equation 2 is hard to obtain. According to [6] thesolution of this differential equation contains an infi-nite series of odd harmonics with respect to the pumpfrequency ωp. Therefore, we use the Runge-Kutta 4method from MATLAB to solve the differential equa-tion numerically. In order to use this numerical method,the differential equation is converted to a set of linearfirst-order differential equations in state space:[α̇1(t)α̇2(t)]=[0 1− S (t)J −RJ] [α1(t)α2(t)]+[0T0J]u(t) (5)whereθ(t) =[1 0] [α1(t)α2(t)]u(t) = cos(ωat) (6)In these equations, α1(t) represents the rotation angleof the hair, α2(t) = α̇1(t) is the angular velocity of thehair, α̇2(t) = α̈1(t) is the acceleration of the hair and theoutput θ(t) is the rotation angle of the hair.To determine the occurrence of parametric amplifi-cation experimentally the pump parameters need to bechosen judiciously. The hair sensor is suspended bytwo silicon nitride beams ideally allowing for torsionalmovement only. However, in practice the beams alsohave a vertical compliance due to finite stiffness in thatdirection, which is illustrated in figure 6.Torsional VerticalFigure 6: Two types of stiffness.Since parametric amplification is implementedby applying voltages, leading to vertical forces andtorques, the entire membrane structure also moves in thevertical direction. Using the Fourier transform methodto determine the effect of parametric amplificationon the gain at the flow frequency component ωa inthe resulting spectrum, contributions from verticalmovement (as a consequence from finite verticalstiffness) are undesired and should be minimized,especially at that frequency component.III – Simulation resultsTo perform the simulations in MATLAB, the data forthe MEMS-based hair flow sensor from tabel 1 is used.MME2010 Workshop 150Back to the ProgrammeTable 1: Properties of a MEMS hair sensor.Quantity Symbol Value UnitTorsional stiffness S 5.2 · 10-9 N ·m/radTorsional resistance R 1.68 · 10-12 N ·m/rad/sMoment of inertia J 1.97 · 10-16 kg ·m2Capacitor gap g 600 nmCapacitor width w 90 µmCapacitor length L 95 µmThe initial values for the input air flow parametersand the AC voltage (which is denoted ”pump”) param-eters, are listed in table 2. The drag torque calculationsfor T0 are based on 40 mm/s flow using Stokes’ me-chanical impedance for oscillating air flow [5].Table 2: Simulation parameters.Quantity Symbol Value UnitDrag torque amplitude T0 4.69 · 10-13 N ·mFlow frequency fa 80 HzPump amplitude Up 1 VPump angle φp 0 degreePump frequency fp 80 HzThe pump frequency fp is chosen in such a waythat the output signal does not contain a component ofvertical movement at the air flow frequency fa, usingthe relationship for the electrically generated force Fpin the vertical direction y:Fp =12U2dCdy=U2p4dCdy(1 + cos(2ωpt + 2φp))(7)To state the effect of parametric amplification on thesensory system performance, the gain G is defined asthe ratio of the amplitudes of the angular rotation withand without pump:G =θ0,with pumpθ0,without pump(8)With the simulation parameters given, first the effectof the pump phase φp with respect to the air flowfrequency on the gain G is determined. The resultingplot is given in figure 7.The effect of changing the pump phase φp can beseen clearly, resulting in a gain varying from about 1.12to 1.44. The sinusoidal shape can be explained as aspring that becomes weaker when the torque is largecompared to the case that the spring is weak when thetorque is small. Note that the case without parametricamplification has a gain of 1 (Up = 0).Also, the pump amplitude Up affects the strength ofparametric amplification, which is also the case whenapplying electrostatic spring softening by a DC voltage.0 9 0 1 8 0 2 7 0 3 6 01 . 01 . 11 . 21 . 31 . 41 . 5GainP h a s e  ( d e g r e e )G a i n  v s .  p u m p  p h a s eFigure 7: Gain vs. pump phase φp.0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 01 . 01 . 11 . 21 . 31 . 41 . 5GainP u m p  a m p l i t u d e  ( V )G a i n  v s .  p u m p  a m p l i t u d eFigure 8: Gain vs. pump amplitude Up.The relationship between the pump amplitude Up andthe resulting gain G is given in figure 8 for φp = 0.In this plot a non-linear relation between the pumpamplitude Up and gain G is observed, where a gain ofabout 1.44 is obtained at a pump amplitude of 1 V. Thepump amplitude cannot be increased too much, becausepull-in occurs when the voltage between the capacitorplates becomes too high.Parametric amplification can lead to selective gainand thus additional filtering of the system, which is de-termined by investigating the relationship between thepump frequency fp and the resulting gain G (figure 9).From this plot we observe that at pumping at halfthe air flow frequency fa the gain is increased a little(from 1.29 to 1.31) with respect to pump frequenciesfp in the same region, but especially when pumpingwith the same frequency fp as the air flow frequencyfa (from 1.29 to 1.44, which is about 12%). Fromequation 2 this means that varying the torsional stiffnessS (t) with twice the frequency of the incoming air flow( fp= fa) leads to significant gain G. Other frequenciesMME2010 Workshop 151Back to the Programme1 0 1 0 0 1 0 0 01 . 01 . 11 . 21 . 31 . 41 . 5H a l f  a i r  f l o w  f r e q u e n c yGainP u m p  f r e q u e n c y  ( H z )G a i n  v s .  p u m p  f r e q u e n c yA i r  f l o w  f r e q u e n c yFigure 9: Gain vs. pump frequency fp.also show a gain larger than 1, as a consequence ofnon-synchronized spring softening and the presence ofthe DC-component in the voltage squared term (seeequation 2).1 0 1 0 0 1 0 0 0F F T  s p e c t r u m  f o r  P a r a m e t r i c  A m p l i f i c a t i o nMagnitudeF r e q u e n c y  ( H z ) W i t h  p u m p W i t h o u t  p u m pFigure 10: FFT spectrum of the rotation angle with (red curve)and without (blue curve) presence of a pump signal.By inspection of the angular rotation spectrumin the presence of the pump signal, we observe theoccurrence of odd harmonics in the solution (figure 10).Therefore, when performing measurements the gaindue to parametric amplification should be determinedbased on the strength of the flow frequency componentby means of a Fourier Transform rather than using theRMS-value for the entire spectrum (also containingundesired vertical movement).IV – ConclusionsBy exploiting the non-linearity of our artificial hairsensory system for measuring oscillating air flow, theperformance of these sensors can be further improved.Parametric amplification is identified to be a usefulmechanism for improving the flow perception andfor allowing filtering by selective gain of the sensorsystem, using appropriate pump amplitude, phase andfrequency.References[1] Tateo Shimozawa, Jun Murakami, and TsenekoKumagai. Sensors and Sensing in Biology andEngineering, chapter 10. Cricket Wind Receptors:Thermal Noise for the Highest Sensitivity Known,pages 145–156. Springer, Wien, Austria, 2003.[2] M. A. Dijkstra, J. J. J. van Baar, R. J. Wiegerink,T. S. J. Lammerink, J. H. de Boer, and G. J. M.Krijnen. Artificial sensory hairs based on the flowsensitive receptor hairs of crickets. Journal ofmicromechanics and microengineering, 15:S132–S138, July 2005.[3] C. M. Bruinink, R. K. Jaganatharaja, M. J. de Boer,J. W. Berenschot, M. L. Kolster, T. S. J. Lammerink,R. J. Wiegerink, and G. J. M. Krijnen. Advance-ments in technology and design of biomimetic flow-sensor arrays. In 22nd IEEE International Confer-ence on Micro Electro Mechanical Systems (MEMS2009), Sorrento, Italy, number CFP09MEM-USB,pages 152–155, Piscataway, January 2009. IEEEComputer Society Press.[4] Dustin W. Carr, Stephane Evoy, Lidija Sekaric,H. G. Craighead, and J. M. Parpia. Parametricamplification in a torsional microresonator. Ap-plied Physics Letters, 77(10):1545–1547, Septem-ber 2000.[5] T. Shimozawa, T. Kumagai, and Y. Baba. Structuralscaling and functional design of the cercal wind-receptor hairs of cricket. J. of Comp. Physiol. A,183:171–186, 1998.[6] G. Floquet. Sur les équations différentielleslinéaires à coefficients périodiques. Annales scien-tifiques de l’É.N.S., 12:47–88, 1883.MME2010 Workshop 152

Advancementsintechnologyanddesignofbiomimeticﬂowsensor arrays.

Artiﬁcial sensory hairs based on the ﬂow sensitive receptor hairs of crickets. Journal of micromechanics and microengineering, 15:S132– S138,

Parametric ampliﬁcation in a torsional microresonator.

Sensors and Sensing in Biology and Engineering, chapter 10. Cricket Wind Receptors: Thermal Noise for the Highest Sensitivity Known,

Structural scaling and functional design of the cercal windreceptor hairs of cricket.

Sur les ´ equations diff´ erentielles lin´ eaires ` a coefﬁcients p´ eriodiques.

http://purl.utwente.nl/publications/74772

Simulation studies of parametric amplification in bio-inspired flow sensors

Simulation studies of parametric amplification in bio-inspired flow sensors

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