318 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 22, NO. 2, APRIL 2013
Parametric Excitation, Amplification, and Tuningof MEMS Folded-Beam Comb Drive Oscillator
Kareem Khirallah
Abstract—This paper proposes a new approach for parametricexcitation, parametric amplification, and linear and nonlineartuning of the folded-beam interdigitated comb drive oscillator.The approach is based on adding a new electrode that generateselectrostatic force on the truss and accordingly generates axialstress on the folded beams. Depending on the application, thisforce can be dc force for resonance frequency tuning or ac forcethat is used for parametric excitation or amplification. Parametricexcitation and amplification are utilized to enhance the effectivequality factor of the system; therefore, they are used in sensorswhere the quality factor is a key parameter in determining theirperformance. The major advantages of this oscillator over tra-ditional types of parametrically excited microelectromechanicalsystems (MEMS) oscillators are its ability to suppress nonlinearityand achieve high amplitude of oscillation. Here, the extendedHamilton principle is used to derive the equation of motion ofthe resonator. Then, an approximate solution is introduced usingperturbation method of multiple scales. Finally, the frequencyresponse of the oscillator in four different excitation conditionswhich are resonance frequency tuning, parametric amplification,parametric resonance, and parametric and forced resonance arepresented. The approach presented here has the potential to en-hance the performance of several MEMS sensor and actuatordevices. [2012-0106]
Index Terms—Comb drive (CD), folded suspension, microelec-tromechanical systems (MEMS) resonators, parametric amplifica-tion, parametric excitation, tuning.
I. INTRODUCTION
E LECTROTHERMAL and electrostatic actuation tech-niques are among the most common methods employed to
achieve actuation in microelectromechanical systems (MEMS)devices [1], [2]. Several electrothermal tuning techniques havebeen developed for frequency tuning of MEMS oscillators suchas localized thermal induced stressing of the mechanical beams[3] and resistive heating to introduce thermal strains in thebeams [4]. These techniques, in general, suffer from high powerdissipation. On the other hand, electrostatic actuation remainsan attractive method for actuation and tuning due to virtuallynonexistent current loss, high energy density, and large forcethat can be generated on the microscale [5].
Despite the aforementioned advantages of the electro-static actuators, there are some drawbacks associated with it.
Manuscript received April 22, 2012; revised August 13, 2012; acceptedSeptember 13, 2012. Date of publication October 17, 2012; date of currentversion March 29, 2013. Subject Editor R. T. Howe.
The author is with the Physics Department and the Yousef Jameel Scienceand Technology Research Center, The American University in Cairo, Cairo11511, Egypt (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JMEMS.2012.2221156
Fig. 1. Conceptual model of the folded-beam CD actuator showing voltagesources and pertinent distances. VP is the voltage applied between both theupper and lower truss electrodes and the upper and lower trusses. The black-shaded area is fixed, and the gray-shaded area is moving. Cl is the capacitancegenerated between the fixed and moving combs, and CP is the capacitancegenerated between the truss electrode and the truss.
In the case of transverse actuators, such as the MEMS variableparallel-plate capacitor, the electrostatic force depends on thedisplacement. This behavior is considered advantageous ordisadvantageous, depending on the application. In the case offrequency tunable resonator, it is considered advantageous be-cause the designer can tune the resonance frequency by control-ling the dc bias voltage. However, this type of oscillator suffersfrom the nonlinearity, which makes the resonance frequencydepends on the amplitude of oscillation. This phenomenon iswidely known as spring softening effect [6]. Therefore, theamplitude of oscillation should be limited to small values toavoid the jump phenomena and the pull-in effect [5]. In the caseof oscillators that are used to generate precision frequency stan-dards, it is considered disadvantageous because the resonancefrequency is sensitive to the bias voltage drift [7].
On the other hand, with the case of interdigitated combdrive (CD) actuators, the pull-in effect is greatly suppressed,and the electrostatic force is approximately independent of thedisplacement [8]. Therefore, generally, it is not tuned electro-statically. In this paper, we suggest adding a new truss electrode(see Fig. 1) that generates electrostatic tensile axial force onthe folded beam and accordingly changes its effective spring
KHIRALLAH: PARAMETRIC EXCITATION, AMPLIFICATION, AND TUNING OF MEMS FOLDED-BEAM CD OSCILLATOR 319
Fig. 2. First parametric instability region of the resonator with parameterspresented in Table I, where σ = (Ω− ω)/ω, VP−ac, Ω, and ω are thedetuning parameter, applied parametric ac voltage, dimensionless excitation,and resonance frequency, respectively.
constant. The same voltage is applied to the top and bottomtruss electrodes to generate an equal and opposite force on eachside of the structure. The high axial stiffness of the folded beamenables stable wide tuning range by applying large axial elec-trostatic dc force. Unlike the parallel-plate actuator, this tuningmechanism can be considered unconditionally stable, and itprovides an electrostatic approach for increasing the effectiveresonance frequency which is desirable in many applications.Meanwhile, this technique preserves the independence of theforce displacement relation of the CD actuator; therefore, largeamplitude of oscillation linear condition is attainable. The de-pendence of the generated parallel-plate electrostatic axial forceon the folded-beam vertical displacement rather than lateraldisplacement reduces the system nonlinearity over the parallel-plate actuator case. In addition, for further suppression of thesystem nonlinearity, a folded in-extensional beam configurationis used rather than the extensional beam such as fixed–fixedbeams [9].
In the parametric excitation condition, ac force with largeamplitude to overcome the linear damping of the system andfrequency that is near double the system resonance frequencyis applied [10]. In other words, the parametric excitation con-dition is in the instability region; see Fig. 2. The parametricresonance of mass spring system results from rapidly changingparameter in the system like the effective spring constant. It hasadvantages over the traditional external excitation resonance insome applications like increasing the sensitivity in the scanningprobe microscopes and mass sensors [11] and enhancing theeffective quality factor of the system [12]. In contrast withthe external excitation condition in which it is modeled as aninhomogeneous differential equation, the parametric excitationis modeled as a homogeneous differential equation with rapidtime-varying coefficient [10]. The parametric excitation bymeans of inducing axial acceleration or force on vertical beamsis one of the most used approaches for parametric excitation inthe bulk scale [13]–[15].
The harmonic axial load using Lorentz force on a U-shapedmicro cantilever beam as an approach for parametric excitationwas studied in [16]. This approach suffers from current dissi-pation and the need for a magnet. Recently, the parametric ex-citation of the noninterdigitated CD resonator has been studiedand tested for tunable filtering applications [17]. For reasonable
Fig. 3. Model of the folded beam, where s is the arc length, l is the total beamlength, P is the applied distributed tensile electrostatic force, Mp is the proofmass, F is the CD electrostatic force, u1(s, t) and v1(s, t) are the verticaland lateral displacements of the beam AB from its static position, respectively,and [u2(s, t)− u1(l, t)] and [v2(s, t) + v1(l, t)] are the vertical and lateraldisplacements of the beam CD from its static position, respectively.
operation, the maximum amplitude of oscillation of this deviceis limited by the spacing between the comb fingers due to thehigh nonlinearity nature of this design. The truss electrode pre-sented in this paper can produce time-varying electrostatic axialforce on the folded-beam resonator and consequently modulateits effective spring constant. The design presented here can sup-press the nonlinearity over the noninterdigitated CD parametricexcitation; thus, large amplitude of oscillation is attainable.
In the case of parametric amplification, external harmonicand parametric force are applied. However, in this case, para-metric excitation condition is in the stable region; see Fig. 2.This phenomenon is used to amplify the resonator responseto external excitation force and increase the system effectivequality factor [12]. The parametric amplification was studiedand tested to amplify the coriolis response of the MEMSgyroscopes to enhance the signal-to-noise ratio and the deviceresolution [18].
As a conclusion, the truss electrode is introduced to generateelectrostatic axial force on the folded beams. Depending on theapplication, this axial force can be dc force, relatively small acforce, or relatively large ac force. In the case of tunable filters,the axial force is constant and generated by dc applied voltage.In the case of principle parametric excitation, this force shouldbe relatively large to exceed the damping threshold, and itsfrequency should be near twice the system resonance frequency.In the case of parametric amplification, this force should berelatively small and near twice the system resonance frequency.Finally, simultaneous parametric and forced resonance is dis-cussed and resulted in interesting frequency response and largeamplitude of oscillation.
II. PROBLEM FORMULATION AND
THEORETICAL SOLUTION
We start the analysis by explaining in detail, with the aidof Fig. 3, a conceptual model for folded-beam CD actuator.The gray area is the vibrating part which contains the movingfingers, the folded beams, and the truss. The outer black area is
320 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 22, NO. 2, APRIL 2013
fixed and holds the fixed right and left combs and the upper andlower new proposed truss electrodes. The capacitances Cl andCp are the varying capacitances generated between the fixedand the moving comb and between the fixed new truss electrodeand the truss, respectively. The middle moving proof mass issuspended with the folded suspension configuration. The dis-tances W1, W2, Y0, l, d, x0, t, and h represent the beamwidth,truss width, the distance separating the truss and the new trusselectrode, beam length, the distance separating two adjacentfingers, initial lateral separation between the fixed and movingcombs, the thickness of the fingers, and the thickness of thestructure, respectively. Due to structural symmetry, only onequarter of the structure is considered, as shown in Fig. 3. Thesame voltage is applied on the upper and lower truss electrodesto generate equal and opposite electrostatic forces on the beamsand the proof mass.
Two symmetrical initially straight uniform beams (AB andCD) of constant length l and mass ρ per unit length connectedwith a rigid truss beam [19] on its end with mass Mt are con-sidered, as shown in Fig. 3. Each cross section of the beam ABexperiences an elastic planar displacement of its centroid at anarbitrary location s, along the x and y axes denoted in Fig. 3 byv1(s, t) and u1(s, t), respectively. Similarly, each cross sectionof the beam CD experiences an elastic planar displacement ofits centroid at an arbitrary location s, along the x and y axesdenoted in Fig. 3 by v2(s, t) and u2(s, t), respectively, whereu2(s, t) and v2(s, t) are the relative vertical and horizontal dis-placements of the beam CD with respect to point C, as shown inFig. 3. Assuming complete overlapping area between the trussand the stationary truss electrode during the oscillation, the newelectrodes generate uniform vertical distributed electrostaticforce P on both the upper and lower trusses, given by
P =ε0A (VP−dc + VP−ac cos(Ωt+ ϕP ))
2
2 (Y0 − u1(l))2
≈ ε0A (VP−dc + VP−ac cos(Ωt+ ϕP ))2
2Y 20
×(1 +
2u1(l)
Y0
)(1)
where A = h×W2 is the truss parallel-plate area, VP−dc isthe dc applied voltage, VP−ac is the ac applied voltage withfrequency Ω, and ϕP is the initial phase of the parametricvoltage VP−ac. The dependence of the parametric force on thevertical displacement of the beam rather than the horizontaldisplacement reduces the resonator nonlinearity over theparallel-plate actuator. The electrostatic force P induces axialtensile force on each beam Pa, given by
Pa=P
4=
ε0A
8Y 20
×[(V 2P−dc+V 2
P−ac/2)+
(V 2P−ac
2
)cos(2Ωt+2ϕP )
]
×[1+
2u1(l)
Y0
]= [Pa−dc+2Pa−accos(2Ωt+2ϕP )]
×[1+
(u1(l)+u2(l))
Y0
]. (2)
Note that the first harmonic force is neglected because, in thecases of interest here, either VP−ac = 0 or VP−dc = 0. Theproof mass Mp is attached to the end of the beam CD atpoint D (see Fig. 3), and it is affected by the CD horizontalelectrostatic force denoted by F . The linear and nonlinearelectrostatic forces F [8] to the third order is given by
F ≈ FL
(ei(Ωt+ϕF ) + e−i(Ωt+ϕF )
)+ FNL
((v1(l) + v2(l))
x0+ 2
(v1(l) + v2(l))3
x30
)(3)
where
FL ≈ −ε0h
d
(N − 1)
4VC−dcVC−ac (4)
and
FNL ≈ Nε0t× h
2x20
V 2C−dc (5)
where N, ε0, VC−dc, and VC−ac are the numbers of fingersin one side of the structure, vacuum permittivity, and dc andac applied voltages to the fixed comb with frequency Ω,respectively, and ϕF is the initial phase of the external appliedvoltage of the CD VC−ac.
It is worth mentioning that the aforementioned expressionsfor the linear FL and the nonlinear electrostatic force FNL
neglect the fringing filed and the effect of the potentiallygrounded substrate beneath the structure. For more accuratecalculation of the linear and nonlinear electrostatic forces ofthe CD with arbitrary dimensions, finite-element simulation isrecommended [20], [21]. The nonlinear electrostatic force ofthe CD adds a degree of freedom to the designer to control theeffective nonlinear spring constant and the resonance frequencyof the structure by choosing the parameters t, x0, or VC−dc.Elshurafa et al. [8] studied the effect of the nonlinear electro-static force of the CD on the dynamics of the structure, and theyfound that it has a softening effect.
To derive the equation of the motion of the structure, we getthe Lagrangian of the system L [22], which is given by
L =
l∫0
[ρ
2v21(s)−
EI
2
(v′′21 + v′21 v
′′21
)
− EI
2
(v′′22 + v′22 v
′′22
)+
ρ
2(v1(l) + v2(s))
2
]ds
+
[[Pa−dc + 2Pa−ac cos(2Ωt+ 2ϕP )]
×[(u1 + u2) + (u1 + u2)
2/Y0
]+
1
2Mtv
21
+1
2MP (v1 + v2)
2 + F × (v1 + v2)
]∣∣∣∣s=l
(6)
where v = ∂v/∂t and v′ = ∂v/∂s.Note that the primary resonance of the system, in which
the beams itself do not resonate, is the case of interest here.Therefore, the inertia nonlinearity of the beams and truss isneglected with respect to the geometric nonlinearity of thebeams.
KHIRALLAH: PARAMETRIC EXCITATION, AMPLIFICATION, AND TUNING OF MEMS FOLDED-BEAM CD OSCILLATOR 321
The inextensibility condition of the Euler-Bernoulli beam isgiven by
u1(l) ≈ −1
2
l∫0
v′21 ds u2(l) ≈ −1
2
l∫0
v′22 ds. (7)
Then, after using the extended Hamilton principle and addingthe linear and the nonlinear dissipation, we get the equation ofthe motion.
where D, C, E, and I are the linear damping coefficient,nonlinear damping coefficient, Young’s modulus, and the sec-ond moment of area of the beams, respectively. Note that thedamping of the beam is neglected with respect to the proof massdamping. The multiplication of the parametric force Pa−ac bya nonlinear term complicates the device frequency response, aswill be shown in Section III.
It is now possible to seek an approximate solution for thedisplacement of the beams using the perturbation method ofmultiple scales [10]; see Appendix A for details. The solution
in terms of the nondimensional parameters and variables isgiven by
v1n(sn, tn) ≈ a(−s3n + 1.5s2n
)cos(Ωntn − γ) (10)
and
v2n(sn, tn) ≈ a(−s3n + 1.5s2n
)cos(Ωntn − γ) (11)
where
v1n =v1l
v2n =v2l
sn =s
l
tn =t√ρl4
(EI)
Ωn = Ω
√ρl4
(EI).
We can get a and γ by solving simultaneously this system ofalgebraic equations
In this section, the resonator frequency response is presentedin four different actuation conditions for applications to reso-nance frequency tuning, parametric amplification, parametricexcitation, and parametric and forced resonance, respectively.The effect of some design parameters on the resonator fre-quency response is discussed. The parameters used in thefollowing calculations are summarized in Table I. We get theseparameters from the resonator presented by Nguyen and Howeat the University of California at Berkeley [7], and the missingparameters are assumed. The effect of the nonlinear electro-static force of the CD FNLn on the resonator dynamics is dis-cussed in [8]. Therefore, in the following figures, we assumed
322 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 22, NO. 2, APRIL 2013
TABLE IPARAMETERS USED TO OBTAIN THE PLOTS IN FIGS. 4–15
Fig. 4. Dimensionless resonance frequency shift (detuning parameter) σ =(Ω− ω)/ω versus dc applied voltage VP−dc. Parameter used: Y0 = 0.5 μm.
zero nonlinear electrostatic force of the CD FNLn to focus onthe effect of other design parameters on the resonator dynamics.
A. Resonance Frequency Tuning
This case is characterized by Vdc−P �= 0,Vac−P = 0, andF �= 0, and it is used for resonance frequency tuning of thefolded-beam CD resonator. Solving (12) and (13) simultane-ously, the frequency response in this case is given by
σ=1
2ωMPn
[−0.25ω2Mtn−0.685ω2+1.157a2+0.6Pdcn
− 0.27y0Pdcna2−FNLn
(x0n+1.5x3
0na2)
±
√(2FLn
a
)2
−(ωDn+0.75a2ω3Cn)2
]. (15)
Equation (15) indicates that the axial dc force Pdcn increasesthe resonance frequency of the system at small amplitude. How-ever, it has softening effect at large amplitude of oscillation.The designer can decrease the softening effect by increasing theinitial distance separating the truss and the truss electrode “Y0.”
Fig. 5 shows the dimensionless resonance frequency shift(detuning parameter) σ with different dc voltages VP−dc atsmall-amplitude linear condition. Figs. 5 and 6 show the fre-quency response with different dc voltages VP−dc at largeamplitude. The figures indicate that, while decreasing the ini-tial space between the truss and truss electrode decreases thedc voltage VP−dc required for tuning, the softening effectincreases.
B. Parametric Amplification
This case is characterized by Vdc−P = 0, Vac−P �= 0, andF �= 0. The stability analysis of the trivial solution in the caseof purely parametric excitation shows that the gray region inFig. 2 is unstable and the white region is stable. The bound-ary between the two regions is given by (16), shown at thebottom of the page. In the case of parametric amplification,the parametric excitation condition is in the stable region, andaccordingly, the parametric resonance will not occur. However,the resonator response to the external excitation is amplified;see Fig. 7.
Fig. 7 shows that, as the parametric voltage increases VP−ac,the amplitude of oscillation increases and the frequency re-sponse becomes more hardening. In this case, the designer canadjust the softening effect of the nonlinear electrostatic force ofthe CD FNLn and the dc parametric force Pdcn to compensatethe hardening effect. From (12), this condition is given by
FNLn6x30n + 1.08y0Pdcn − 4.63 = 0. (17)
There is a strong dependence between the parametric ampli-fication gain and the phase difference between the parametricapplied voltage VP−ac and the external applied voltage VC−ac.For more understanding of the effect of the phase difference(ϕP − ϕF ) on the parametric amplification gain and to sim-plify the analysis, we neglect the nonlinear terms in (12) and(13). Moreover, we study the parametric amplification gainat the shifted resonance frequency in which the maximumamplitude of oscillation occurs. In other words, we assume that
2ωMPnσ+0.25ω2Mtn+0.685ω2+FNLnx0n−0.6Pdcn=0.(18)
Substituting (18) into (12) and (13) and neglecting the nonlinearterms, we obtain
a =2FLn cos(γ + ϕF )
0.6Pacn cos(2γ + 2ϕP )(19)
and
a =2FLn sin(γ + ϕF )
0.6Pacn sin(2γ + 2ϕP ) + ωDn. (20)
Note that both (19) and (20) must be satisfied simultaneously.From (20), the maximum amplitude of oscillation amax occurswhen the numerator has maximum value and the dominator hasminimum value. Since, in the parametric amplification at theshifted resonance frequency, 0.6Pacn < ωDn, the maximumamplitude of oscillation occurs when
γ + ϕF = π/2 + nπ (21)
and
2γ + 2ϕP = −π/2 + 2mπ (22)
where n and m are integers. Substituting (21) and (22) into (19),we get undefined amplitude of oscillation amax. This quantitycan be defined using (20). Therefore, both (19) and (20) aresatisfied using (21) and (22), and the maximum amplitude ofoscillation is given by
amax =2FLn
ωDn − 0.6Pacn. (23)
Using (21) and (22), the maximum parametric gain and maxi-mum amplitude of oscillation amax can be achieved when
2ϕP − 2ϕF = −3π/2 + 2k π (24)
where k is an integer. Likewise, from (20), the minimumamplitude of oscillation amin occurs when the numerator hasminimum value and the dominator has maximum value. Sinceboth (19) and (20) must be satisfied, the minimum amplitude ofoscillation amin occurs when
γ + ϕF = π/2 + nπ (25)
and
2γ + 2ϕP = π/2 + 2mπ. (26)
The minimum amplitude of oscillation amin is given by
amin =2FLn
ωDn + 0.6Pacn. (27)
Note that the parametric gain in this case is less than one, andthe oscillation is attenuated.
Using (25) and (26), the minimum parametric gain andminimum amplitude of oscillation amin are achievable when
2ϕP − 2ϕF = −π/2 + 2k π (28)
where k is an integer. Note that the condition for gettingthe maximum or minimum parametric gain given by (24) or(28) is derived based on assuming that the parametric force2Pa−accos(2Ωt+ 2ϕP ) has a cosine time dependence andinitial phase given by 2ϕP . Moreover, from the external force,2FLcos(Ωt+ ϕF ) has a cosine time dependence and initialphase given by ϕF . In the derivation of (24) and (28), if weassumed that the parametric force and the external force havea sine time dependence and initial phase given by 2ϕP andϕF , respectively, the condition for getting the maximum orminimum parametric gain will be different. In other words, ifthe parametric force has the form 2Pa−acsin(2Ωt+ 2ϕP ) andthe external force has the form 2FLsin(Ωt+ ϕF ), the conditionfor the maximum or minimum parametric gain will be differentthan (24) or (28), respectively. To derive this condition, wereplace the initial phase of the parametric force 2ϕP by 2ϕP −π/2 and the initial phase of the external force ϕF by ϕF − π/2in (24) and (28), we get, for maximum parametric gain
2ϕP − 2ϕF = 2k π (29)
324 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 22, NO. 2, APRIL 2013
This strange behavior can be explained by noting that the maxi-mum or minimum parametric gain does not depend on the phasedifference between the parametric force and the external force(2ϕP − ϕF ); however, it depends on the difference betweenthe parametric force phase and double the external force phase(2ϕP − 2ϕF ).
Hu et al. [23] and Harish et al. [12] studied experimentallyand theoretically the parametric amplification and the para-metric damping in MEMS. Their theoretical analysis predictedthat the highest gain is achievable at zero phase differencebetween the parametric force and the external force and thesmallest gain is achievable when the phase difference is 180◦.However, they found a discrepancy between the theory andthe experimental results. The analysis presented here can givereasonable explanation to the source of this discrepancy; seeAppendix B for details.
Fig. 8 shows the frequency response in three different para-metric amplification conditions. The first condition is whenboth initial phases are zero (i.e, 2ϕP − 2ϕF = 0). In this con-dition, an intermediate parametric gain is observed. The secondand third conditions are when the maximum and minimumparametric gain conditions given by (24) and (28) are satisfied,respectively.
C. Parametric Excitation
This case is characterized by Vdc−P = 0, Vac−P �= 0, andVC−ac = 0. In this case, the parametric excitation condition isin the unstable region as shown in Fig. 2, and the external force
Fig. 9. Dimensionless amplitude of oscillation (a) versus detuningparameter σ. Cn = 5, and Y0 = 0.5 μm.
is zero. Using (12) and (13), the frequency response in this caseis given by
σ=1
2ωMPn
⎡⎢⎣− 0.25ω2Mtn − 0.685ω2+1.157a2+0.6Pdcn
− 0.27y0a2Pdcn − FNLn
(x0n+1.5x3
0na2)
+[0.6− 0.36y0a2]
×
√P 2nac −
[ωDn+0.75a2ω3Cn
0.18a2y0 − 0.6
]2 ⎤⎥⎦. (31)
Fig. 9 shows the frequency response for different parametricactuation voltages VP−ac. We note that the bandwidth dependson the applied voltage, which is one of the parametric excitationcharacteristics.
The existence of the nonlinear parametric force in the equa-tion of motion results in complex characteristic of the sys-tem nonlinearity (i.e., hardening or softening). A completedescription of its effect on the dynamics of a purely parametricexcitation in MEM resonator was explained in [24]. For moreunderstanding of the effect of the design parameters on thesystem nonlinearity and to simplify the analysis, we assumedzero damping in (31). In this case, the amplitude of oscillationis given by (32) and (33), shown at the bottom of the page.The sign of the quantity under the square root determines thefrequency range over which the amplitude of oscillation is realvalued and has a physical meaning. Note that the numerator ofthe two solutions can take positive or negative sign by sweepingthe excitation frequency (i.e., detuning parameter σ). Therefore,the sign of the dominator determines the bending of the twosolutions in the frequency response plot, either to the left
KHIRALLAH: PARAMETRIC EXCITATION, AMPLIFICATION, AND TUNING OF MEMS FOLDED-BEAM CD OSCILLATOR 325
Fig. 10. γ3, λ3 parameter space. Regions designated I and III exhibit hard-ening and softening nonlinear characteristics, respectively. Regions designatedIIa and IIb exhibit mixed hardening and softening nonlinear characteristics. Theblue line shows the transition between the response regimes under varying theparametric voltage amplitude VP−ac.
(i.e., softening) or to the right (i.e., hardening) for certain deviceparameters.
We introduce two parameters γ3 and λ3 from the denomina-tor of (32) and (33), where
γ3 = 4.63− 1.08y0Pdcn − 6x30nFNLn (34)
and
λ3 = 1.44y0Pacn. (35)
To describe the effective nonlinearity, we introduce two param-eters η1 and η2, where
η1 = γ3 + λ3 (36)
and
η2 = γ3 − λ3. (37)
The case η1 > 0 and η2 > 0 result in the hardening behavior.Similarly, the case η1 < 0 and η2 < 0 result in the softeningbehavior. However, due to the existence of the nonlinear para-metric term in the equation of motion, mixed nonlinearity canexist (i.e., simultaneous softening and hardening). This case ischaracterized by η1 > 0 and η2 < 0 or η1 < 0 and η2 > 0.
Fig. 10 shows the division of the parameter space of γ3 andλ3 into four regions. When these parameters are located inregion I, the system nonlinearity will be the hardening behavior.Fig. 9 shows the frequency response in this region for differentparametric VP−ac voltages. When these parameters are locatedin region III, the system nonlinearity will be the softeningbehavior. However, when these parameters are located in regionIIa or IIb, the system nonlinearity will be a mixed softeningand hardening response. The blue line in Fig. 10 shows thetransition between the response regimes under varying theparametric voltage amplitude VP−ac. From (34), the designercan adjust the nonlinear electrostatic force of the CD FNLn,which has a softening type, to control the device operatingregion under certain parametric excitation voltage Vp−ac. Forexample, when 6x3
0nFNLn > 4.63, the device will have a soft-ening nonlinearity, and it will operate in region III, as shown inFig. 10, regardless of the amplitude of the parametric excitation
Fig. 11. Frequency response plot showing the amplitude of oscillation versusdetuning parameter (σ) at zero damping. Parameters used: VP−ac = 20 V andthe parameters γ3 and λ3 located in region IIa as shown in Fig. 10.
Fig. 12. Dimensionless amplitude of oscillation (a) versus detuning parame-ter σ. Cn = 5, and Y0 = 0.5 μm.
voltage VP−ac. Fig. 11 shows the frequency response at zerodamping assumption when the parameters γ3 and λ3 are locatedin region IIa. Note that, from (13), we can find a third solutionwith constant amplitude, given by
a3 =
√0.6
0.18y0. (38)
The intersection of the constant amplitude solution with theother two solutions (a1, a2) corresponds to a bifurcation pointat which the stability of these solutions changes [24].
To gain a better understanding of the dynamics of a realdevice, we consider the effect of the damping on the frequencyresponse. Fig. 12 shows the frequency response of the resonatorwhen the parameters γ3 and λ3 are located in regions I, IIa, andIIb by applying corresponding parametric voltages VP−ac thatare equal to 14, 20, and 25 V, respectively. The figure indicatesthat the designer can adjust the parametric voltage VP−ac toavoid the jump phenomenon by locating the parameters γ3 andλ3 in region IIa or IIb.
The effect of the linear and the nonlinear damping on theamplitude of oscillation can be derived from (31). The sign ofthe quantity under the square root determines the amplitude ofoscillation range that can be observed physically. This quantitywill have a negative sign if the amplitude of oscillation islocated in the range given by
2
√0.6Pnac−ωDn
3ω3Cn+0.72y0Pnac≤a≤2
√0.6Pnac+ωDn
3ω3Cn−0.72y0Pnac. (39)
326 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 22, NO. 2, APRIL 2013
Fig. 13. Dimensionless amplitude of oscillation (a) versus detuning pa-rameter (σ) showing the lower amplitude of oscillation solution, the higheramplitude of oscillation solution, and the forbidden region. Parameters used:Cn = 5, Y0 = 0.5 μm, and VP−ac = 20 V.
Therefore, in the presence of damping, this range of theamplitude of oscillation is forbidden physically, as shown inFig. 13. From (39), we can conclude the following. First, thetrivial solution at certain excitation frequency will be unstablewhen 0.6Pnac > ωDn. Second, higher amplitude of oscillationsolution will exist when 0.72y0Pnac > 3ω3Cn. We will see inSection III-D that, using simultaneous forcing and parametricresonance and in the presence of damping, the forbidden am-plitude of oscillation range can disappear.
D. Parametric and Forced Resonance
This case is characterized by Vdc−P = 0, Vac−P �= 0, andF �= 0. In this case, the parametric excitation condition is inthe unstable region, and the external harmonic force is not zero.The dynamic behavior of the parametric and forced resonanceis studied in a number of publications [25], [26]. Fig. 14 showsthat, at certain frequency, there are three possible solutions.The stability analysis of the nontrivial solution shows that thedashed curve is unstable and cannot be observed experimen-tally, while the initial condition determines the steady-statestable solution that will be observed experimentally.
As mentioned in Section III-C, in the case of purelyparametric excitation and when 0.72y0Pnac > 3ω3Cn, higheramplitude of oscillation solution and a forbidden range willappear as shown in Fig. 13. However, in the case of parametric
and forced resonance and when a high external force is applied,the forbidden range disappears, and a continuous solution forthe amplitude of oscillation exists. Fig. 15 shows the amplitudeof oscillation versus the detuning parameter when this caseis satisfied and when the parameters γ3 and λ3 are located inregion IIa as shown in Fig. 10. Note that, in this figure atcertain range of excitation frequencies, there is no existingstable solution. A higher order nonlinear analysis is requiredto calculate the expected amplitude of oscillation that willbe observed physically at this range of frequencies, which isoutside the scope of this paper. Fig. 16 shows the same casebut when the parameters γ3 and λ3 are located in region I, asshown in Fig. 10.
IV. CONCLUSION
The dynamic analysis of a novel design of MEMSfolded-beam interdigitated CD oscillator that is capable of para-metrically actuated, parametrically amplified, or tuned electro-statically has been presented. The extended Hamilton principleis used to derive the equation of motion of the oscillator. Then,an approximate solution is introduced using the perturbationmethod of multiple scales. One of the advantages of this designover traditional types of oscillators that have the ability to beparametrically actuated or tuned electrostatically is its ability tosuppress nonlinearity and achieve high amplitude of oscillation.It was shown that the performance of the devices that utilize
KHIRALLAH: PARAMETRIC EXCITATION, AMPLIFICATION, AND TUNING OF MEMS FOLDED-BEAM CD OSCILLATOR 327
folded suspension configuration can be improved by addingthe truss electrode. This design has the potential to be usedfor increasing the amplitude of oscillation in the driving modeof the gyroscope, increasing the gyroscope response to thecoriolis force, and matching the driving and sensing mode of thegyroscope. Moreover, it can be used as a tunable filter [27] or asa standard frequency generator [28]. Moreover, the parametricresonance condition can be used for mass sensing [28].
APPENDIX A
In this appendix, the steps used to solve the equation ofmotion given by (8) and (9) are introduced. To indicate thesignificance of each term in the equations of motion, we intro-duce the bookkeeping parameter ε, where 1 � ε > 0. Using thenondimensional parameters presented in Section II, the nondi-mensional equation of the motion can be written as follows. ForAB beam,
εv1n + viv1n + ε(v′1n (v
′1nv
′′1n)
′)′
− ε[Pndc + Pnac
(ei(2Ωt+2ϕP ) + e−i(2Ωt+2ϕP )
)]
×
⎡⎣v′′1n − 2y0
⎛⎝ 1∫
0
v′21ndsn
⎞⎠ v′′1n
⎤⎦ = 0 (A.1)
with boundary condition
[v′′1n − εDn(v1n + v2n)− εCn(v1n + v2n)
3
−MPn(v1n + v2n)− εMtn(v1n)
− ε
1∫0
(v1n(1) + v2n) dsn
+ εFLn
(ei(Ωntn+ϕF ) + e−i(Ωntn+ϕF )
)+εFNLn
(x0n(v1n + v2n) + 2x3
0n(v1n + v2n)3)]
sn=1
=0
v1n|sn=0=v′1n|sn=0=v′1n|sn=1=0.
For CD beam,
εv2n + εv1n(1) + viv2n + ε(v′2n (v
′2nv
′′2n)
′)′
− ε[Pndc + Pnac
(ei(2Ωt+2ϕP ) + e−i(2Ωt+2ϕP )
)]×
⎡⎣v′′2n − 2y0
⎛⎝ 1∫
0
v′22ndsn
⎞⎠ v′′2n
⎤⎦ = 0 (A.2)
with boundary condition[v′′′2n − εDn(v1n+v2n)− εCn(v1n+v2n)
where T0 = tn is a fast time scale that characterizes the motionat the excitation frequency and T1 = εtn is a slow time scalethat characterizes the time variation of the amplitude of oscilla-tion and the phase. We substitute
d
dt=D0 + εD1 + · · · (A.5)
d2
dt2=D2
0 + 2εD0D1 + · · · (A.6)
where Di = ∂/∂Ti and D2i = ∂2/∂T 2
i .In this paper, we assume that any mode that is not directly
excited will decay to zero due to the presence of damping. Inother words, we assume a single mode approximation; thus, thesolution can be expressed as
v11 =ϕ11(sn)(A(T1)e
iωT0 +A(T1)e−iωT0
)(A.7)
v12 =ϕ12(sn)(A(T1)e
iωT0 +A(T1)e−iωT0
)(A.8)
v21 =ϕ21(sn, T1)(eiωT0 + e−iωT0
)(A.9)
v22 =ϕ22(sn, T1)(eiωT0 + e−iωT0
)(A.10)
where the bar denotes a complex conjugate. To express thenearness of the excitation frequency Ω to the resonance fre-quency ω, we introduce the detuning parameter σ defined byΩ = ω + εσ.
Substitute (A.3)–(A.10) into (A.1) and (A.2); then, equate thecoefficients of like powers of ε. On the order of O(ε), we notethat the variables v11 and v21 are interchangeable; therefore,we assume that they have the same solution. We multiply theresults by e−iωT0 , and integrating over the interval from 0 to2π/ω, we get for order ε0
ϕiv11 = 0 (A.11)
with boundary condition
ϕ′′′11 + ω2MPn(ϕ11 + ϕ21) = 0
ϕ11
∣∣sn=0
= ϕ′11
∣∣sn=0
= ϕ′11
∣∣sn=1
= 0
328 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 22, NO. 2, APRIL 2013
and
ϕiv21 = 0 (A.12)
with boundary condition
ϕ′′′21 + ω2MPn(ϕ11 + ϕ21) = 0
ϕ21
∣∣sn=0
= ϕ′21
∣∣sn=0
= ϕ′21
∣∣sn=1
= 0.
The solution of v11 and v21 is given by
v11 = v21 =(−s3n + 1.5s2n
) (A(T2)e
−iωT0 +A(T2)eiωT0
).
(A.13)
For order ε1,
ϕiv12 =ω2Aϕ11 − 3A|A|2
(ϕ′11 (ϕ
′11ϕ
′′11)
′)′
+ Pndc ×
⎡⎣Aϕ′′
11 − 6A|A|2y0
⎛⎝ 1∫
0
ϕ′211dsn
⎞⎠ϕ′′
11
⎤⎦
+ Pnac ×
⎡⎢⎣Aϕ′′
11ei(2σT1+2ϕP ) − 2y0
⎛⎝ 1∫
0
ϕ′211dsn
⎞⎠
× ϕ′′11
(A3e−i(2σT1+2ϕP )
+ 3A|A|2ei(2σT1+2ϕP ))⎤⎥⎦
= H1(sn, T1) (A.14)
with boundary condition
ϕ′′′12 + ω2MPn(ϕ12 + ϕ22)
= i2ωMPnD1A(T1) + iωADn
+ i3ω3A|A|2Cn − 0.5ω2AMtn − 0.75ω2A
− Fnei(σT1+ϕF ) − FNLn
(x0nA+ 6x3
0nA|A|2)
= Θ1(T1)
ϕ12
∣∣sn=0
= ϕ′12
∣∣sn=0
= ϕ′12
∣∣sn=1
= 0
and
ϕiv22=ω2Aϕ21+ω2Aϕ11(l)−3A|A|2
(ϕ′21 (ϕ
′21ϕ
′′21)
′)′
+Pndc ×
⎡⎣Aϕ′′
21−6A|A|2y0
⎛⎝ 1∫
0
ϕ′221dsn
⎞⎠ϕ′′
21
⎤⎦
+Pnac×
⎡⎢⎣ϕ′′
21
(Aei(2σT1+2ϕP )
)−2y0
⎛⎝ 1∫
0
ϕ′221dsn
⎞⎠
×ϕ′′21
(A3e−i(2σT1+2ϕP )
+3A|A|2ei(2σT1+2ϕP ))⎤⎥⎦
=H2(sn, T1). (A.15)
with boundary condition
ϕ′′′22 + ω2MPn(ϕ12 + ϕ22)
= i2ωMPnD1A(T1) + iωADn
+ i3ω3A|A|2Cn − FLnei(σT1+ϕF )
− FNLn
(x0nA+ 6x3
0nA|A|2)
= Θ2(T1)
ϕ22
∣∣sn=0
= ϕ′22
∣∣sn=0
= ϕ′22
∣∣sn=1
= 0.
Since (A.11), (A.12) and (A.14), (A.15) are self-adjoint [30],the solvability condition is given by
[Θ1φ11 +Θ2φ12]∣∣sn=1
=
1∫0
(H1φ11 +H2φ12) dsn. (A.16)
Substitute A(T1) = 0.5a(T1)eiB(T1) on the solvability condi-
tion, then multiply the result by e−iB(T1), and introduce thenew variable γ, where γ(T1) = σT1 −B(T1); then, separatingthe real and imaginary parts, we get the following. For the realpart,
ωMPna (−σ+γ′(T1))−0.125ω2aMtn−0.1875ω2a
−Fncos(γ+ϕF )−0.5FNLn
(x0na+1.5x3
0na3)
=0.09ω2a−0.579a3−0.3a(Pdcn+Pacncos(2γ+2ϕP ))
+ 0.125a3(1.08y0Pdcn+1.44y0Pacncos(2γ+2ϕP ))
+ 0.0625aω2. (A.17)
For the imaginary part,
ωMPna′(T2)+0.5ωaDn+0.375ω3a3Cn−Fnsin(γ+ϕF )
+0.3aPnacsin(2γ+2ϕP )−0.09a3y0Pnacsin(2γ+2ϕP )=0
(A.18)
where prime denotes differentiation with respect to T1. Dedi-cating the steady-state condition γ′(T2) = a′(T2) = 0, we get
ωMPnaσ+0.125ω2aMtn+0.3425ω2a+Fn cos(γ+ϕF )
+ 0.5FNLn
(x0na+1.5x3
0na3)
− 0.3a (Pdcn+Pacn cos(2γ+2ϕP ))−0.578a3
+ 0.125a3 (1.08y0Pdcn+1.44y0Pacn cos(2γ+2ϕP ))
=0 (A.19)
and
−0.5ωaDn−0.375ω3a3Cn+Fnsin(γ+ϕF )
−0.3aPnacsin(2γ+2ϕP )+0.09a3y0Pnacsin(2γ+2ϕP )=0.
(A.20)
KHIRALLAH: PARAMETRIC EXCITATION, AMPLIFICATION, AND TUNING OF MEMS FOLDED-BEAM CD OSCILLATOR 329
Fig. 17. Parametric gain (Gain) versus the initial phase of the parametric orexternal excitation in degrees, where φ = φP = φF and Gain = a ∗ 2μ/F .Parameters used: F = 0.1, P = 3e− 3, and μ = 1e− 3.
APPENDIX B
In this appendix, we will discuss the effect of the phasedifference between the parametric force and the external forceon the parametric amplification gain. The equation of motionof a simple parametrically amplified system can be modeled bythe forced Mathieu equation given by
x+2μx+(1+P cos(2Ωt+φP ))x=F cos(Ωt+φF ). (B.1)
This equation is solved using the standard perturbation methodof multiple scales. For the steady-state solution, we obtain
x ≈ a cos(Ωt+ γ). (B.2)
We can obtain a and γ by solving simultaneously this systemof algebraic equations
(Ω−1)a−0.25Pa cos(2γ+φP )+0.5F cos(γ+φF )=0 (B.3)
μa+0.25Pa sin(2γ+φP )−0.5F sin(γ+φF )=0. (B.4)
When the excitation frequency Ω is equal to the resonancefrequency, we get
From (B.8) and (B.9), we note that the amplitude of oscil-lation depends on the difference between the initial phaseof the parametric force φP and double the initial phase ofthe external force 2φF . In other words, the parametric gaindepends on both the phase difference between the parametricforce and the external force φP − φF ) and the initial phaseof the parametric force or the external force. An interestingcase that can elaborate the aforementioned discussion is shownin Fig. 17. In this case, the parametric gain is shown at zero
Fig. 18. Parametric gain (G1, G2) versus the initial phase of the parametricforce φP when φF = 0. Parameters used: F = 0.1, P = 3e− 3, and μ =1e− 3.
phase difference between the parametric force and the externalforce (i.e., φP − φF = 0) with respect to different initial phasesof the parametric force or the external force. This result wasverified numerically using Matlab/Simulink environment andhas excellent agreement with the numerical solution.
Harish et al. [12] studied experimentally and theoreticallythe parametric amplification and the parametric damping inMEMS gyroscope. In their theoretical analysis, they modeledthe system by the forced Mathieu equation given by
x1+2μx1+(1+P sin(2Ωt+φP ))x1=F sin(Ωt). (B.10)
The solution of this equation predicts that the highest gain isachievable at φp = 0 and the smallest gain is achievable atφp = 180◦. However, their experimental results found that themaximum parametric gain occurs when the phase differencebetween the parametric force and the external force is π/4 andthe minimum parametric gain occurs when this difference isequal to 3π/4. From this experimental result, the system can bemore accurately modeled using the forced Mathieu equation ofthe form
The difference between (B.10) and (B.11) is the initial phase ofthe parametric and the external force.
The solution of (B.10) is assumed to have the form
x1 ≈ a1 cos(Ωt+ γ1). (B.12)
Likewise, the solution of (B.11) is assumed to have the form
x2 ≈ a2 cos(Ωt+ γ2). (B.13)
Fig. 18 shows the parametric gain G1 of the system describedby (B.10) and the parametric gain G2 of the system describedby (B.11) versus the phase difference between the parametricforce and the external force φP , where
G1 = a1 × 2 μ/F (B.14)
and
G2 = a2 × 2 μ/F. (B.15)
330 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 22, NO. 2, APRIL 2013
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Kareem Khirallah received the B.Sc. degree (withhonors) in electrical engineering from Ain ShamsUniversity, Cairo, Egypt, in 2008, and the M.Sc.degree in physics from the American University inCairo (AUC), Cairo, in 2011.
He is currently a Research Assistant with theYousef Jameel Science and Technology ResearchCenter, AUC. His research interests include MEMStechnology, applied mathematics, and modelingof nonlinear phenomena in MEMS devices andstructures.
