Open access peer-reviewed chapter - ONLINE FIRST

Ultra-Precise MEMS Based Bio-Sensors

By Vinayak Pachkawade

Submitted: July 26th 2020Reviewed: September 8th 2020Published: October 6th 2020

DOI: 10.5772/intechopen.93931

Downloaded: 22

Abstract

This chapter evaluated the state-of-the art MEMS sensors used for bio sensing applications. A new class of resonant micro sensor is studied. A sensor structure based on the array of weakly coupled resonators is presented. It is shown that due to the weak coupling employed between the resonators in an array manifest ultra-high sensitivity of the output to the added analytes/biomolecules. Due to the highly-precise output of such bio-sensors, minimum detectable mass in the range of sub-actogram is also possible using such MEMS sensors. Analytical modeling of such micro biosensors is presented in this chapter to understand the key performance parameters. Furthermore, role of these new classes of MEMS resonant biosensors operating at ambient temperature and/or pressure is also discussed.

Keywords

  • MEMS biosensor
  • bioelectronics
  • micro sensors for biological applications
  • biochemical analyte detection and identification using MEMS

1. Introduction

In recent years, there has been a growing interest in the development and implementation of innovative solutions in the form of a miniaturized bio-sensors. In this regard, in the MEMS community emphasis has been given to design and fabricate highly sensitive, and precise biomass sensors. These bio-sensors are used for detection, identification and measurement of either single and/or multi-analyte/s at lower cost, size, weight, and power consumption. Moreover, resonant devices are widely popular as a sensor for various chemical/biological applications [1]. In the context of biomass sensing, typical examples of resonant sensing include mass identification or detection [2, 3, 4, 5, 6, 7]. A key attribute of these sensors is that the output signal is the variation/shift in the resonant frequency (∆f) of a vibrating structure that is subjected to small perturbations in the structural parameters i.e. effective mass/stiffness. Additional features of this method of detection are simple mechanical design, semi-digital nature of the signal (thus using simple frequency measurement system such as frequency counter and not requiring additional analog-to-digital (A/D) conversion circuit), ultra-high resolution [8, 9, 10], (up to 10−15 grams scale [11, 12, 13] and up to 10−18 grams scale [14, 15, 16, 17]). There are however also a challenge associated with the resonant sensor employing only one resonator; firstly, maximum theoretical frequency shift based parametric sensitivity, ∆f/f is limited to 1/2 [18]. Sensors of these types are prone to environmental shifts such as pressure and/or temperature. Furthermore, these types of sensors, when used as a mass sensor are able to detect only one type of material (target analyte) at a time thus avoiding the possibility to rapidly detect and differentiate multiple biochemical analytes in parallel [19].

In the past few years, in the MEMS community, a paradigm shift is observed in the design and implementation of micromechanical resonating sensors. A new perspective is presented in using 1-d chain of a coupled resonating proof masses, more familiarly refereed as multi degree-of-freedom (m-DoF) array, coupled resonator (CR) array, weakly coupled resonators (WCR) or mode-localized sensors [20, 21, 22, 23, 24, 25, 26, 27, 28]. Figure 1 shows a representative schematic for such system. In this class of a sensors, coupling between the vibrating proof masses is constituted either electrostatically or mechanically. These sensors attribute an ultra-high parametric sensitivity (up to three to four orders high in magnitude) [29]. Such elevated levels of sensitivities are manifested via a novel transduction principle, i.e. sensing magnitude of vibrational energy exchange between the moving proof masses that are subject to a small disruption introduced into the system. This disruption serves to alter an effective mass, ∆m of one of the proof mass element in a chain (see Figure 1). Primarily, due to relatively higher parametric sensitivities, m-DoF coupled resonators are emerging as an alternative and promising resonant sensing solution.

Figure 1.

A schematic representation of a 2-DoF coupled resonant mass-spring-damper system to be used as a high-sensitivity biomass detector/identifier sensor. In a symmetric design, it is assumed that M1 = M2 = M, K1  = K2  = K. A mass perturbation, ∆m is added onto the mass M2 , causing system imbalance, leading to the energy localization/confinement and mode shape change. Mode-shape change is utilized as the output of the sensor.

Other acknowledged advantages of weakly-coupled resonating sensors are linearity (attributed to high sensitivity, relative immunity against responding to common mode noise for example, ambient pressure and/or temperature [30, 31, 32] and the parallel detection capability in the context of the mass sensing applications [33, 34, 35, 36]. These characteristics make mode-localized coupled resonators effective. For the obvious advantages as given, m-DoF coupled resonant sensors are being pursued over conventional method of resonant sensing, i.e. sensing the frequency shift, ∆f of a single resonating device.

2. Theory

A viable method to understand the operation of the mode-localized CR is its analysis through the transfer function model. The transfer function analysis enables to understand the system-level behavior of a biosensor unit. Figure 1 shows a lumped parameter model of a 2 DoF mass-spring-damper system in the context of CR biosensor. It shows proof masses, Mi , mechanical spring constant, Ki and the damping coefficients, ci , (i = 1, 2). The cc models the damping force between the two resonating proof masses. Two proof masses are coupled through another spring, Kc as shown. A displacement of the proof mass, xi (i = 1, 2) in response to the applied force, Fi (i = 1, 2) is also indicated. Based on the free body diagram, a set of governing differential equations of motion for 2-DoF mass-spring-damper can be used to derive the theoretical transfer function. Subsequently, an expressions for mode-frequencies, ωi and modal amplitudes/amplitude ratio (AR) as a function of applied disorder in the mass, δm can be obtained.

Following assumptions hold true for a symmetric device- M1 = M2 = M, K1  = K2  = K and c1 = c2 = c. Forcing vectors Fi acting on the proof masses are the harmonic excitation (drive) forces that cause displacements, xi , assumed to occur at one frequency. Quantities δm and κ are normalized perturbation to the mass and normalized coupling factor, given as δm  = ∆m/M and κ = Kc /K, respectively. When system experiences imbalance into the initial symmetry i.e., ∆m ≠ 0, governing set of equations of motion for the two-coupled proof masses is given as follows:

Mx¨1+c+ccẋ1+K+Kcx1cẋ2Kcx2=F1tE1
M+Δmx¨2+c+ccẋ2+K+Kcx2cẋ1Kcx1=F2tE2

By operating the system in vacuum, the impact of the following can be reduced, i) damping force of individual proof mass and ii) damping force that occurs between two proof masses, hence c1=c2=cc=0can be assumed for the simplified analysis. Therefore, (1) and (2) can be modified as below:

Mx¨1+K+Kcx1Kcx2=F1tE3
M+Δmx¨2+K+Kcx2Kcx1=F2tE4

By applying a Laplace Transform to Eqs. (3) and (4), following expressions are obtained:

G11sX1sG12sX2s=F1sE5
G22sX2sG21sX1s=F2s,E6

where

G11s=s2M+K+KcE7
G12s=G21s=KcE8
G22s=s2M+Δm+K+KcE9

In Eq. (6), setF2s=0, and derive an expression for X1sand X2sto use these values back in Eq. (5). An output transfer function is then obtained as follows:

G1s=X1sF1s=G22sG11sG22sG12sG21sE10
G2s=X2sF1s=G21sG11sG22sG12sG21sE11

Similar procedure can be applied to obtain an expression for G3s=X1sF2sand G4s=X2sF2s. Using the values of G11s, G12sG21sand G22sderived earlier in Eq. (7) through (9), following equations are obtained

G1s=X1sF1s=s2M+Δm+Kαs4MM+Δm+s2KαM+M+Δm+Kα2Kc2E12
G2s=X2sF1s=Kcs4MM+Δm+s2KαM+M+Δm+Kα2Kc2E13

Here Kα=K+Kc. Using s = jω, Eqs. (12) and (13) can be modified to attain

G1=X1F1=ω2M+Δm+Kαω4MM+Δmω2Kα2M+Δm+Kα2Kc2E14
G2=X2F1=Kcω4MM+Δmω2Kα2M+Δm+Kα2Kc2E15

A denominator of Eqs. (14) and (15) is given by

ω4MM+Δmω2Kα2M+Δm+Kα2Kc2=0E16

Eq. (16) is the characteristic equation of 2 DoF coupled system. Roots of Eq. (16) can be given as

ωip2Kα2M+ΔmΔm2Kα2M2+4Kc2M2+Δm2M2E17
ωop2Kα2M+Δm+Δm2Kα2M2+4Kc2M2+Δm2M2E18

Here, ωip2and ωop2are in-phase and out-of-phase natural mode frequencies of the device. With ∆m = 0, Eqs. (17) and (18) take the form ωip2=KMand ωop2=K+2KcM. Dividing (14) by (15), amplitude ratio (AR) is obtained as

H1H2=ω2M+Δm+KαKcE19

By substituting the values of ω in Eq. (19), the expression for mode AR as a function of mass perturbation, ∆m is obtained as follows:

H1jωipH2jωip=Kα2M+ΔmΔm2Kα2M2+4Kc2M2+Δm2M2M+Δm+KαKcE20
H1jωipH2jωip=Kα2M+Δm+Δm2Kα2M2+4Kc2M2+Δm2M2M+Δm+KαKcE21

With ∆m = 0, Eq. (20) and (21) take the form as

H1jωipH2jωip=1;H1jωopH2jωop=1E22

Eq. (22) represents initial balanced condition of a two CR.

2.1 Analytical models

Figure 2 represents the analytical plots of the 2-DoF CR sensor system. In Figure 2(a), mode-frequencies as a function of induced mass disorder is shown. Lower resonant frequency of the out-of-phase mode indicates that a design uses an electrical coupling between the two resonators. With δm  = 0, two resonant frequencies are closely spaced apart (frequency separation determined by the value of the coupling spring constant, Kc used in the system). However, when δm  ≠ 0, resonant mode-frequencies veer away from each other as the magnitude of the δm is increased. In Figure 2(b), similar trend can be observed, where, amplitude ratio (AR) as a function of induced mass disorder is shown. Slopes of the curves in each graphs determine the sensitivity of mode-frequencies and AR to the normalized mass perturbation into the system. The theoretical sensitivity norms (in the context of the mass perturbations, ∆m) for the amplitude ratio (AR), eigenstate and the resonant frequency used in CR sensors are expressed as below:

Figure 2.

Mode-frequencies and AR veering phenomenon observed in CR mode-localized mass sensors. δm is the applied normalized mass perturbation. Slope of the AR curves determined the sensitivity. Lower value of the coupling spring, Kc enhances sensitivity to the mass perturbation for the AR output.

x1x2ji=rnr0r0Δm2KcE23
xji=ana0a0Δm4KcE24
fji=fnf0f0Δm2meffE25

for j th resonator (j = 1, 2) at i th mode of the frequency response. (i = 1, 2), respectively. For the electrostatic coupling between the two resonators, an effective value of the coupling spring is given by Kc=Δv2ε0Ag3, where ∆v refers the potential difference between the two masses, g is capacitive gap, ε is permittivity and A is the cross sectional area of the parallel-plate capacitor.

Figure 3 shows a plot of AR variation as a function of mass perturbation. Two different values of coupling spring, Kc are used. For lower effective value of the coupling spring, Kc  = 100 N/m, (coupling factor, κ = 0.00075), higher changes in the AR can be extracted (relatively higher slope). This aspect shows the tunable characteristic of a sensitivity in a CR biosensor unit. In Figure 4, different forms of the outputs (as expressed in (23) through (25)) are compared against the values of applied mass perturbations in a 2-DoF CR sensor system. As seen, the AR output offers the highest achievable sensitivity. Therefore, AR sensing is the preferred method of the sensing in CR biosensor system. It can be written that SRi>SaiSfi. Here, SRi, Sajiand Sfidenote the theoretical maximum sensitivity for AR, amplitude and frequency for j th resonator (j = 1, 2) at the i th mode of the frequency response (i = 1, 2), respectively. Another way to understand the operation of a 2-DoF CR biosensor is through its output frequency response. Figure 5 shows an output response of resonator 1 and 2 in a 2-DoF coupled resonant (CR) mass biosensor system. As two resonators are used in the CR system, two peaks appear in the output response. Initially, assuming a symmetric system (i.e. M1 = M2 = M, K1  = K2  = K), jth resonator (j = 1, 2) vibrate with equal amplitudes at the ith mode (i = 1, 2) of the output response. After the mass disorder, δm is introduced, it is observed that vibration amplitudes of the jth resonator (j = 1, 2) at the ith mode (i = 1, 2) of the output response change. An amplitude shift is denoted by ∆a . The frequency shift, ∆f at both the modes is also seen. After the mass perturbation, for the resonator 1, vibration amplitude is seen decreased at the first mode, whereas amplitude is seen increased at the second mode.

Figure 3.

Amplitude ratio (AR) curve veering in 2-DoF mode-localized coupled resonators mass sensors. The lower coupling factor κ leads to higher changes in the AR when the normalized perturbation δm is applied.

Figure 4.

Different types of the outputs with coupled resonators (CR) sensors. The amplitude ratio (AR) shift shows the highest percentage changes as a function of the mass perturbation, δm .

Figure 5.

The output response of a 2-DoF coupled resonator mode-localized design used for biomass sensing applications. Due to the mass addition or removal process, initial eigenmodes and eigen-frequencies of the jth resonator (j = 1, 2) change due to the mode-localization. Changes in the amplitudes, ∆a are found to be 2 to 3 orders high in magnitude than that of the corresponding changes into the frequencies, ∆f at the ith mode (i = 1, 2) of the output response.

2.2 Frequency response and a finite element model (FEM)

Figure 6(a) shows a frequency response (bode diagram) of a 2-DoF CR system. Figure 6(b) shows a COMSOL mode shape simulation for a designed geometry of a two mechanically coupled resonators. A structural mechanics module of COMSOL Multiphysics [37] can be used to design CR sensor and simulate for the mode shape and eigen-frequencies of the design. As seen from the FEM results, for mode 1, vibrating elements (cantilevers) move in the same direction (in-phase mode) and the same amplitudes [1, 1]. For mode 2, both the cantilevers move in opposite direction (out-of-phase mode) and the same amplitudes [1,-1]. The two simulated mode frequencies are f1 and f2 for mode 1 (in-phase) and mode 2 (out-of-phase), respectively. As the FEM illustrates, the coupling used in the design is mechanical. If the coupling used in the design is electrical, out-of-phase mode precedes the in-phase mode as seen in the bode diagram (note the phase difference of the two resonators).

Figure 6.

Output frequency response (bode diagram) of the CR sensor showing the two modes of a 2-DoF sensor. A finite element model is also depicted to determine the mode shape and resonant frequencies of the design. Note the swapping of the two modes as per the coupling used in the sensor.

3. Case studies in m-DoF resonant mass sensing

In this section, different types of MEMS ultra-precise sensors based on the m-DoF CR architecture are discussed. In MEMS resonant biosensors, a surface of the micromechanical resonator is coated with a sensitive thin film. A resonant frequency shift is monitored as a result of adsorption/absorption of the target analyte/s [19]. In the same framework, CR structures are used as a mass sensors owing to the enhanced mass sensitivity and parallel monitoring of multiple analyte/s.

3.1 Study I

For the first time, it was proposed that a vibration mode localization can be used to demonstrate an elevated mass sensitivity [36]. A fabricated prototype is shown in Figure 7(a). In this work, two nearly identical mechanically coupled gold-foil microcantilevers were used. For the experimentation, borosilicate microspheres (mean diameter of 4.9 μm with a mass of ≈154 pg were added on cantilever 2. Piezoelectric shaker was used for the driving scheme for the sensor. A laser doppler vibrometer was used to capture the tip velocities at different locations of individual cantilevers. An output plot as seen in Figure 7(b) was obtained to show eigenstate variation as a function of normalized mass perturbation, δm . With ∆m = 0, vibration amplitudes of cantilevers 1 and 2 at two mode frequencies (i.e. two distinct modes as in-phase and out-of-phase) are seen. Uneven amplitudes for both the cantilevers at both the modes are result of fabrication mismatch. With ∆m ≠ 0, vibration amplitudes of both the cantilevers change at both the modes. With mass added to cantilever 2, amplitudes of both the resonators at both the modes are seen to be increased. Amplitude of resonator 1 is relatively higher than amplitude of resonator 2 at the first mode (at lower frequency in the response). Amplitude of resonator 2 is relatively higher than amplitude of resonator 1 at the second mode (at the higher frequency in the response). A relatively larger shift (either in amplitude or resonant frequency) indicates vibration energy is localized to that particular cantilever at the mode of operation. With a mass differential (∆m) in the system, resonant frequencies of both the cantilevers at both the modes also change. However, relative changes in the amplitudes are orders of magnitude higher than the changes in frequencies. This work experimentally demonstrated about two orders higher in magnitude relative changes into the eigenstates (5–7%) than relative changes in the frequencies (0.01%). Enhanced sensitivity of eigenstates to the added mass was attributed to the decreased scaled coupling strength, κ between the two cantilevers. Each eigenstate is the normalized vector formed by the amplitudes of the two vibrating elements (cantilevers) at a corresponding resonance frequency. In the same work, mass removal from the cantilever surface resulted in return of eigenstates to their original values.

Figure 7.

Ultrasensitive mass sensor using a mode localization in coupled microcantilevers (a) fabricated prototype and (b) amplitude-frequency response of a fabricated prototype before and after the mass imbalance introduced into the system [33]. Reprinted from [33] with the permission of AIP Publishing.

3.2 Study II

Moreover, using an array of polysilicon microcantilevers (up to 15) it is possible to record up to 3 orders higher changes in eigenstate based output of the sensor [35]. In an array of cantilevers, each pattern of eigenmode shifts is unique. Therefore, by examining an experimentally measured pattern of eigenmode shifts it is possible to determine to which cantilever a target analyte particle has adhered. A mass sensitivity of up to two orders higher was found as opposed to the previous work [36] reported by the same group. A mass sensitivity of up to three orders higher was found as opposed to relative frequency shifts. It is therefore feasible to design coupled resonant (CR) microstructures and use eigenmode as an output metric for enhanced parametric sensitivity over resonant sensors that use frequency shift output. However, it is also evident that merely adding the number of resonators in a 1-dimensional (1-d) chain does not necessarily increase the parametric sensitivity in proportion.

3.3 Study III

In a study, an array of four micro beams, S1-S4 are attached to a common shuttle mass, SM for the detection and identification of multiple analytes [19, 38]. Geometrical asymmetry in the micro beams (length mismatch) assured sufficient separation of individual resonant peaks (as seen in Figure 8(a)) at the corresponding eigenmode frequencies in the output response. An output response of the fabricated prototype along with mode shapes are shown in Figure 8(a). A capillary tube containing the specific polymer solutions was interfaced to one or all of the micro beams to functionalize them for vapor detection. Specifically, toluene and methanol, and toluene/methanol mixtures were used with the polymers to prepare analyte concentration for the functionalization of the surface of the microbeam/s in an array. The functionalized prototype was excited to motion by the piezoelectric actuator operating at a pressure level of 200 Torr. As shown in Figure 8(b), a single output signal (resonant frequency shifts of all micro beams as a function of analyte concentration in %) was measured optically from the SM to obtain the composite response due to the mode-localization. Detection process is as follows: i) measure the resonance frequencies, M1-M4 (without added mass) in the pure nitrogen gas, ii) introduce an analyte and wait for the absorption to reach steady state, iii) measure the resonance frequencies once more, and determining the resulting frequency shifts. In this experiment, the frequency shift of each localized microbeam mode (M1-M4) was determined for various analyte concentrations (Figure 8(b)). As observed from Figure 8(b), M3 shows the highest sensitivity to the toluene, whereas, M4 exhibits highest sensitivity to the added mass concentration of the methanol. Knowing the shifts of the two most sensitive microbeam modes M 3 and M 4 and their sensitivities from Figure 8(b), the concentrations of methanol and toluene in the vapor were estimated to be 2.8% and 2.0%, respectively, and found in good agreement with the actual concentrations of 2.3% methanol and 2.3% toluene.

Figure 8.

Frequency response of the single input single output scheme in an array of coupled micro beams. (a) Composite response indicated by B and the individual mode-frequencies (M1-M4) of the mode-localized micro beams upon mass absorption due to the added analyte concentration. Also shown are the resonant mode shapes of the system. (b) Resonant frequency shifts of the four individual modes as a function of added mass concentration of the analyte. Slopes of the curve determine the sensitivity to the analyte, also making it possible to identify particular vapor concentration. Reprinted from [38] with the permission of AIP Publishing.

3.4 Study IV

In the recent study, ‘Fano resonances’ were observed in purely mechanical systems constituted by an array of nano and microcantilever resonators [39]. An array of micro/nano-scale cantilevers were used for mass sensing. A fabricated prototype and the output response is shown in Figure 9. Nanoscale cantilevers (thickness of 100 nm) fabricated of silicon nitrate were excited with piezoelectric disk shaker. A laser optical based detection was performed with sub-picometer resolution. Cantilevers here are directly connected to the bulk silicon without any suspended overhang at the base. As a consequence, each cantilever resonance is said to be identified by the main resonance curve (Lorentzian curve). No other resonant peak (entangled states due to coupling) are observed in the output response. Instead, due to an interference between the tails of the Lorentzian curve and the resonance peaks of the other cantilevers in an array, it was said that fano-peaks appeared at the corresponding resonant frequencies of the other cantilevers in an array (Figure 9(b)). Increased sensitivity (i.e. being able to measure smallest change in mass, ∆m of up to 5 pg, ∆m ≈ m/Q) of fano-resonances was attributed to their exhibited relatively higher quality factor, Q (10× than the main resonance peak). A negative shift in the resonant frequencies of all the ‘Fano-resonances’ was observed as the results of parallel mass loading on all the cantilevers in an array. Therefore, a single, fast measurement scheme (parallelization) was proposed to characterize the frequency response of one cantilever in an array. Several configurations of micro/nano scale cantilever arrays were fabricated and experimental results were given along with the theory model.

Figure 9.

Fabricated prototype (a) and (b) vibration spectra of one cantilever in an array. Up to 9 cantilevers, not shown here were used in this work to propose a single and fast measurement scheme for cantilever arrays using Fano-resonance analysis [39]. Reprinted from [39] with the permission of Springer Nature under creative commons licenses.

4. Common-mode-rejection in resonant mass sensors

Mode-localized sensors that utilize an array of ideally identical, weakly coupled, vibrating microstructural elements are shown to be relatively immune to the false output. Environmental factors (such as ambient pressure [30, 32], temperature [31, 40]) and/or nonspecific bindings (in case of mass sensing) influence all the vibrating elements uniformly. These factors as mentioned ideally does not affect the eigenmodes of the system, while shifts in the resonance frequencies still occur. One of the early work addressing the common-mode rejection of mode-localized sensor is given in [30]. As shown in Figure 1, in a 2-DoF WCR sensor, induced mismatch (∆m) affects one of the resonator. This is called a mass differential mode of operation. However, environmental variable such as temperature or pressure affects both the resonators simultaneously, which is termed as common-mode behavior of the micromechanical mode-localized sensors. It is found that, the mode-localized sensor is efficient in rejecting a common-mode noise (temperature and pressure) to the first order.

Table 1 provides a comparative performance summary (in terms of the attainable mass sensitivity and minimum detectable mass) of the ultra-precise and highly sensitive resonant mass sensors.

ReferenceOutput usedRelative sensitivity aMinimum detectable mass, ∆mDoFMaterial used
[36]eigenstates5% to 7% (higher than frequency shift output)154 pg2borosilicate microspheres
[26]eigenstates3 orders higher13.5 pg2platinum patches
[19, 38]Frequency shiftNANA4polymer solutions
[41]Frequency shiftNA1.42 × 10−14 kg3NA
[42]eigenstates700010 pg3polystyrene micro-spheres
[39]Frequency shiftNA5 pg4NA
[43]AR34,3612.1 ng2NA
[24]AR2.5%/pg (estimated via numerical model)6 pg2NA
[44]AR (Atmospheric pressure test)25.31 (two orders higher than frequency shift output)180 ng3Nanoparticles used
[34]Eigenstates/frequency shiftsNANA4Thermally killed bacteria

Table 1.

Mode-localized mass sensors summary.

Relative sensitivity is the ratio of sensitivity of eigenstate/s or AR output to the sensitivity of frequency shift output.


5. Conclusion

In this chapter, the state-of-the-art in MEMS resonant sensor is studied. Numerical models were presented to understand the operation of the ultra-precise, high sensitivity devices used for the bio applications. Key performance parameters such as mass sensitivity was derived for the different available outputs in the CR resonant mass sensors. From the recent case studies and a comparative analysis as provided in Table I, it can be concluded that CR resonant biosensor is emerging as a new sensing standard in the MEMS community.

Abbreviations and nomenclature

MEMS

Micro-electromechnical systems

m-DoF

multi-degree of freedom

CR

Coupled resonators

WCR

Weakly coupled resonators

A/D

Analog-to-digital converter

AR

Amplitude ratio

Mi

Proof mass gram

Ki

Mechanical spring constant N/m

ci

Damping Ns/m

F

Forcing term N

x

Displacement of the proof mass m

Kc

Coupling spring constant (electrical) N/m

κ

Coupling factor Unitless

∆m

Mass perturbation gram

δ

Normalized perturbation Unitless

δm

Normalized mass perturbation Unitless

ωip

In-phase mode frequency rad/sec

ωop

Out-of-phase mode frequency rad/sec

x1x2ji

Amplitude ratio of the j th resonator at the i th mode of the frequency response Unitless

xji

Amplitude of the j th resonator at the i th mode of the frequency response V

fji

Resonant frequency of the j th resonator at the i th mode of the frequency response Hz

∆f

Frequency shift Hz

∆a

Amplitude shift V

∆f/f

Frequency shift (normalized) Unitless

SRi

Sensitivity of AR shift to the normalized mass perturbation V/V/δm

Saji

Sensitivity of amplitude shift to the normalized mass perturbation V/δm

Sfi

Sensitivity of frequency shift to the normalized mass perturbation Hz/ δm

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Vinayak Pachkawade (October 6th 2020). Ultra-Precise MEMS Based Bio-Sensors [Online First], IntechOpen, DOI: 10.5772/intechopen.93931. Available from:

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