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MICROMECHANICAL FILTERS

The basic principles described so far for mechanical filters can be used in the

design their micro-sized counterparts. However, the accuracy of these formulations

is plausible at the micro scale for reasons such as the structural dimensions being

not large enough compared with wavelength, nonidealities of boundary conditionsand other nonlinear effects.

Nonetheless these formulations are definite indicators for understanding the

operational principles of these micro devices.

The goal is to fabricate devices such as filters so small that they can be integrated

into rest of the circuit in a single chip leading to a ‘system-on-a-chip’.

Conventional filters using crystal oscillators are not amenable to such

miniaturization. Hence a larger emphasis has been put on miniaturization of

mechanical filters, using standard IC fabrication techniques, so that they can beintegrated with other circuits easily.

The performance for micromachined filters is enhanced by using a series of

resonator tanks connected together with coupling networks. In general, the number

of such tanks used is equal to the (order of filter is the order of its polynomial

transfer function).

The higher this number the better the frequency selectivity for the filter. But the

insertion loss is simultaneously degraded, which can, however, be improved by

designing the filter with very large Q factors.

Only electrostatically actuated devices are discussed in this section, although thepossibility of using other schemes may not be ruled out. A parallel plate capacitor

configuration is common for such large electromechanical filters. However, an

additional possibility exists for micro devices, where electrostatic comb drives

vibrating in a plane parallel to the substrate can be fabricated

1.Electrostatic comb drive

An electrostatically driven parallel plate actuator has a clamped-clamped beam

configuration.

This configuration has nonlinear response characteristics. This nonlinearity can

cause frequency instability in the filter operation . Hence another electrostatically

actuated structure is preferred at the microscale.



The layout of such a laterally driven electrostatic resonant structure is shown in

Figure

Two resonator configurations are possible with this structure

In the first, a two-port configuration, the structure is driven on one of the comb

structures and sensed at the other, for capacitance variations. In the second

configuration, both comb structures are used to drive differentially, while sensing

is achieved by monitoring shift in impedance at resonance.

The folded beam truss suspension has large compliance and is capable of

reducing the residual strain in the structural film.

In the two-port configuration, the driving force and the sensitivity of the output are

both proportional to the variation of capacitance with lateral displacement ∂C/∂x.

static displacement at the drive port, for an applied drive voltage vD,

x=Fx/k s=(1/2 k s) vD2(∂C/∂x.)

Lateral electrostatic comb actuator

where Fx is the x component of

the electrostatic force and k s is

the spring constant of the

system. Assuming the trusses

that join the folded beam are

rigid, the spring constant is

obtained analytically a

To ensure stability, the drive voltage consists of an ac voltage of amplitude vd

superimposed on a dc bias V P such that

vD = V P + vd sin ωt

It may be mentioned at this point that realization of a small interelectrode gap is

essential to reduce the drive voltage requirements of this actuation mechanism. A



fabrication technique using combination of oxidation machining with a suitable

post-release positioning has been developed to address this issue

Submicron gaps can be achieved by this approach.

time derivative, we get

∂x/∂t = (1/2k s) (∂C/∂x) ( 2V Pvdω cos ωt + v2dω sin 2ωt)

For ac voltages much smaller than the dc bias, the second harmonic term on the

right-hand side can be neglected. At resonance, the magnitude is multiplied by the

quality factor, to get the magnitude of the electromechanical transfer function

which relates the phasor displacement X to the phasor drive voltage V d:

The quality factor for this structure is estimated to be

The sensed current is at the output port is

The resonant frequency of the structure is determined by Rayleigh method as

Their fabrication uses a single mask for most of the critical features; this eases the

process design and can potentially reduce cost.

The parasitic capacitive coupling between the input and output ports is minimized



by including a grounded planar electrode, which also helps suppress excitation of

undesired modes

2.Micromechanical filters using comb drives

A number of resonant structures can be coupled together in either series or parallel

configuration to obtain high-quality filter characteristics. Schematics for these

configurations are shown . In the series filter, a square truss coupling

spring connects the two resonators. In the parallel configuration of band pass filter,

the input and output terminals of the resonators are connected in parallel such that

the output

(a) Series and (b) parallel combination of resonators

(b)currents are added up. A notch filter can also be realized in a similar way, by

adding the

(c) currents in opposite phases.

(d)In this analysis, mass of the coupling beam is assumed negligible. The

bandwidth

(e) of the filter depends on the ratio of stiffness of the coupling beam (k sij ) to

that of the(f) resonator beam (k r).

Bandwidth=f L k sij/ k nij k r



where f L is the filter center frequency and k nij is the normalized coupling

coefficient used in filter design. These resonators are designed to have slightly

different resonan frequencies such that their difference is related to the Q factor

f 2 − f 1=f1/Q1

This ensures flat and symmetrical band pass characteristics, provided the

individual resonators have identical 3-dB bandwidths and resonance amplitudes. In

other words, the difference in frequencies is equal to the 3-dB bandwidth of the

filter. To obtain a steep roll-off and flat pass band characteristics, a large number

of resonators should be connected in parallel. In terms of the highest and lowest

resonant frequencies, f u and f L,

respectively, the number of resonators is obtained as

N = Q(f u – f L)/f L

filter with n resonators coupled in series, the overall transfer function

A mechanical model and the corresponding electrical equivalent circuit for the

filter configuration. The equivalent mass, spring constant and dampingconstant for ith resonator in the mechanical model are expressed in terms of the

physical parameters

where M pi is the mass of the plate and M bi is that of the folded beams at the ith

resonator; wi and Li are, respectively, the width and length of the folded

suspension in the ith resonator; h is the thickness of the polysilicon structures and



E p is its Young’s modulus. The stiffness of the coupling spring is similarly

obtained as:

Making use of the electromechanical mobility analogy described in

parameters in the electrical equivalent circuits are related to the above mechanical

parameters



Photograph and (b) equivalent circuits of micromechanical filters

where the transformation parameter η for the filter is defined as

The amplification factor in the electrical equivalent circuit

To avoid the problems associated with stiction and deterioration of electrodes

during operation, an electrostatic repulsive actuator has been recently developed

The repulsive force in this is generated by the asymmetry of the in-plane electric

field causing the movable electrode to slide in the direction shown. The



force generated, resonant frequency and quality factor of this configuration are

generally derived using finite elements analysis

3.Micromechanical filters using electrostatic coupled

beam structures

Lateral drive actuators have a linear transfer function between displacement and

voltage and hence have significant advantages on filter performance. However,

these are relatively large structures. It may be recalled that the resonant frequency

of a simple spring mass system is

f =1/2π(k/m)1/2

This shows that in order to increase the resonant frequency, the structure should

have equivalent higher spring constant and/or very low mass. Reducing the mass of the relatively large comb structure may not be feasible. An alternative

configuration for a high-frequency filter is based on coupled clamped-clamped

beam resonators, a schematic of which is A similar resonator is also developed

with free-free beam configuration

Although inherently nonlinear, their operation can be assumed linear for the small

signal case. On the actuator side, a voltage applied between an electrode below the

beam and a fixed electrode on the substrate causes the beam to move down, by

electrostatic attractive

Perspective view and equivalent circuit of a resonator with two clamped beams



This movement of the beam is coupled towards the next beam, which operates as

a capacitive transducer that senses the displacement of the beam.

The dynamic analysis of clamped-clamped beam presented earlier in this chapter is

valid for these micro structures as well. The input voltage consists of dc bias V P

and a dynamic ac signal vd.

The resonant frequency can be obtained

The quality factor of the resulting filter is

The dynamic spring constant k c of the beam varies with distance from anchor

points.

Thus to improve the quality factor of the filter, the coupling beam is not attached at

the center of the beams. Instead it is attached at a point closer to the anchor point,

where the dynamic spring constant is higher and thus the filter Q factor. As with

the previous case, the length of the coupling beam is taken as quarter acoustic

wavelength. The coupling in this case is therefore modeled as:

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