INTRODUCTION
Miniaturization of the constituent components of superheterodyne wireless
transceivers is a field of research that has received considerable attention
recently. To date, several of these components can already be miniaturized using
integrated circuit transistor technologies. Reduced size constitutes the most
obvious incentive for replacing SAWs and crystals by equivalent devices. Typically,
the frontend of a wireless transceiver contains a good number of offchip highQ
components that are potentially replaceable by micromechanical versions. Among
the components targeted for replacement are RF filters, including image reject
filters (with center frequencies ranging from 800 MHz to 2.5 GHz), IF filters
(with center frequencies ranging from 455 kHz to 254 MHz) and highQ low phasenoise
local oscillators with frequency requirements in the 10 MHz to 2.5 GHz range
(Nguyen, 1998, 1999; De
Los Santos, 2002). One of the challenging issues which has hindered deployment
of the microelectromechanical resonators and filters is large motional resistance
R_{x} of these devices with electrostatically and capacitively transduction.
Among methods for lowering the motional resistance R_{x} of electrostatically
and capacitively transduced micromechanical resonators presented so far are:
(1) decreasing the electrodetoresonator gap (Bannon III
et al., 2000), (2) increasing the dcbias voltage V_{P} and
(3) summing together the output currents of an array of identical resonators
(Demirci et al., 2003). Unfortunately, each of
these methods comes with some drawbacks. In particular, although the first two
methods are very effective in lowering R_{x} (with fourth power and
square law dependencies, respectively), they do so at the cost of linearity
(Navid et al., 2001; Alastalo
and Kaajakari, 2006). On the other hand, method (3) actually improves linearity
while lowering R_{x}. However, this method is difficult to implement;
since it requires resonators with precisely identical responses and consumes
large area of silicon chip; since it uses an array of resonators instead of
a single resonator.
This study presents a novel method for lowering motional resistance based on
a technique which utilizes true potential of a single square frame resonator
in threedimensions and raises the linearity as well. Using this new technique,
a simplysupported square frame microelectromechanical resonator (with an effective
R_{x} of 478 kΩ) was designed and simulated at 71 MHz. This effective
resistance is about 75X smaller than the 35.9 kΩ exhibited by a 72 MHz
ClamedClamped beam resonator (Varadan and Vinoy, 2002),
73X smaller than the 34.8 kΩ demonstrated by a 71 MHz freefree beam resonator
(Wang et al., 2000) and 8.4X smaller than the
4kΩ presented by a 68.1 MHz mechanicallycoupled 11resonators array (Demirci
et al., 2003).
This R_{x}reduction method is superior to methods based on scaling
down of electrodetoresonator gaps, dcbias increases, or using an array
of identical resonators; because it allows a reduction in R_{x}
without sacrificing linearity and consuming large chip area.
MATERIALS AND METHODS
Resonator structure and operation: Figure 1
shows schematic top view of the proposed micromechanical resonator (along
with appropriate bias, excitation and sensing circuitry). This resonator
consists of a simplysupported square frame resonators, supported by four
tethers and attached to substrate only at anchors, all suspended above
the substrate.
To operate this resonator, a dcbias V_{P} is applied to the
suspended resonator structure and two complement ac input voltages v_{i}
and v_{i} are applied to the input electrodes, which are placed
by a 100 nm gap from the structures, as shown in Fig. 1.
The application of input voltages v_{i} and v_{i} creates
x and ydirected electrostatic forces between input electrodes and the
conductive resonator that induce x and ydirected vibration of the input
resonator when the frequency of the input voltage comes equal to the resonant
frequency of the mechanical resonator. Vibration of the square frame resonator
creates a dcbiased, timevarying capacitor between the conductive resonator
and electrodes, which then sources an output current given by:
where, ∂C/∂x is the change in resonatortoelectrode capacitance
per unit displacement at input ports. The output current i_{o }is
then directed to resistor R_{L}, which converts the current to
output voltage.
Since, center frequency of a given mechanical filter and oscillation frequency
of a given oscillator are determined primarily by the resonance frequencies
of its constituent resonators, careful mechanical resonator design is imperative
for successful device implementation. The selected resonator design must not
only be able to achieve the needed frequency but must also do so with adequate
linearity and tunability and with sufficient Q.
As shown in Fig. 2, the microresonator is formed of
four polysilicon beams which are attached to each other and organized
in a square frame. Other polysilicon tethers attach corners of this frame
to anchors.

Fig. 1: 
Top view schematic of a square frame micromechanical
resonator, along with the preferred bias, excitation and sensing circuitry 

Fig. 2: 
Fundamental mode shape of the micromechanical resonator
simulated by ANSYS 
The anchors are tightly placed on substrate and cause the
whole structure to suspend above the substrate with a little space between
them. This polysilicon square frame can freely move parallel to substrate
in x and y orientations.
The resonance frequency of this simplysupported square frame depends
upon many factors, including geometry, structural material properties,
stress, the magnitude of the applied dcbias voltage V_{P} and
surface topography.
Accounting for these while neglecting finite width effects, an expression for
resonance frequency can be written as (Timoshenko et al.,
1974):
where W_{r} and L_{r} are the width and effective length
of the beam, respectively, E is the Young’s modulus, ρ is the density
of the structural material, β_{n} = 3.1415, 6.2831, 9.4247
for the first three modes of a simplysupported beam, f_{nom}
is the nominal mechanical resonance frequency of the resonator if there
were no electrodes or applied voltages and κ is a scaling factor
that models the effects of surface topography.

Fig. 3: 
Mechanical frequency response of the microresonator 
Mechanical frequency response of the designed resonator is shown in Fig.
3.
To properly excite this device, a voltage consisting of a dcbias V_{P}
and an ac excitation v_{i} is applied across one of the
resonatortoelectrode capacitors (i.e., the input transducer). This creates
a force component between the electrode and resonator proportional to
the product V_{P}v_{i} and at the frequency of v_{i}.
When the frequency of v_{i} nears its resonance frequency, the
microresonator begins to vibrate, creating a dcbiased timevarying capacitor
C_{0}(x,t) at the input transducer. A current is then generated
through the output transducer and serves as the output of this device.
When plotted against the frequency of the excitation signal v_{i},
the output current i_{o} traces out the bandpass biquad characteristic
expected for a highQ tank circuit.
It must be noted that from the discussion associated with Eq.
1, the effective input force (~V_{p}v_{i}) and output current
can be nulled by setting V_{P} = 0. Thus, a micromechanical resonator
(and oscillator or filter constructed of such resonators) can be switched
in and out by the mere application and removal of the dcbias voltage
V_{P}. Such switchability can be used to great advantage in receiver
architectures.
Design of the micromechanical resonator in form of Fig.
2, has some advantages as follows:
• 
The fullydifferential electrode configuration cancels the second
harmonic distortion term (HD_{2}); therefore, improves the
powerhandling capability and dynamic range of the resonator 
• 
The present structure design, decreases input and output impedances
significantly and in filter design case, matches the impedance of
filter to the impedances of the stages before and after the filter,
properly 
• 
Since, the proposed resonator structure has four almost motionless
node points, the quality factor due to energy loss mechanisms of support
loss (Q_{Support}) is high and hence Q of the whole structure
is high 
• 
Since, the resonator vibrates in x and y directions, the electrodes
are placed besides the structure instead of beneath it. So, the fabrication,
mask defining, pattern generation and manufacturing of the device
will be done more easily and inexpensively 
Frequency tuning: Resonance frequency of the microresonator is
a function of the dcbias voltage V_{P}. Thus, frequency of this
device is tunable via adjustment of V_{P} and this can be used
advantageously to implement filters and oscillators with tunable center
frequencies, or to correct for passband distortion caused by finite planar
fabrication tolerances.
The dcbias dependence of resonance frequency arises from a V_{P}dependent
electrical spring constant k_{e} that subtracts from the mechanical
spring constant of the system k_{m}, lowering the overall spring
stiffness k_{r} = k_{m}k_{e}, thus lowering the
resonance frequency according to the expression:
where, k_{m} and m_{r} denote values at a particular location
(usually the beam center location) and the quantity k_{e}/k_{m}
must be obtained via integration over the electrode width W_{e} due
to the location dependence of k_{m} as follows (Nathanson
et al., 1967):
where, range of integration is from L_{e1} = 0.5(L_{r}W_{e})
to L_{e2} = 0.5(L_{r}+W_{e}) and h_{r}
and d(x) are structure thickness and gap between electrode and vibrating
beam, respectively.
It must be noted that Eq. 3 is only valid for a case
in which four outer electrodes are used. If in addition of four outer
electrodes, four inner electrodes are also used (in order to reduce R_{x}
further), then the term k_{e}/k_{m} in Eq.
3 should be multiplied by a factor of 2. In that case, the frequency
f_{0} decreases further, with increasing V_{P}.

Fig. 4: 
Simulated frequency versus applied dcbias V_{P}
for the present microresonator 
The dependence of the resonance frequency to dcbias voltage V_{P},
is shown in Fig. 4.
MOTIONAL RESISTANCE CALCULATION
An electrical model with a core RLC circuit was defined for the microresonator
based on massspringdamper system. Of the elements in the equivalent
RLC circuit, the series motional resistance R_{x} is the most
influential in both oscillator and filter circuits. In oscillators, R_{x}
generally governs the gain needed to instigate and sustain oscillation;
whereas in bandpass filters, it dictates the ease of matching the designed
filter to low impedance stages before and after the filter (e.g., the
antenna).
where, η_{e} is the transduction parameter for a capacitive
transducer and is calculated theoretically as follows:
and D_{r} is the damping factor which is given by:
where k_{m} and m_{r} denote the mechanical spring constant
value and the equivalent mass of the system at the beam center location
on the resonator and Q_{nom} is the quality factor of the resonator
without the influence of applied voltages and electrodes.



Fig. 5: 
Simulated plots presenting R _{x }values obtained
via., Eq. 8. (a) R _{x }versus electrodetoresonator
overlap area A _{0}. (b) R _{x} versus electrodetoresonator
gap spacing d _{0}. (c) R _{x }versus dcbias V _{P} 
To having better insight into what parameters governs R_{x},
a closed form expression can be obtained by substituting Eq. 6 into 5 and using
lumped terms for integrated parameters, which yields:
where, A_{0} is the effective electrodetoresonator overlap
area of the resonator.
Figure 5 present plots of R_{x} versus various
parameters in Eq. 8, showing that small values of R_{x}
are feasible if small values of electrodetoresonator gap spacing d_{0}
and large values of dcbias V_{P} and sufficiently large electrodetoresonator
overlap area are used. However, the use of such values should not sacrifice
linearity.
As indicated in Fig. 5a, by choosing d_{0}
= 100 nm, V_{P} = 35 V and using eight electrodes (four outer
electrodes and four inner electrodes), motional impedance of R_{x}
is reduced to 478 Ω; which is a desirable value for a micromechanical
resonator. Furthermore, as it was explained before, use of special design
for the support beams prevents lowering quality factor of the flexural
mode beam resonators. Hence, there will not be any decrease in Q and increase
in R_{x} as a result.
It must be noted that in Fig. 5b and c,
total electrodetoresonator overlap area is equal to 540 μm^{2}
and overlap area for each electroderesonator pair is only 75 μm^{2},
which leads to a large pullin voltage (~180 V).
SUPPORT STRUCTURE DESIGN
The designed square frame mechanical resonator is supported by four flexural
beams attached at its fundamentalmode node points (Fig.
2). Since, these beams are attached at node points, the support springs
sustain no translational movement during resonator vibration (ideally)
and thus, support (i.e., anchor) losses due to translational movements
are greatly alleviated. Furthermore, with the recognition that the supporting
flexural beams actually behave like acoustic transmission lines at the
VHF frequencies of interest, flexural loss mechanisms can also be negated
by strategically choosing support dimensions so that they present virtually
no impedance to the simply supported beam. In particular, by choosing
the dimensions of a flexural support beam such that they correspond to
an effective quarterwavelength of the resonator operating frequency,
the solid anchor condition on one side of the support beam is transformed
to a freeend condition on the other side, which connects to the resonator.
In terms of impedance, the infinite acoustic impedance at the anchors
is transformed to zero impedance at the resonator attachment points. As
a result, the resonator effectively sees no supports at all and operates
as if levitated above the substrate, devoid of anchors and their associated
loss mechanisms.
Through appropriate acoustical network analysis, the dimensions of a
flexural beam are found to correspond to a quarterwavelength of the operating
frequency when they satisfy the following expression:
where, W_{s} and L_{s} are the width and length of the
support beams, respectively, β_{n} = 4.730, 7.853, 10.996
for the first three modes of a clampedclamped beam and f_{0}
is the resonance frequency of the microresonator.
Estimating quality factor: The mechanical quality factor (Q) of
a resonator is:
where, ΔW denotes the energy dissipated per cycle of vibration and
W denotes the maximum vibration energy stored per cycle.
Many dissipation mechanisms exist in microelectromechanical resonators, such
as air damping, thermoelastic damping (TED), surface loss and support loss.
Unloaded Q of a microresonator is mainly the combination of these dissipation
mechanisms, expressed as (Bannon III et al., 2000):
Thus, to determine Q of the designed resonator, it was necessary to calculate
Q of each dissipation mechanisms as following:
Q_{air} denotes the quality factor due to energy loss mechanisms of
air damping and is determined as follows (Blom et al.,
1992; Cho et al., 1993):
where, k is stiffness of vibrating spring and b is damping coefficient of a
rectangular parallelplate geometries and has been derived from a linearized
form of the compressible Reynolds gasfilm equation as follows (Rebeiz,
2003; Cheng et al., 2002):
where μ = 1.78x10^{5} kg m^{1} sec^{1} (for
air in standard temperature and pressure (STP) conditions) is coefficient of
viscosity and proportional to gas pressure and consequently mean free path of
gas molecules (Abdelmoneum et al., 2003). A and
d_{0} are area of the device and gap between the two plates, respectively.
Q_{TED} denotes the quality factor due to energy loss mechanisms of
thermoelastic damping and is expressed as (Hao et al.,
2003; Braginsky et al., 1985; Lifshitz
and Roukes, 2000):
where α_{T} and C_{p} denote thermal expansion coefficient
and specific heat at constant pressure of the material used for the beam,
respectively; T_{0 }is the environmental temperature and where
C_{T} denotes thermal conductivity of the beam material and ω
denotes the angular frequency of the beam resonator.
Q_{Surface} denotes the quality factor due to energy loss mechanisms
of surface loss and the following expression it has been suggested for it (Hao
et al., 2003; Braginsky et al., 1985):
where, δ denotes the characterized thickness of the surface layer and
E_{ds} is a constant related to the surface stress (Yasumura
et al., 2000).
Q_{Support} denotes the quality factor due to energy loss mechanisms
of support loss and it can be calculated as follows (Mihailovich
and MacDonald, 1995):
where, KE_{tot} is the stored flexural vibration energy for each resonant
mode of a beam resonator can be expressed as (Cross and Lifshitz,
2001):
where, ω_{0} and U_{0} denote the fundamental angular
frequency of the vibration and the vibration amplitude, respectively and E_{loss}
is the energy dissipated per cycle of vibration through supports via., anchors
to substrate and for a clampedfree beam is calculated as follows (Hao
et al., 2003; Hao and Ayazi, 2005):
where, υ is Poisson ratio of the support material and Γ_{0}
is a fundamental vibrating shear force on support where attached to substrate,
which can be achievable by finite element analysis.
Plots of quality factor of the proposed microresonator versus electrodetoresonator
gap and versus ambient pressure are shown in Fig. 6
and 7, respectively.

Fig. 6: 
Plot of Q versus electrodetoresonator gap 

Fig. 7: 
Plot of Q versus ambient pressure of the microresonator 
MICROMECHANICAL RESONATOR CHARACTERISTICS
It must be noted that the simulated spectrum for the 71 MHz micromechanical
resonator shown in Fig. 3 was related to the not properly
terminated resonator and the transmission gain at the peak of the simulated
frequency characteristic was determined from an impedancemismatched single
resonator circuit. So, the simulated transmission gain was not the same
as insertion loss. Indeed, the transmission gain was calculated via.,
the following formula:
where, R_{p} is the polysilicon interconnect series resistance.
The material properties used in this study and surrounding conditions of the
resonator are shown in Table 1 (Hao
et al., 2003; Abdolvand et al., 2006).
Table 1: 
Material properties and surrounding conditions of the
resonator 

Table 2: 
VHF micromechanical resonator summary 

Table 3: 
Calculated Q of each dissipation mechanisms and total
Q of the resonator 

Table 2 shown the simulated micromechanical resonator characteristics
and finally, the calculated quality factor of the resonator and each of dissipation
mechanisms are presented in Table 3.
CONCLUSION
Design and simulation of a VHF micromechanical resonator based on the
new structure square frame suitable for operating around 71 MHz is reported.
The proposed microresonator exhibits series motional resistances considerably
smaller than that of other beam resonators by a factor equal to the number
of electrodes used in each resonator. The present method for R_{x}reduction
does not degrade linearity of the resonator and in contrast to arrayed
microresonators does not consume chip area anymore. This technique alleviates
some of remaining challenges that slow the advancement in integration
resonators, filters and oscillators into communication systems and helps
realization of a singlechip, fully integrated communication system based
on RF MEMS technology.
ACKNOWLEDGMENT
This research was supported by Iran Telecommunication Research Center
(ITRC).