Nonlinear Modeling and Investigating the Nonlinear Effects on Frequency Response of
Silicon Bulk-mode Ring Resonator
Abstract: This paper presents a nonlinear analytical model for micromechanical silicon ring
resonators with bulk-mode vibrations. A distributed element model has been developed to
describe the dynamic behavior of the micromechanical ring resonator. This model shows the
nonlinear effects in a silicon ring resonator focusing on the effect of large amplitudes around
the resonance frequency, material and electrical nonlinearities. Through the combination of
geometrical and material nonlinearities, closed-form expressions for third-order nonlinearity
in mechanical stiffness of bulk-mode ring resonators are obtained. Using the perturbation
method and the method of harmonic balance, the expressions for describing the effect of
nonlinearities on the resonance frequency and stability are derived. The results, which show
the effect of varying the AC drive voltage, initial gap, DC applied voltage and the quality
factor on the frequency response and resonant frequencies, are discussed in detail. The
nonlinear model introduces an appropriate method in the field of bulk-mode ring resonator
design for achieving sufficient power handling and low motional resistance.
Keywords: Micromechanical ring resonator, nonlinearity, bulk-mode, frequency response,
power handling
Introduction:
Today, in this rapidly-developing world of micro-communications, how to develop
micromechanical resonators and filters with high quality factor, high power handling
capability and low motional resistance, is a matter of debate. The silicon micromechanical
resonators due to their small size, low cost and compatibility with integrated circuit (IC)
technology are a promising alternative to surface acoustic wave (SAW) and quartz crystal
resonators. However, low motional resistance and high power handling make them difficult
1
to handle with linearity and small size. And, due to their small size, these resonators cannot
store high energy and, therefore, they should be driven at high excitation value which causes
them to be easily driven into nonlinear regimes [1]. Moreover, the most direct methods for
lowering the motional resistance in micromechanical resonators, like scaling down the
electrode to resonator gap and raising the DC applied bias voltage, decrease the linearity [2].
The ability to accurately model nonlinearity and investigate its effect on frequency response
is, therefore, a key requirement to optimum design of silicon micromechanical resonators.
Several mechanical and electrical nonlinearities exist in silicon micromechanical resonators.
Depending on the resonator design and operating conditions, different nonlinearities may be
dominant and result in hardening or softening behavior in the dynamic behavior of
micromechanical resonator [3]. A lot of research has been conducted on modeling the
nonlinear effects in micromechanical resonators. Gui et al. [4] investigated the nonlinear
effects on the performance of a micro-bridge, using Rayleigh’s energy method, and showed
the dependency of the hysteresis criterion on the quality factor, operating conditions,
geometric and material properties of the micromechanical resonator. Mestrom et al. [3]
studied the frequency responses and the nonlinear dynamic properties of clamped-clamped
beam resonator and predicted the hardening behavior. They also compared their analytical
results with the results from experiments and found a reasonable agreement. Alastalo et al.
[5] presented a very useful study on the longitudinal mode resonators with respect to
mechanical and electrical nonlinearity. Assuming a high quality factor, some of the
aforementioned works have ignored damping coefficient in calculating the shift resonance
frequency caused by nonlinearities. Moreover, the study on the nonlinearities in bulk-mode
micromechanical resonators is inadequate, and previous works have only focused on the
longitudinal and square extensional mode resonators [6]. This paper deals with the effects of
nonlinearities on frequency response of silicon bulk-mode ring resonators. First, a
2
comprehensive nonlinear model of this resonator is derived. The model includes the effect of
large amplitudes around the resonance frequency, material and electrical nonlinearities.
Then, the effect of key parameters on nonlinear frequency response is evaluated applying
perturbation techniques and the method of harmonic balance.
1. Bulk-mode Ring Resonator
Among silicon micromechanical resonators, bulk-mode ring resonators offer lower motional
resistance and higher quality factor due to their high structural stiffness, ring geometry and
having four quasi-nodal points at their outer periphery in some non-axisymmetric bulkmodes like wine glass and extensional wine glass. Therefore, they are more extensively
developed in RF transceiver front-end architectures [7].
r
θ
(a)
(b)
Figure 1: The schematic views of the bulk-mode shapes of the ring resonator a) wine glass (WG) b) extensional
wine glass (EWG)
In the bulk-mode ring resonators which are shown in Figure 1, assuming that the width of the
ring is much larger than its thickness (Ro-Ri >> t), the 2-D linear equations governing the
vibration of a ring, without body force and internal heat source can be given by [8,9]:
 2u
 E A
 E  A0



u





u


A
0

t 2
 12
 2(1   )
(1)
where E, υ, ρ and A are Young’s modulus, Poisson’s ratio, density of structural material and
transduction area of the resonator, respectively. The displacement vector u=xer+weθ is
defined in terms of the radial (x) and circumferential displacements (w) in polar coordinates
through the time and mode shape functions as follows:
3
x (t , )  X r (t )  cos(n ) , w (t , ) W r (t )  sin(n )
(2)
where
X r (t ) 


X rq cos(q t ) , W r (t ) 
q 1

W rq cos(qt )
(3)
q 1
where at the natural resonance frequency of ωnm=hnm√(E/ρ(1-υ2), the radial displacement at
inner and outer radius can be assumed as:
x o (t , )  X o (t )  cos(n ) , x i (t , )  X i (t )  cos( n )
(4)
and
X o (t ) U o (t )  U o  cos(t ) , X i (t ) U i (t )  U i  cos(t )
(5)
where Uo=Ur=Ro and Ui=Ur=Ri can be expressed by following equation:
Ur 
An
hnm
d 
 n C

Bn
D
Y n (hnm r )    n J n (hnm r )  n Y n (hnm r )  
  J n (hnm r ) 
An
An
 r  An
 
 dr 
(6)
In the above equations, Jn and Yn are Bessel functions of the first and second kind,
respectively. hnm and knm are mode consonants of non-axisymmetric mode shape of (n,m),
while knm is related to hnm by knm/hnm=√2/(1-υ). It should be noted that the relative constants
of elastic waves (Bn/An, Cn/An, Bn/An) can be found by solving the equation of det (M4×4) =0.
The element of M was explained in detail in Ref. [7,10].
1
NODAL SOLUTION
STEP=1
SUB =6
FREQ=.220E+09
USUM
(AVG)
RSYS=0
DMX =431715
SMN =8020
SMX =431715
8020
55097
Quasi-nodal points
MN
MX
Y
Z X
102174
196329
290483
384638
149251
243406
337560
431715
Figure 2: ANSYS simulation of EWG mode shape ring resonator
In the rest of the paper, the extensional wine glass (2,4) mode ring resonator with the
specifications presented in Table 1 has been used to evaluate the frequency response caused
4
by nonlinear effects. Therefore, the subscript nm is omitted and the subscript n=2
representing the second order mode are used in the following equations.
Table 1: The related EWG mode parameters for silicon ring resonator
Parameter
Inner radius (Ri)
Outer radius (Ro)
Thickness (t)
Electrode angle (θe)
Resonance frequency (fnm)
Linear Young’s modulus (E0)
Poisson ratio (υ)
Density (ρ)
First order corrections (E1)
second order corrections (E2)
Value
30.16
50
2
35
220
170
0.28
2330
-2.6
-8.1
Unit
μm
μm
μm
deg
MHz
GPa
Kgm-3
2. Modeling Approach
Since both actuation and detection of the micromechanical ring resonator is realized using
the inner and outer electrodes, it can be assumed as a two separate ring resonator attached at
the radius of (Ri+Ro)/2 and each ring resonator is also modeled as a distributed element.
Considering an infinitesimal element, the gap variations in capacitive transducers are
assumed along the radial direction.
Input electrodes
(Ro+Ri)/2
Ko
xo
Fo
Mo
fi(θ)
θe
fo(θ)
bo
Kc
Ki
mi(θ) mo(θ)
Mi
bi
Fi
xi
Output electrodes
(b)
(a)
Figure 3: a) The schematic view of the infinitesimal element b) The equivalent mass-spring damper system of
the ring resonator
Thus, the governing equation of a second order mechanical system (mass-spring-damper) for
an infinitesimal element of the ring resonator can be expressed as:

mo ( )x o   bo ( )x o   k mo ( )x o  k c  x o  x i   2 f do ( )  f so ( )
5

(7)

mi ( )x i   bi ( )x i   k mi ( )x i  k c  x i  x o   2 f d i ( )  f s i ( )

(8)
where x represents the radial displacements of the effective lumped mass of m and b, km and
kc are damping coefficient, mechanical stiffness and coupling stiffness, respectively. The
subscripts i and o denote the inner and outer ring resonators, respectively. The effective
masses for infinitesimal element in inner and outer resonators are given by:
Ro
mo ( ) 
  t  R i  Ro U r2  rdr
2
U o2
, m i ( ) 
R i  Ro
2 U2
r
Ri
U i2
 t  
 rdr
(9)
Assuming that the first mode is the dominant mode of the system, the inner to outer
displacement ratio can be expressed as:
x i (t , )
x o (t , )
Ui

Uo
(10)
Substituting equations (4) and (10) into equations (7) and (8) and combining them together
results in the following expression:
 mo ( )  m i ( )    cos(2 )  X o  (t )  bo ( )  bi ( )    cos(2 )  X o  (t ) 
k m
o
(11)

( )  k m i ( )   cos(2 )  X o (t )  f ( )
where

 
f ( )  2 f d o ( )  f d i ( )  2 f so ( )  f s i ( )

(12)
3. Modeling the Origins of the Nonlinearity
Nonlinearity in micromechanical resonators generally arises from electrical and mechanical
origins. The electrical nonlinearities include the electrostatic force nonlinearity caused by
variable gap capacitive transducers and fringing effect. The large structural deformation can
be contributed to geometrical and material nonlinearities, classified as mechanical
nonlinearities.
3.1. Electrical Nonlinearity:
6
In many micromechanical bulk-mode resonators, the initial gap between electrodes and the
resonator is not negligible as compared to the thickness of structure (small aspect ratio).
Also, the fringing fields are considerable. Therefore, the electrostatic force nonlinearity can
be modeled using the parallel plate capacitor approximation together with the correction term
of Cf representing fringing effects as follows:
f d
f s

( )  f d i ( ) 
o
o
C ( ) C ( )
1
V P  v i (t ) 2  o  i 
2
x d
 x
1
 C ( ) C i ( ) 
( )  f s i ( )  V P2  o

2
x s
 x

(13 a)
(13 b)
where
 C i ,o ( )  C f C 0i ,o ( ) 
2
3
4

  x i ,o  2 x i2,o  3 x i3,o  

 

d
d
d
d


 x
s
C f C 0i ,o ( ) 
 C i ,o ( ) 
2
3
4

1  x i ,0  2 x i2,o  3 x i3,o  

 

d
d
d
 d

 x
d
(14)
(15)
where C0o(θ)=εRot/d and Coi(θ)=εRit/d imply the capacitance over the outer and inner gaps
when the outer and inner radial displacements are equal to zero and d is the corresponding
initial gap width. According to sense (or drive) electrodes configuration, the value of β would
be equal to 1 or -1 as the sense (or drive) electrodes are placed at the same side or different
side, respectively. The correction term of capacitances for the fringing effect depends on the
dimensions of the structure as per the following equation [11]:
1  4.246 ; 0    0.005

C f  1  11.0872 2  0.001097 ; 0.005    0.05

0.8258
; 0.005  
1  1.9861
(16)
where λ=d/t is the ratio of initial gap per thickness of the ring structure.
3.2. Mechanical Nonlinearity
Mechanical (material and geometrical) nonlinearities are caused by the stresses developing
inside the resonator structure under large displacements. Depending on the operation mode,
7
the material or geometrical nonlinearities can be dominant in micromechanical resonators.
Unlike the electrical nonlinearities, which can be easily represented by Taylor series with
respect to displacement, the mechanical nonlinearities can be approximated for the bulkmode ring resonator by deformation analysis. For this purpose, the mechanical nonlinearity is
first modeled by adding the higher order terms to the mechanical stiffness as follow:

k mi ,o ( )  k m0i ,o ( ) 1  k 1i ,o x i ,o  k 2i ,o x i2,o

(17 a)
2
k m 0i ,o ( )  m i ,o ( )  nm
(17 b)
In order to consider the geometrical nonlinearities, the geometrical deformation caused by
large radial displacement in infinitesimal element could be given by:
e
A
2
e
A ( ) d 
(18)
2
where
 R
A ( )  R    t  1 
R

x 


  A0 ( ) 1  

r 


(19)
Figure 4: The geometrical variation in infinitesimal element of the ring resonator
Moreover, the large displacement develops internal stresses inside the structure of resonator
and therefore, nonlinear elastic behavior of silicon result in nonlinear Young’s modulus as
follows [12]:
2

x
 x  
E  E 0 1  E 1
 E2 
 


r

r

 

(20)
8
where E1 and E2 denote the first and second order corrections to linear Young’s modulus of
E0, respectively. The values of E1 and E2 are listed in Table 1 for bulk-mode silicon oriented
along [100] [13]. Substituting equations (18) and (20) into equation (1) and multiplying by
cos(2θ) and then integrating both sides, from θ=0 to θ=2π results in the approximate
expressions for third order nonlinearity in mechanical stiffness, k2o and k2i, as follow:
k 2o 

1
U o2
Ro
R i  Ro
2
Ro
R i  Ro
2

 (r )dr
(r )dr
, k 2i 
1
U i2
R i  Ro
2
Ri
R i  Ro
2
Ri

 (r )dr

(r )dr
(21)
where
2

  U 2   2U
U  U r 
r
r
 (r )  U r  E 2 (2  3)  E1  r 

(
E

E

)
2
1 

 
2

r

r

r


  r


   2U
r
(r )  U r  
  r 2


U
 U r 
  2  3 r
 r  

r
 r 

3

 U r   
r


 r   

  





(22)
(23)
In fact, the second-order nonlinearity term caused by symmetry of the ring structure and
mode shape is omitted. Using the self-written numerical codes in MATLAB software, the
values of k2o and k2i for the case study of this paper were calculated to be -2.842×10-11 Nm-1
and -1.024×10-10 Nm-1, respectively.
4. Analysis of the Nonlinear Effect on Frequency Response
Based on the electrostatic and mechanical nonlinearities and multiplying equation (11) by
cos(2θ) and then integrating over the range of 0 to 2π, the governing equation of a second
order mechanical system of micromechanical ring can be rewritten as:
X o  (t ) 
t
Q

 X o  (t )  t2  X o (t )   2 X o2 (t )  3X o3 (t )  g 0  v i (t ) g 1  g 2 X o (t )  g 3X o2 (t )

(24)
where ωt is the resonance frequency caused by electrical nonlinearity and Q denotes the
mechanical quality factor. The notations used in the above equation are presented in Table 2
and θe denotes the electrode to resonator overlap angle.
9
Table 2: Definitions used in the equation (24)
Parameter
Definition
t
K re M 1
K re
K m0  k e
K m0
 K m 0o   K m 0i
Ke
V P2d 2   C 0i  C 0o  sin(2e )  2e  
K m 0o ,i
0
Q
B 1 MK re
B
0
g0
V P2d 1M

2
2

k m 0o ,i ( )  cos 2 (2 ) d 
bo ( )  bi ( )     cos2 (2 ) d 
   1 C 0i  C 0o  sin(e )
2V P d 1M 1 C 0i  C 0o  sin(e )
V P d 2 M 1   C 0i  C 0o  sin(2e )  2e 
g1
g2
1

cos2 (e )  2 sin(e )
V P2d 3M 1    1  2C 0i  C 0o  cos2 (e )  2  sin(e )
2V P d 3M 1  2C 0i  C 0o
g3
2
3
M
4
3
1
K m
0o




1

1
k 2o   3 K m 0i k 2i V P2d 4  3C 0i  C 0o  sin(2e )  cos(e )    e 
3
2




The DC term in equation (24) has no effect on the nonlinear analysis of frequency response
and it could be ignored:
X o  (t ) 
t
Q

 X o  (t )  t2  X o (t )   0  d 1X o (t )   2 X o2 (t )   3X o3 (t )

(25)
where
 0  g 1v i (t ) , 1 
g 2v i (t )
d
, 2 
 g 3v i (t )  2 
d
, 3 
3
d
(26)
The approximate analytic method can be used to solve the equation above. Assuming weak
nonlinearities, this problem can be solved by perturbation technique. This method assumes
that the resonance frequency along with the solution varies as a function of perturbation
terms and the solution is given by:
X o (t )  X o0 (t ) 

d n X o
n 1
n
(t )
(27)
10
t  t 

(28)
d n n
n 1
The second order approximation of this solution can be used to achieve a more exact
evaluation as:
X o (t ) X o0 (t )  dX o1 (t )
(29)
t  t  d 1
(30)
Substituting equations (29) and (30) into equation (25) and grouping the terms in power of d,
the two following expressions are obtained:
X o0  (t ) 
X o1 (t ) 
t
X o0  (t )  t2 X o0 (t )   0
Q
t
Q
X o1 (t )  t2 X o1 (t ) 
1
Q
(31)
X o0  (t )   2t1  1  X o0 (t )   2 X o20 (t )   3X o30 (t )
(32)
When the input drive voltage vi(t)=|vi|cos(ωt) is used, the solution of Xo0(t) can be obtained as


X o0 (t )  Re g 1 v i e jt  H ( )  g 1 v i  H  cos(t )  H  sin(t ) 
(33)
where
H ( ) 
1
t    j
2
2
t
 H ( )  jH ( )
(34)
Q
where H΄ω and H˝ω denote the real and imaginary parts of the transfer function of H(ω),
respectively. Substituting equation (33) into the right hand side of equation (32) and setting
the secular terms to zero, the expression of ω1 is approximately given by:
2
3 3 g1 v i
1  
8
2
 H 
2
 H  2

(35)
t
Thus, substituting equation (35) into (30), the resonance frequency caused by mechanical and
electrical nonlinearities, ώt, is expressed as follows:
t 
t
2

(36)
1
3
t2  3 g12v i2 (H  2  H  2 )
2
2
11
As shown in the equation above, ώt is a function of the amplitude of the first order
approximation of the solution. By Substituting equation (33) into (32), the solution of Xo1(t)
is calculated and, therefore, Xo(t) can be expressed as follows:
X o (t )
g1 v i
G1 cos(t ) G1sin(t )  G 2 cos(2t ) G 2 sin(2t )  G3 cos(3t ) G3sin(3t )
(37
)
where
3
G1  H   g 1 v i
4
2
3
G1  H   g 1 v i
2
2





4
4
2 1
2 
 g 13 H   H   g 3H   H   H   
3




(38 a)


5 
1

H   g 13H  H  2  H  2  g 3  H  2  H  2  
6
5






(38 b)

1
G 2   v i g 12 2H 2 H  H   H 2 H  2  H  2  g 2  H  H 2  H  H 2 
2
1
G 2   v i g 12 2H 2 H  H   H 2 H  2  H  2  g 2  H  H 2  H  H 2 
2
1
G 3   g 1 v i
4




(39 a)
(39 b)
2
1 
3
2
 g 13H 3 H   3H 3 H   g 13H   g 3 
3 


(40 a)

2 

3H 3 H  H   g 13H   g 3   H  2 H 3  g 13H   g 3  
3 


1
G 3   g 1 v i
4
21
1 
3
2
 g 13H 3 H   3H 3 H   g 13H   g 3 
3 

2
(40 b)

3
2 

 H 3 H  H   g 13H   g 3   H  2 H 3  g 13H   g 3  
2
3 


where H΄nω and H˝nω (n= 2,3), denote the real and imaginary parts of the transfer function of
H(nω), respectively. Substituting equation (37) into (4a), the output current for infinitesimal
element of micromechanical ring resonator can be expressed using the following equation as:
i om (t , ) 
2V P x o (t , )  2
(41
d  C 0o     C 0i     2d C 0o   2C 0i x o (t , )  3 C 0o   3C 0i x o2 (t , ) 
3



t
d




)
Multiplying by cos(2θ) and then integrating both sides, from 0 to 2π, the total output current
due to applied drive voltage at resonance frequency, is finally derived as:
i o (t ) 

 i oq cos(qt  q )
(42)
q 1
12
The experimental results of the study conducted by Xie et al. [7] were used to experimentally
verify derived expression for presented model in linear regime and electrical stiffness. As
shown in Figure 5 (a) and (b), the frequency response is measured using the mixing approach
[7]. To adapt this approach to the proposed model, the equations of |vi|=(VLO|vRF|)/2 and
Po=(RL|io|2)/2, (RL=50 Ω) [14] are used in the derived expressions.
-50
Exprimental [7]
Exprimental
-80
linear model [7]
linear model [7]
Nonlinear model
Nonlinear model
Power (dBm)
Power (dBm)
-60
-70
-80
-90
-100
-90
220.5
220.8
221.1
435.3
Frequency (MHz)
435.6
435.9
436.2
Frequency (MHz)
(a)
(b)
Figure 5: The frequency response of polysilicon EWG mode ring resonator a) 220.8 MHz b) 435 MHz
It can be seen from Figures 5 and 6 (a) that the frequency response and resonance frequency
change of the nonlinear model agree well with experimental data. However, the low
difference between their amplitude and resonance frequency may be arise from the parasitic
capacitance or resistance in experimental measurement.
0
-40
(ft-fnm)/fnm (ppm)
(ft-fnm)/fnm (ppm)
0
-80
Experimental [7]
Analytical [7]
Analytical (This work)
-120
0
2
4
6
8
10
DC applied voltage (V)
-1000
-2000
-3000
-4000
2
1.5
12
14
x 10
-7
1
d (m)
(a)
0.5
50
60
70
80
90
100
Vp (V)
(b)
Figure 6: The fractional resonance frequency change caused by electrical nonlinearity for
a) Polysilicon EWG ring resonator with f0=220.8 MHz b) Silicon EWG mode ring resonator (as Table 1)
13
According to the expression of electrical stiffness, Ke, when applied DC voltage (VP),
increases or the initial gap between the electrodes to resonator (d) decreases, the resonance
frequency caused by electrical nonlinearity (ωt), shifts down as shown in Figure 6(b).
In order to address the nonlinear effects on frequency response, assuming that the first mode
is the dominant mode and using the first order approximation, the solution is assumed as:
X o (t )  X o cos(t )
X o  g1 v i
(43)
H  2  H  2
(44)
Thus, the phase difference between the drive applied voltage vi(t), and solution Xo(t), is
maintained as the phase of applied voltage and is fixed as the phase of solution as follows:
v i (t )  v i cos(t   )  v 1 cos(t )  v 2 sin(t )
(45)
Substituting equations (28) and (29) into (25) and using the method of harmonic balance, the
following equations are obtained:
v1 


X o 33X o2  4  2  t2
3g 3X o2

 4g1
, v2 

4t  X o
Q g 3X o20  4 g1
(46)

Substituting the above equation into v12+v22=vi2, leads to the relation between the amplitude
of drive applied voltage and vibration amplitude as:


 X 3 X 2  4  2   2
3 o
t
 o

3g 3X o2  4 g1


 
2

4t  X o

 
2
  Q g 3X o  4 g 1


2


 v
i


2
(47)
To study the effect of drive voltage on the frequency response, the DC bias voltage VP, is
assumed constant and the drive voltage varies. In Figure 7(a), it is shown that as drive
voltage (vi) increases, the resonator frequency response changes from an almost linear curve
to a completely nonlinear one (hysteresis). In fact, higher drive voltage leads to larger
vibrations and eventually results in a jump in frequency response. This behavior is evident
when there are two amplitudes of vibration for a given forcing frequency at high drive
14
voltage. Moreover, the peak frequency tilts to a lower frequency as the vibration amplitude
increases and the ring resonator, therefore, exhibits a softening behavior.
10
180
vi=3 Vpp
vi=4 Vpp
ώt
150
vi=5 Vpp
vi=6 Vpp
6
4
(degree)
(
Xo x10
-8
)
8
120
90
vi=3 Vpp
60
vi=4 Vpp
2
vi=5 Vpp
30
vi=6 Vpp
0
219.70
219.72
219.74
219.76
219.78
0
219.70
219.80
219.72
Frequency (MHz)
219.74
219.76
219.78
219.80
Frequency (MHz)
(b)
(a)
Figure 7: Effect of AC drive voltage on frequency response of a EWG mode ring resonator with Q=8000, d=90
nm and VP=100 V
10
10
VP=90 v
VP=100v
8
d=80 nm
d=90 nm
d=100 nm
d=110 nm
8
Xo x10
-8
)
)
-8
4
6
(
(
Xo x10
)
-8
VP=120v
6
(
Xo x10
(
Xo x10
-8
)
VP=110v
2
4
2
0
219.55 219.60 219.65 219.70 219.75 219.80 219.85
Frequency (MHz)
0
219.6
219.7
219.8
219.9
Frequency (MHz)
(a)
(b)
Figure 8: Effects of DC bias voltage and initial gap variations on frequency response of a EWG mode ring
resonator with Q=8000 a) d=90 nm and vi=5 VPP b) VP=100 V and vi=5 VPP
It can also be seen from Figure 8 that an increase of the VP or a decrease in d, tilts the
resonance peak more, and the softening behavior of the micromechanical ring resonator
becomes more pronounced. Quality factor is a fundamental parameter to design of
micromechanical resonators for MEMS-based filters with a low insertion loss. There exist
several intrinsic and extrinsic energy-loss mechanisms in the bulk-mode resonators like air
damping, anchor loss and thermoelastic damping (TED); for a bulk-mode resonator operating
15
in high vacuum, the air damping mechanism can be ignored. Also, in these resonators, the
thermoelastic damping is negligible as compared to the anchor loss and, therefore, anchor
loss is the dominant energy-loss mechanism [15]. It is demonstrated that, anchor loss is
strongly dependent to the support beam dimension, and it can be reduced by mechanically
isolating the support beams from the substrate and optimum design of support beams
dimension [16,17]. As shown in Figure 9 (a), the nonlinear effects on frequency response
become more apparent by increasing quality factor. The greatest vibration amplitude before
hysteresis, called the critical vibration amplitude, Xc, can be used to estimate the limit for
power handling as given by Ec=(Km0Xc2)/2 [12]. Where Ec is the maximum stable energy
stored in the resonator. In Figure 9 (b), it is shown that under the same condition, a higher
quality factor decreases the critical vibration amplitude and therefore, lowers maximum
stable energy stored.
10
8
Q=6000
Q=8000
Q=10000
Q=12000
)
6
(
Xo x 10
-8
6
(
Xo x 10
-8
)
8
4
Q=6000
Q=8000
Q=10000
Q=12000
7
Xc=51.6 nm
5
4
Xc=45.3 nm
Xc=40.8 nm
Xc=37.7 nm
3
2
2
1
0
219.70
219.72
219.74
219.76
219.78
0
219.70
219.80
Frequency (MHz)
219.72
219.74
219.76
219.78
219.80
Frequency (MHz)
(b)
(a)
Figure 9: The Effect of Q-factor on a) Frequency response of a EWG ring resonator with vi=4 VPP, d=90 nm and
VP=100 V b) Critical vibration amplitude (XC) of a EWG ring resonator with d=90 nm and VP=100 V
5. Conclusion
This paper deals with the nonlinear modeling of micromechanical silicon ring resonators with
electrostatic actuation and detection on both inner and outer radius. The origins of
nonlinearities in the micromechanical resonators were completely modeled as a distributed
16
element using Taylor series and deformation analysis. Based on the derived expressions for
third-order nonlinearities in mechanical and electrical stiffness, the expression of the
resonance frequency was obtained using the perturbation method. The nonlinear effects on
frequency response were addressed in details, as well. It was shown that the micromechanical
ring resonator exhibits the softening behavior. Moreover, there is a trade-off between the
conventional methods to reduce motional resistance and stability of the frequency response in
micromechanical ring resonators. While the applied DC voltage is increased or the initial gap
is decreased, the ring resonator becomes more susceptible to nonlinear effects. In addition,
increasing the resonator quality factor was shown to tilt the peak frequency toward lower
frequencies and decrease the maximum energy stored. The presented model and the results of
this paper allow designers to optimize the design of micromechanical ring resonators and
improve the performance of MEMS-based filters and oscillators.
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18
[14] Wang, J., Ren, Z. and Nguyen, C. T. C., “1.156-GHz Self-Aligned vibrating
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19