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Multiple Timescales and Modeling of Dynamic
Bounce Phenomena in RF MEMS Switches
Ryan C. Tung
Purdue University, Birck Nanotechnology Center, rtung@purdue.edu
Adam Fruehling
Purdue University, Birck Nanotechnology Center, soimems@purdue.edu
Dimitrios Peroulis
Purdue University, Birck Nanotechnology Center, dperouli@purdue.edu
Arvind Raman
Purdue University, Birck Nanotechnology Center, raman@purdue.edu
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Tung, Ryan C.; Fruehling, Adam; Peroulis, Dimitrios; and Raman, Arvind, "Multiple Timescales and Modeling of Dynamic Bounce
Phenomena in RF MEMS Switches" (2014). Birck and NCN Publications. Paper 1520.
http://dx.doi.org/10.1109/JMEMS.2013.2271252
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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 23, NO. 1, FEBRUARY 2014
137
Multiple Timescales and Modeling of Dynamic
Bounce Phenomena in RF MEMS Switches
Ryan C. Tung, Adam Fruehling, Dimitrios Peroulis, and Arvind Raman
Abstract— Electrostatically operated RF-MEMS switches are
known to suffer from discrete switch bounce events during
switch closure that increase wear and tear and lead to increased
switching times. Here, we use laser Doppler vibrometer to analyze
the switch response of three types of cantilevered dc-contact
switches at a 200 ns time resolution. We find that bounce events
are multiple time scale events with distinct motion occurring
at 10−1 and 101 s timescales in effect high frequency bounces
within a bounce. To understand the origin of this effect, we
develop a multiple eigenmode model of a cantilever switch with
electrostatics, repulsive and adhesive contact forces, and rarefied
gas damping and find that the high frequency bounce arises from
the transient excitation of the 2nd eigenmode of the cantilever
structure of the RF-MEMS switch. This phenomenon not only
describes the multiple time scales involved in bounce events, but
also shows that the transient excitation of the second mode leads
to complex drum roll like dynamics, leading to a series of closely
spaced impacts in each actuation cycle. A careful study of the
dependence of the phenomenon on contact stiffness and adhesion
shows how the landing pad stiffness, adhesion, and actuation
voltage in dc contact switches can increase or diminish repeated
impacts during actuation.
[2012-0228]
Index Terms— RF MEMS, switch bounce, laser Doppler
vibrometer (LDV), switch wear.
I. I NTRODUCTION
M
ICROMECHANICAL RF MEMS switches can
increase signal isolation, signal linearity, and shock
resistance while decreasing power consumption and insertion
losses [1], however, their reliability remains a concern [2].
While several key failure mechanisms can affect RF-MEMS
switches [3], the contact wear due to repeated actuation is
considered an important source. Switch bounce refers to the
phenomenon where the switch, upon electrostatic actuation,
fails to immediately close the air gap but rather suffers
intermittent bounces before closing completely. This leads to
a higher potential for electrical arcing, increased contact wear
which may lead to premature stiction, and an overall lose of
signal integrity [2]–[4].
Manuscript received August 6, 2012; revised April 24, 2013; accepted
April 30, 2013. Date of publication July 24, 2013; date of current version
January 30, 2014. This work was supported in part by the Department of
Energy and the National Nuclear Security Administration under Award DEFC52-08NA28617. Subject Editor C. Rembe.
R. C. Tung and A. Raman are with the School of Mechanical Engineering,
Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907
USA (e-mail: tung.ryan@gmail.com; raman@purdue.edu).
A. Fruehling and D. Peroulis are with the School of Electrical and
Computer Engineering, Birck Nanotechnology Center, Purdue University,
West Lafayette, IN 47907 USA (e-mail: adam.fruehling@gmail.com; dperouli@purdue.edu).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JMEMS.2013.2271252
The phenomenon of switch bounce has been observed since
the fabrication of the earliest MEMS switches [5]. In fact its
existence pre-dates MEMS and has been observed in macroscale electromagnetic relays [6]. Since Petersen [5] reported
microscale switch bounce in 1979 much work has been done
to understand the nature of switch bounce in MEMS [7].
McCarthy et al. [8] developed a dynamic MEMS switch
model using a finite difference scheme in both time and
space: a Reynolds based squeeze film damping model was
used and a piecewise linear contact stiffness was assumed.
Switch bouncing was captured, but its origin and dependence on system parameters was not discussed. Decuzzi [9]
et al. used a similar approach, but included an adhesive
tip force in their model. Ostasevicius [10] used a 2D FEA
model to study MEMS cantilever switch closure and bounce.
Guo et al. [11] used a 3D nonlinear FEA model, along with a
finite difference scheme in time to simulate MEMS switch
closing events. Krylov and Maimon [12] used a Galerkin
procedure to study the pull-in instability dynamics of a MEMS
switch. All these works have helped improve the modeling of
switch closing events, but the underlying physics to explain,
predict bounces, and design against bounces remains elusive.
Additionally, in [13], we present a new method to monitor
in situ the bounce in RF MEMS switches.
In this work we investigate the origin of switch bounce
through a compact multiple eigenmode model, incorporating
electrostatics, contact forces, and squeeze film gas damping.
Specifically we focus on DC-contact RF MEMS switches, with
cantilevered structures whose contacts can be approximated
by point contact models. Switches whose contact is better
represented by area-type contact models, such as capacitive
RF MEMS switches, will be neglected. In the latter cases,
switch bouncing is highly suppressed due to large contact area,
electrostatic, adhesive, and gas damping forces [14].
The origin of switch bounce is studied through the use
of efficient reduced order models based on cantilever eigenmodes, thus avoiding the more computationally intensive
approaches using spatial finite difference schemes, or FEM
analyses [15]. This also allows for computationally efficient
studies of parameter variation and optimal selection of system
parameters. MEMS switch analysis based upon reduced order
models generated from component eigenmodes of the system
has been studied [16], [17] in the context of resonators, here
we present a reduced order model that accurately explains the
origin of multiple timescale switch bouncing in DC-contact RF
MEMS switches. Experimental data are obtained using a laser
Doppler vibrometer (LDV) on three distinct cantilevered RF
MEMS devices. In addition we provide a parametric analysis
1057-7157 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
138
JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 23, NO. 1, FEBRUARY 2014
based on key non-dimensional parameters in an attempt to
minimize switch bouncing, in contrast other researchers have
attempted to mitigate switch bounce events through tuning of
the input actuation voltage waveform [4], [18].
(a)
II. E XPERIMENTS
We begin with experiments performed with different DCcontact switches. These devices are single crystal silicon RF
MEMS switches previously reported in [19], with gold to
gold contact. A Polytec Microsystems Analyzer (MSA) 400
LDV was used to measure the response of the beams to
various step voltage inputs. All data for this analysis was
collected synchronously with the experiments conducted in
[13] to ensure consistency.
Figure 1 shows scanning electron microscope images of
the three types of switches tested. Type I switches refer
to cantilevered switches with a contact pad fixed on three
sides, Type II switches refer to cantilevered switches with a
cantilevered contact pad, Type III switches refers to U-shaped
cantilevered switches with a cantilevered contact. Additional
information about the fabrication together with the details of
the low noise capacitive measurements that were performed
in parallel with laser-Doppler-vibrometery measurements are
available by direct request from the authors, and have also
been submitted for publication [13].
All measurements were conducted at atmospheric pressure
and room temperature in air. The laser spot of the LDV
was focused as close to the tip of the switches as possible,
while avoiding the contact pad. Figure 1 also shows typical
measurement data for the various switch types at different bias
voltages, notice that the bouncing phenomenon is apparent in
every switch measured. In Fig. 1(a) we see a slight change in
the displacement over time caused by a constant velocity bias
in the measurement, which we have attempted to remove in
all cases. Note that the raw measurement data is velocity vs.
time, which we integrate to provided displacement vs. time.
Figure 2 shows the measured response, sampled at
5.12 MHz, of a 240 μm long Type I RF switch actuated
at 230 V, again the bouncing phenomenon is quite apparent.
The inset picture in Fig. 2 shows a zoomed in portion of
the measured response. Notice that the ratio between the time
period of the fundamental bouncing, T 1, and the time period
of the secondary bouncing, T 2, corresponds to approximately
6.3, which reflects the analytical frequency ratio of the second
transverse eigenmode to the first, f 2 / f 1 , for a cantilevered
beam.
To elucidate this phenomenon we utilize the continuous
wavelet transformation [20]. Additionally, the resolution in the
time-frequency plane is variable, as opposed to the short-time
Fourier transform (STFT) which has a fixed time-frequency
resolution. Here we have chosen to use the “Ricker” wavelet
function for the analysis, which is proportional to the second
Hermite function, due to its geometric similarity with the
bouncing phenomenon. Other wavelets, such as the “Meyer”
wavelet, can give similar results. Figure 3 shows the continuous wavelet transformation of the signal in Fig. 2, the high
frequency oscillations of the second mode are clearly visible
(b)
(c)
Fig. 1. Characteristic responses for : a.) 240 μm long Type I RF switch
actuated at 260 V, b.) 240 μm long Type II RF switch actuated at 260 V, c.)
240 μm long Type III RF switch actuated at 220 V.
at each impact event in scales 1–3. Recall that the scale is
inversely proportional to the frequency, larger scales represent
low frequencies, while smaller scales represent higher frequencies. Figure 4 shows the evolution of the experimentally
measured response and wavelet transformation of a typical
Type I RF switch to increasing input bias voltages. The multiple timescale bouncing signature apparent in the wavelet data
increases in intensity as the applied voltage bias is increased.
As the voltage is further increased the duration of continued
bouncing also increases. With additional voltage, not shown in
Fig. 4, the double bouncing phenomenon transitions to triple
bouncing. We now develop a compact model to explain this
TUNG et al.: MULTIPLE TIMESCALES AND MODELING OF DYNAMIC BOUNCE PHENOMENA
139
(a)
(b)
Fig. 2. Characteristic response of type I RF switch to an applied voltage
measured by the Polytec MSA-400 LDV at a sampling rate of f s =
5.12 MHz. Data reflects experiment for a 240 μm long cantilevered switch
with cantilevered contact at an applied voltage equal to 230 V. Inset image
shows an expanded view of the data, the bouncing phenomenon is clearly
visible. The bouncing frequency corresponds to a ratio of approximately 6.3
time the fundamental frequency, indicating excitation of the second eigenmode
of the RF switch.
(c)
Fig. 3.
Continuous Wavelet transformation of data in Fig. 2 using the
“Ricker” wavelet. The LDV displacement vs. time data is represented by
the blue curve. The high frequency components of the signal can be see in
the range of scales from 1–3, notice that these components are stimulated at
each contact event.
phenomenon and understand the key system parameters on
which it depends.
III. T HEORY
The simple geometry of the cantilevered DC contact
switches allows their dynamics to be modeled with the EulerBernoulli equation given by:
∂ 4 w(x, t)
+ C f ẇ(x, t) = FC + FE , (1)
∂x4
where w(x, t) is the transverse displacement of the beam,
ρ is the beam density per unit length, A is the cross sectional
area, E is the elastic modulus, I is the second moment of
area, overdots represent differentiation with respect to time,
FC represents contact forces, and FE represents electrostatic
forces. The list of symbols used in the model description
equations are summarized in Table I for brevity. A simplified
diagram is shown in Fig. 5.
The damping coefficient, C f , is estimated using the compact
model of Guo and Alexeenko [21]. This model accurately
ρ Aẅ(x, t) + E I
Fig. 4. Continuous Wavelet transformation of the response of a type I RF
switch to various applied voltages using the “Ricker” wavelet. The LDV
displacement vs. time data is represented by the blue curve. a.) shows the
response and wavelet transformation for the measured switch response at a
220 V applied voltage bias. b.) shows the response and wavelet transformation
for the measured switch response at a 250 V applied voltage bias. c.) shows
the response and wavelet transformation for the measured switch response at
a 280 V applied voltage bias. Notice that the intensity of the bounce signature
described in Fig. 3 increases as the applied voltage is increased, signifying
greater excitation of the 2nd eigenmode.
captures the sub-continuum squeeze film damping between a
vibrating microcantilever and nearby wall, and is based on the
numerical solution to the Boltzmann kinetic equation. C f is
given by [21]:
c
A bg h
(2)
Cf =
d e ,
b
λ
1+ B g
b
where g represents the gap distance between the beam and
nearby wall, λ is the mean free path of the gas, b is the beam
width, A, c, d, and e are fitting coefficients. For this problem
there are two relevant gaps, g0 and gs , the gap between the
beam and bias bridge and the gap between the beam and
the bottom substrate, respectively. Additionally, we assume
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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 23, NO. 1, FEBRUARY 2014
TABLE I
L IST OF M ODELING PARAMETERS
Fig. 6. Schematic of linear hysteretic contact adhesion force model. The
y-axis represents the contact force, the x-axis represents the tip displacement.
As the switch approaches contact, the force follows the blue “approach” curve.
As the switch leaves contact the red “retract” path is followed. gc is the
distance to the contact, ga is the parameter that defines the hysteretic contact
adhesion zone, and ks is the contact stiffness.
direct solid contacts or due to the formation of a capillary neck
under elevated humidity conditions. Figure 6 shows a diagram
of the force model used, which has been experimentally
observed in many cases [22], [23]. As the switch comes into
contact at gc it experiences the linear spring with stiffness
ks , as the switch leaves contact it experiences an additional
force proportional to ks for a distance ga . This force model is
meant to qualitatively simulate the overall contribution of van
der Waals forces, the meniscus or capillary force, and forces
due to chemical bonds [23].
The electrostatic force per unit length applied to the beam
from the bias bridge is given by:
Fig. 5. Schematic of RF MEMS switch used in simulation. w(x, t) is the
transverse displacement of the switch, g0 is the distance between the beam
and bias bridge, gs is the distance between the beam and the substrate, gc is
the distance between the beam and contact, ks is the stiffness of the contact,
and L is the length of the beam.
that damping forces due to each gap can be superimposed.
With this assumption we replace g in Eq. 2 with g0 − w(x, t)
for damping caused by the gap g0 and gs +w(x, t) for damping
caused by the gap gs .
The contact force experienced at the tip of the beam is given
by:
FC = −ks [gc − w(L, t)]δ(x − L)H [(w(L, t) − gc )]
−ks [gc − w(L, t)]δ(x − L)
× [H [w(L, t) − (gc − ga )] − H [w(L, t) − gc ]]
×H [−Ẇ (L, t)],
(3)
δ is the Dirac delta function and H is the Heaviside step
function,
0, n < 0,
H [n] =
1, n ≥ 0.
Equation 3 represents a hysteretic linear contact force model.
This model will be used to explore the role of adhesion due to
1 2 ∂C
V
,
(4)
2 ∂w
where C, accounting for fringing field corrections, is given by
[24]:
1
4
b
b
C = 0
+ 0.77 + 1.06
g0 − w(x, t)
g0 − w(x, t)
1 2
h
+ 1.06
,
(5)
g0 − w(x, t)
FE =
where 0 is the free space permittivity. In order to model the
voltage ramp applied to the device in an experimental setting
−t
we replace the constant V with V (t) = Vdc (1 − e τV ). Where
τV represents the voltage ramp time constant, and Vdc is the
desired constant voltage bias applied. To aid in the solution
of Eq. 1 the electrostatic force is represented as a 4t h order
Taylor series expansion in the form of:
FE = α0 + α1 w + α2 w2 + α3 w3 + α4 w4 .
(6)
This approximation is accurate to within 5% of the analytical
expression across the entire calculation range. The damping
factor C f is also represented in a 2nd order Taylor series for
each gap, in the form of:
C f = γ0i + γ1i w + γ2i w2 .
(7)
TUNG et al.: MULTIPLE TIMESCALES AND MODELING OF DYNAMIC BOUNCE PHENOMENA
141
TABLE II
L IST OF S IMULATION PARAMETER VALUES
This approximation has a mean error of 5% across the entire
calculation range. i is an index that represents which gap
the damping coefficient is modeling. The coefficients of the
expansion will be dependent on the gap being modeled, for
instance γ00 represents the first Taylor coefficient for damping
caused by the gap g0 and γ0s represents the first Taylor
coefficient for damping caused by the gap gs .
The equation of motion is then non-dimensionalized as
follows:
ŵ =
t
x
2π
w
τ=
x̂ =
T0 =
,
gc
L
T0
ω1
where ω1 is the first natural frequency of the beam and is
given by:
EI
ω1 = β12
,
(8)
ρ AL 4
where β1 is the first root of the dispersion relation for the
beam. The equation of motion is then re-written as:
2
¨ x̂, τ ) + T0 E I ŵ + C f (w)T0 ŵ˙
ŵ(
ρ AL 4
ρA
2
T
= 0 ( FˆC (w) + FˆE (w)),
ρ Agc
(9)
primes denote differentiation with respect to x̄.
We then discretize the transverse displacement using as a
functional basis the eigenmodes of a free cantilevered beam,
which constitute a complete set for the transverse displacement
field:
ŵ(x̂, τ ) =
∞
qn (τ )φn (x̂),
(10)
n=1
where φn (x̂) represents the n t h transverse bending mode of the
cantilevered beam, qn (t) represents the temporal component
of the solution. The modes are normalized such that φn (1) =
1 [25]. Using the relation φn (x̂) =
∞
q̈n φn + T02
n=1
=
∞
ρ AL 4 2
E I ωn ,
Eq. 9 becomes:
∞
C f (qn )T0 q̇n φn
ρA
n=1
F̂c (qn ) + FˆE (qn ) ,
qn ωn2 φn +
n=1
T02
ρ Agc
(11)
here we have left C f , F̂c , and FˆE unexpanded for clarity.
A Galerkin projection [26] is then performed on the above
equation, in which the inner product is taken with each
respective mode shape to obtain n, coupled 2nd order ordinary
differential equations in qn . Through modal orthogonality one
can show that:
1
φi φ j d x̂ = 0 for i = j.
(12)
0
This system is then decomposed into a set of coupled
first order differential equations for solution in an ordinary
differential equation solver such as MATLAB [27].
The coupled nonlinear ordinary differential equations resulting from the Galerkin projection procedure for n = 2
are provided in Appendix A. These coupled equations are
then solved numerically, from which they can readily be redimensionalized by inserting appropriate parameters gathered
from measurement.
Table II provides the parameter values used in the simulation. As indicated by Table II the parameter values used were:
based on nominal parameter values, measured experimentally,
or empirically tuned. For the gas damping model we have
used room temperature and an ambient pressure of 1 atm.
The thickness of the beam was calculated from the measured
first natural frequency and Eq. 8 using nominal values for the
remaining parameters. The frequencies of the first and second
transverse bending modes were obtained experimentally using
a ring-down analysis [28]. The contact stiffness and voltage
time constant were empirically tuned, to match the experimental data.
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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 23, NO. 1, FEBRUARY 2014
Fig. 8. Decomposition of total response into constitutive eigenmodes, for
n = 2 simulation. Here we have simulated the response using a large contact
stiffness. Notice that at every contact event the second eigenmode experiences
an impulsive forcing. Without the contribution of the second eigenmode, the
bounce phenomenon would not occur.
Fig. 7. Measured and simulated tip displacement for initial bounce events.
Here we show results for simulations for which n = 1, 2, and 3 modes.
It can be seen that for n = 3 the results are qualitatively similar to n = 2.
IV. S IMULATION R ESULTS
Here we present the simulation results obtained from the
previously derived model. We first begin with the case where
adhesion is negligible, ga = 0. For each simulation a bias
voltage is simulated with an exponential rise to the desired
DC value. Zero displacement and zero velocity initial conditions are used. The coupled differential equations are then
numerically solved.
Figure 7 shows the simulated transient dynamics of the
switch vs. the experimentally measured data for a Type I
device. Here we have conducted three simulations for n = 1,
n = 2, and n = 3, where n is the number of degrees of
freedom in the n DOF reduced order model.
For n = 1 the fundamental period of bouncing is accurately
captured, however, the multiple impact events are not present.
The n = 1 system lacks sufficient degrees of freedom to
quickly rebound at the stiff contact and bounce, and thus
remains in contact until the trajectory of the periodic motion is
reversed and the contact spring has fully relaxed. For n = 2 we
see that the intermittent contact phenomenon is now present,
and it is due to the excitation of the second eigenmode of
the structure. For n = 3 the results are qualitatively similar to
n = 2. The added complexity of a 3 DOF reduced order model
does not provide new information to describe the multiple
impact phenomenon observed in the experimental data, and
henceforth we utilize the n = 2 model.
It can be seen that in Fig. 7 the simulations for n ≥ 2 agree
very well with the experimentally measured data. Note that
for the n = 3 simulation the third transverse bending mode
frequency was analytically calculated, rather than experimentally measured, this is because this frequency is beyond the
bandwidth of the LDV detector (1.5 MHz).
We now repeat the above n = 2 simulation, with a
much larger contact stiffness, in an attempt to elucidate the
multiple timescale bounce phenomenon and demonstrate that
it is caused by the excitation of the second eigenmode.
Figure 8 shows the calculated total response, along with the
individual modal responses. The total response shows the
characteristic bounce phenomenon. Examining the response
of the 1st mode the fundamental period of bouncing is
apparent, along with the individual contact events of the
switch, indicated by the saturation of amplitude of the 1st
mode as contact is made. Examining the 2nd mode we see
that at each contact event apparent in the 1st mode the
second mode responds sharply, as if excited by an impulse.
The impulsive excitation of the 2nd eigenmode of the RF
switch is in fact, the basis of the multiple timescale bounce
phenomenon.
Now that the origin of the multiple timescale bounce
phenomenon has been identified, a parametric analysis using
key non-dimensional parameters of the system is performed to
understand this phenomenon. This parametric analysis is easily
afforded by the compact 2 DOF reduced order model, which
is computationally inexpensive. Similar analysis performed
using FEA would be computationally expensive and complex.
The key non-dimensional parameters are as follows, the nondimensional electrostatic force, F˜E , given by:
F˜E =
12π 2 12 V 2 0 L
gc2 β14 kc
,
and the non-dimensional contact stiffness, k̃, given by:
k̃ =
12π 2 ks L
.
β14 kc
For this initial parametric analysis we neglect adhesion,
ga = 0. The non-dimensional electrostatic force, F˜E , can be
TUNG et al.: MULTIPLE TIMESCALES AND MODELING OF DYNAMIC BOUNCE PHENOMENA
Fig. 9. Non-dimensional bounce parameter space for an individual actuation
event, for n = 2 simulation. The x-axis represents the non-dimensional ratio of
contact stiffness to switch stiffness. The y-axis represents the non-dimensional
electrostatic force. The overlayed black curves represent the characteristic
bouncing response in the region they are located. Note that number of bounces
was only computed for the initial contact event, subsequent contacts as the
beam settles to permanent contact during the actuation cycle my have more
or less bounces.
thought of as the ratio of electrostatic force to elastic restoring
force of the RF switch. The non-dimensional contact stiffness,
k̃, is the ratio of the contact stiffness to the static stiffness of
the RF switch. The static stiffness, kc is given as:
kc =
3E I
.
L3
(13)
Each of the non-dimensional parameters are varied independently, and the previously discussed simulation is performed.
To vary the non-dimensional electrostatic force in experiments one would vary the applied bias voltage. To vary the
non-dimensional contact stiffness independently of the nondimensional electrostatic force one would increase the contact
stiffness by increasing the thickness of the contact pad, for
example. For each simulation the number of bounces in the
first contact period is recorded. Figure 9 shows the results
of the numerical simulations. “Zero bounces” refers to the
situation in which the electrostatic force is insufficient to
cause the switch to close permanently. “n” bounces, where
n is an integer, refers to the number of bounces recorded
in the first contact event. “Permanent contact” refers to the
situation in which the beam closes with no bounces, and
remains closed. We can see from Fig. 9 that for contact
stiffnesses large in comparison to the RF switch stiffness (large
k̃) we have an increased propensity for multiple bouncing
events, up to four bounces for the parameter space studied.
It is evident that for permanent contact to occur, and thus
no bouncing, it is favorable to have cantilever stiffnesses
on the same order as, or greater than, the contact stiffness and to keep the electrostatic forces large compared to
cantilever elastic restoring forces. A similar simulation was
performed using n = 3 eigenmodes for the reduced order
model, in a smaller subset of the parameter space. Figure
10 shows the results. The “permanent contact” area remains
unaffected, however the organized structure of the bouncing
phenomenon appears to break down as higher eigenmodes
are excited.
143
Fig. 10. Non-dimensional bounce parameter space for an individual actuation
event, for n = 3 simulation. The x-axis represents the non-dimensional ratio of
contact stiffness to switch stiffness. The y-axis represents the non-dimensional
electrostatic force. Note that number of bounces was only computed for the
initial contact event, subsequent contacts as the beam settles to permanent
contact during the actuation cycle my have more or less bounces.
Exploring the same parameter space we now examine the
static contact force applied by the spring, ks , at steady state
equilibrium, for n = 2 eigenmodes in the reduced order
model. To do this Eq. (15) and Eq. (16) are solved with
all time dependence removed. That is, the static solutions
are obtained for a given parameter pair, F˜E and k̃. Once the
steady state displacements are obtained, the resulting steady
state spring force can be calculated. Figure 11 shows the
results of the calculations. It can be seen that in the desired
“permanent contact” area, seen in Figs. 9 and 10, the static
contact forces are low, compared to the remaining values in the
parameter space. This indicates that the “permanent contact”
region is still the optimal choice for efficient operation. It
was surmised that the static contact forces in the permanent
contact area would be quite large, and adversely effect switch
lifetime. Here we see that the contact forces in this area are
quite small, compared to the remaining parameter space. By
operating in the “permanent contact” area we also reduce
the dynamic forces caused by intermittent bouncing, which
will be large compared to static contact forces. Additionally,
switch settling time is reduced, due to the mitigation of overall
bouncing.
Finally we discuss the effects of surface adhesion to the
switch dynamics. It is known that as a MEMS switch is
actuated in low or zero current conditions the overall adhesion
forces will increase [3]. Here we have run simulations for the
parameters given in Table II, but have added the adhesion
force discussed in Eq. 3. Mathematically, for n = 2, it can be
represented as:
FC = −ks [gc − (q1 + q2 )]δ(x − L)H [(q1 + q2 − gc )]
−ks [gc − (q1 + q2 )]δ(x − L)
× [H [q1 + q2 − (gc − ga )] − H [q1 + q1 − gc ]]
(14)
×H [−q̇1],
ga defines the region in which the adhesive force exists.
We have used q̇1 to control the hysteretic behavior of the force:
If the cantilever tip is within the contact adhesion region and q̇1
144
JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 23, NO. 1, FEBRUARY 2014
Fig. 11. Static contact force from the spring, ks , for the non-dimensional
parameter space studied in Fig. 9. It can be seen that the “permanent contact”
area in Fig. 9 results in low static contact forces.
(a)
We have chosen three cases to simulate: no adhesive forces,
ga = 1 nm, and ga = 10 nm. Here an increase in ga represents
an increase in the adhesion force, and these values are on
the order of what can be observed in experimentally obtained
atomic force microscope (AFM) force-distance curves [23].
Figure 12 shows the results. It is clear that as the adhesive
forces increase the overall duration of bouncing near the
fundamental period decreases. As the adhesive force increases
further all bouncing will cease. The overall magnitude and
frequency of the smaller timescale bounces is weakly affected.
Here we have presented simulation results and experimentally measured data for DC-contact RF MEMS switches.
A parametric study was performed to investigate the effect of
system parameters on switch bounce. These results would be
applicable for similar MEMS devices with cantilevered structures whose contacts can be approximated by point contact
models. Devices with area-type contact, such as capacitive RF
MEMS switches, would not be suitable to this analysis due to
the nature of the contact mechanics.
V. C ONCLUSION
(b)
(c)
In this paper we have discussed the origin of the multiple
timescale bounce phenomenon in RF MEMS cantilevered
switches. In summary a compact analytical model was developed and numerical simulations were conducted to explain
the multiple timescale bounce phenomenon in RF MEMS,
and develop strategies to mitigate this phenomenon. A LDV
was used to experimentally measure the transient behavior of
RF MEMS switch closing for three types of devices. It was
determined that multiple timescale bouncing is the result of
the intermittent excitation of the 2nd eigenmode of the system
during contact events. A non-dimensional parameter study was
conducted revealing that in order to mitigate bouncing it is
favorable to have cantilever stiffnesses on the same order as, or
greater than, the contact stiffness and to keep the electrostatic
forces large compared to cantilever elastic restoring force.
Additionally, the role of contact adhesion was studied and
shown to decrease overall bounce height and duration with
increasing adhesion force. We expect that the mathematical
models and analysis methods described here will find use to
optimize or better understand the bounce and bounce related
wear in RF-MEMS switches.
A PPENDIX
A PPENDIX A
Fig. 12. Effect of hysteretic adhesion force on bounce duration. As the
adhesion force is increased, the bounce duration decreases and thus the overall
switch settling time decreases.
is directed away from the surface the adhesive force is active.
In general this assumption is valid and precludes the inclusion
of complicated event monitoring algorithms in the simulation.
Here we present the fully expanded coupled nonlinear
ordinary differential equations resulting from the Galerkin
projection procedure, for n = 2.
The expression for the first mode is(15):
1
1
φ12 d x̂ + T02 q1 ω12
φ12 d x̂
q¨1
0
0
βi
D
T0
˙ 00 + γ0s )
q1(γ
+
φ12 d x̂ + (γ10 + γ1s )
i
ρA
αD
βi
βi
D
D
φ13 d x̂ + q˙2 q1
φ12 φ2 d x̂
× q˙1 q1
α iD
α iD
TUNG et al.: MULTIPLE TIMESCALES AND MODELING OF DYNAMIC BOUNCE PHENOMENA
+ q˙1 q2
β iD
α iD
+ (γ20 + γ2s ) q˙1 q12
φ12 φ2 d x̂ + q˙2 q2
β iD
+ q˙2 q12
=
T02
α iD
+ q˙1 q22
=
φ13 φ2 d x̂ + 2q˙1 q1 q2
+ 2q˙2 q1 q2
φ1 φ22 d x̂
β iD
α iD
+ q˙1 q22
β iD
φ14 d x̂
α iD
β iD
α iD
β iD
αi
D
ks (1 − q1 − q2 )
1
β iD
α iD
φ13 φ2 d x̂
φ12 φ22 d x̂
φ12 φ22 d x̂ + q˙2 q22
β iD
φ1 φ23 d x̂
αi
D
φ1 δ(x̂ − 1) d x̂ H [q1 + q2 − 1]
ρA
0
2 2
−t
T0
1
+
Vdc (1 − e τV )
2
ρ Ag02
βE
βE
× α0
φ1 d x̂ + α1 q1
φ12 d x̂
αE
αE
βE
+ α2 q12
φ12 φ1 d x̂
αE
+ 2q1 q2
+ α3 q13
βE
αE
+ 3q1 q22
+α4 q14
βE
αE
+ 4q1q23
βE
βE
αE
βE
αE
βE
αE
d x̂
+ 4q13 q2
βE
φ22 φ1
αE
d x̂
βE
αE
φ33 φ1 d x̂
φ13 φ2 φ1 d x̂
αE
φ12 φ22 d x̂
φ1 φ23 φ1 d x̂ + q24
βE
αE
φ24 φ1 d x̂
.
(15)
1
0
φ22 d x̂ + T02 q2 ω12
˙ 00 + γ0s )
× q2(γ
β iD
α iD
1
0
β iD
α iD
+ q˙1 q2
β iD
β iD
α iD
+ 2q˙2 q1 q2
φ12 φ2 d x̂ + q˙2 q1
β iD
α iD
β iD
φ1 φ22 d x̂
α iD
φ1 φ22 d x̂ + q˙2 q2
β iD
α iD
φ23 d x̂
φ13 φ2 d x̂
α iD
+ q˙2 q12
T0
ρA
φ22 d x̂
+ (γ10 + γ1s ) q˙1 q1
+ (γ20 + γ2s ) q˙1 q12
φ22 d x̂ +
φ12 φ22 d x̂ + 2q˙1 q1 q2
β iD
α iD
φ1 φ23 d x̂
β iD
αi
φ24 d x̂
D
φ2 δ(x̂ − 1) d x̂ H [q1 + q2 − 1]
βE
αE
+ 6q12q22
βE
αE
αE
φ1 φ22 φ2
φ14 φ2 d x̂ + 4q13q2
βE
φ12 φ22
d x̂ + q23
d x̂ +
αE
βE
+ q24
φ24 φ2 d x̂ .
βE
αE
βE
αE
φ33 φ2 d x̂
φ13 φ2 φ2 d x̂
4q1 q23
βE
αE
φ1 φ23 φ2 d x̂
(16)
Here we have written the equations in compact form, for
integrals involving the damping terms γi0 and γis . The integral
i take the values α i = α 0 = 0.25 and
bounds α iD and β D
D
D
i
0
β D = β D = 0.75 when the integrals are multiplied by terms
i = β s = 1 when the integrals
γi0 , and α iD = α sD = 0 and β D
D
s
are multiplied by terms γi . Additionally, α E = 0.25 and
β E = 0.75. This revision of integral boundaries accounts for
the fact that the electrode bias bridge only partially covers the
RF MEMS switch.
ACKNOWLEDGEMENT
The expression for the second mode is (16):
q¨2
1
αE
φ12 φ2 φ1 d x̂
φ1 φ22 φ1 d x̂ + q23
αE
φ14 φ1
+ 6q12q22
φ13 φ1 d x̂ + 3q12q2
q22
αi
φ1 φ23 d x̂ + q˙2 q22
ρA
0
2 2 β E
−t
T0
1
+
Vdc (1 − e τV )
α0
φ2 d x̂
2
ρ Ag02
αE
βE
+ α1 q1
φ22 d x̂
αE
βE
βE
2
2
+α2 q1
φ1 φ2 d x̂ + 2q1 q2
φ1 φ2 φ2 d x̂
αE
αE
βE
+ q22
φ22 φ2 d x̂
α
β EE
βE
+α3 q13
φ13 φ2 d x̂ + 3q12 q2
φ12 φ2 φ2 d x̂
+ α4 q14
βE
β iD
D
ks (1 − q1 − q2 )
+ 3q1q22
φ1 φ2 φ1 d x̂ +
αE
T02
αE
βE
145
β iD
α iD
φ12 φ22 d x̂
This report was prepared as an account of work sponsored
by an agency of the United States Government. Neither the
United States Government nor any agency thereof, nor any of
their employees, makes any warranty, express or implied, or
assumes any legal liability or responsibility for the accuracy,
completeness, or usefulness of any information, apparatus,
product, or process disclosed, or represents that its use would
not infringe privately owned rights. Reference herein to any
specific commercial product, process, or service by trade
name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation,
or favoring by the United States Government or any agency
thereof. The views and opinions of authors expressed herein
do not necessarily state or reflect those of the United States
Government or any agency thereof.
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Ryan C. Tung received the B.S. degree in mechanical engineering from the University of Nevada,
Reno, NV, USA, in 2006, and the M.S. and Ph.D.
degree in mechanical engineering from Purdue University, West Lafayette, IN, USA, in 2008 and 2012,
respectively.
He is currently a National Research Council PostDoctoral Fellow at the National Institute of Standards and Technology, Boulder, CO, USA. His
current research interests include the dynamics of
contact resonance atomic force microscopy.
Adam Fruehling received the bachelor’s and Ph.D.
degrees in electrical engineering from Purdue University, West Lafayette, IN, USA, in 2005 and 2012,
respectively.
He is currently working as a MEMS Designer with
Texas Instruments, Plano, TX, USA, with a focus on
new technology development.
Dimitrios Peroulis (S’99–M’04) received the Ph.D.
degree in electrical engineering from the University
of Michigan, Ann Arbor, MI, USA, in 2003. He
has been with Purdue University, West Lafayette,
IN, USA, since August 2003, where he is currently
leading a group of graduate students on a variety
of research projects in the areas of RF MEMS,
sensing and power harvesting applications as well
as RFID sensors for the health monitoring of sensitive equipment. He has been a PI or a co-PI in
numerous projects funded by government agencies
and industry in these areas. He has been a key contributor on developing very
high quality (Q>1000) RF MEMS tunable filters in mobile form factors.
He has been investigating failure modes of RF MEMS and MEMS sensors
through the DARPA M/NEMS S&T Fundamentals Program, Phases I and II)
and the Center for the Prediction of Reliability, Integrity and Survivability
of Microsystems funded by the National Nuclear Security Administration.
He received the National Science Foundation CAREER Award in 2008. His
students have received numerous student paper awards and other student
research-based scholarships. He is a Purdue University Faculty Scholar. He has
received ten teaching awards, including the 2010 HKN C. Holmes MacDonald
Outstanding Teaching Award and the 2010 Charles B. Murphy Award, which
is Purdue University’s highest undergraduate teaching honor.
Arvind Raman is a Professor of mechanical engineering and a University Faculty Scholar with Purdue University, West Lafayette, IN, USA. His current
research interests include applied nonlinear dynamics, nanomechanics, and fluid-structure interactions.
His group has significantly advanced the understanding of complex dynamics in nanotechnology
applications, such as atomic force microscopy and
micro- and nano-electromechanical systems, in gyroscopic systems for data storage and manufacturing,
in electronics cooling, and in biomechanics. He has
mentored 15 Ph.D. students, coauthored more than 100 peer-reviewed journal
articles, held visiting positions with the Universidad Autonoma de Madrid,
Madrid, Spain, University of Oxford, Oxford, U.K., and Darmstadt University
of Technology, Darmstadt, Germany, and secured funding from the NSF, NIH,
NASA, NNSA, and several national and international industrial sponsors. He
is a Fellow of the ASME and a recipient of the Gustus Larson Memorial
Award from the ASME, a Keeley fellowship (Wadham College, Oxford),
College of Engineering Outstanding Young Investigator Award, and the NSF
CAREER Award. He has pioneered the use of cyber-infrastructure in the AFM
community for research and education through advanced simulation tools and
online classes, which are used by thousands around the world and led College
of Engineering strategic initiatives for global engagement in Latin America.