Download Slide 1

Mechanical Modeling
The dynamic response is give by d’Alembert principle and is
d 2x
dx
m 2  b  kx  f ext
d t
dt
From this equation the switching time, mechanical bandwidth and
and effect of thermal noise can be predicted. The parameter that
affect the these parameter are resonant frequency and quality factor.
The actual mass of the beam is 0.35-0.45 which is moving. 0.5Q 2 is
desirable for a better operation.
Gas fundamental and Quality factor
The mean free path , Kundsen number K and viscosity  is given by
a 
P0

1
0 , k n  , e 
Pa
g
1  9.638k n1.159
A low kundsen number means that there are many collusion and
the gas is viscous (liquid).
The damping and Quality factor is given by
E t 2
3 A2
12 A2 p p 2 ln( p) 3
k
3
b
* 3 ,b 
* 3 ( 

 ), Q 
, Qcant 
*
g
0
2 g 0
N g 0 2 8
4
8
0 b
 ( wl ) 2
So from the equation above it is evident that by slot (p) and low pressure
low damping increase the Quality factor. Damping also change with height.
Switching Time
The switching time can be calculate form eq. 3.16 and with little
damping and with damping the switching time is given by.
t s  3.67
Vp
Vs0
, t sd 
9V p
2
40QVs
2
The release time can be calculated using nonlinear dynamic equation
from the restoring force with actuation force keeping zero.
The velocity, acceleration and Current of the beam during switching
is shown in the figure 3.7,3.8,3.9.
The fringing capacitance is given by equation 3.27 and force 3.28
and it is seen duo to fringing capacitance the switching time reduces.
Effect of damping resistance and Taylored Actuation voltage
(consult with Trond)
Switching energy, Responses to waveform and self actuation
The energy consumed in switching process is sum of electrical and
mechanical energy of the beam and given by eq 3.29, 3.30.
For a typical switch the switching energy is around 5 nJ.
The response to single wave form is shown in fig 3.13 and the double
or multiple wave form is shown in fig 3.14 . Also the amplitude and
Frequency modulated signal response is shown.
For high power the dynamic analysis also show the same trend as static
and self actuate the beam and collapse to down position for high power.
The intermodulation product power is given by eq 3.44 and increase
with Cup as the product depends on the upstate capacitance. So inter
modulation product are much larger in varactors than switch.
It also depends on K and g0.
The brownian noise mainly come from damping so a high Q switch has less noise.
Fn=sqrt(4bkT), It also depends on K.
Electromagnetic Modeling
A MEMS shunt switch can be modeled RLC series resonator in shunt
connection to Transmission line shown in fig 4.1.
The switch shunt impedance and LC series resonant frequency given by
Z s  Rs  jL 
1
1
1
, f0 
*
jC
2
LC
The Up-state capacitance with fringing field is given in Table 4.1. The
Rule of thumb is that the hole diameter should be less that 3g, not
to affect the up-state parallel plate capacitors.
The down state capacitance is degraded if the MEMS bridge layer
and dielectric layer is rough. It can be used high-dielectric to increase
the downstate capacitance.
Current distribution, Resistance, Inductance and Loss
The Current distribution is shown in figure 4.3 and it is seen that the
current is concentrated on the edge of the transmission line and also
at the edge of beam, i.e. skin effect.
Series resistance is combination of Rs1 duo to t-line loss and Rs due
to MEMS bridge only. For thin beam the bridge resistance is constant.
Inductance: The inductance can be modeled from down state position
it’s mainly depends on the gap in CPW line and is higher for low spring
constant beam due to mender.
Loss: The loss is given by eq 4.9 and 4.10 and the switch loss is
given by eq 4.14 and 4.15 for both up and down position.
Fitting CLR parameter
The upstate capacitance can be found from S parameter and eq. 4.16
and 4.17.
Down state capacitance and inductance: Below resonant frequency
and s parameter and eq. 4.19 the downstate capacitance can be
measured, and from resonant frequency the inductance can be
measured which is dominant after f0.
Series resistance: The series resistance is best fitted around LC resonant
frequency and given by eq 4.20. For upstate resistance a no of switch has
to be series connected and using eq. 4.14