Page 167 - Principles and Applications of NanoMEMS Physics

4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 155

4.2.2.2.3 Parametric Amplification in Torsional MEM Resonator
This device was experimentally demonstrated by Carr, Evoy, Sekaric,
Craighead and Parpia [179]. A torsional resonator of quality factor Q, Figure
3-5, is excited at a fundamental driving frequency, ω , which applies

END VIE W
TO P VIE W END VIE W
TO P VIE W
Torsion
Torsional al
+V
V V + V _ω + +V 2ω
V
ω
ω
AC_
DC
AC_
Resonator r
Resonato DC AC AC _2
w w
ϕ ϕ
0 0
Paddle le
Padd
SID E VIE W
SID E VIE W
g 0 g 0
L L
Substrate te
Substra
Transm ission Lin
Transm ission Lines es
Figure 4-5. Schematic of torsional parametric amplifier [22].
a torque ( ) t ωτ . If the device is driven at resonance, with an applied torque
given by () ττ t = sin (ω +t ) θ , where θ is the phase angle between
0
2
excitations at ω and ω , then the torsional spring constant exhibits a
κ ′
modulation, () t = κ ′ cos (2ω ) t . Under these circumstances, the angular
0 0
amplitude response, ϕ , adopts the form
0
/ 1 2
2
2
τ Q ª cos θ sin θ º
ϕ = 0 « + » . (1)
κ « ¬ (1+ κQ ′ 0 2κ ) (1− κQ 0 ′ 2κ ) » ¼
0 2 2
Accordingly, with zero signal amplitude at ω , κ ′ () 0=t , and the
2
angular response is τ Q κ . Otherwise, the square-root factor acts as a
0
phase-dependent gain and, becoming infinity when θ = π 2 , and
′
<
κ = 2 κ Q . For 0 θ < π 2 , the angular response may be approximated
0
by,