74
G. Chakraborty and N. Jani
compared to the resonant frequency. This scheme is also known as phase feedback
oscillator. Similar noise reduction has been experimentally demonstrated for a 2
DOF-coupled nonlinear oscillator [34].
b. Frequency stability
In timing devices, where stability of the output frequency is of greatest concern, the
change in frequency, caused by increase in amplitude (i.e. amplitude-frequency,
A-f effect), is a problem. However, the nonlinearity can help in achieving temperature compensation in silicon micromechanical resonator [35]. The main challenge
in Si resonator is that Si has a temperature coefficient of frequency (TCF) near
to 30 ppm/
◦ C, which is large compared to that of a quartz oscillator (18 ppm/
◦ C
at 25
◦ C). With increasing temperature, both the resonant frequency and quality
factor increase as follows:
f r = f 0 (1 + TCF(T − T 0 )) ,
Q r = Q 0 (1 + TCQ(T − T 0 )) ,
(31)
where f 0 and Q 0 are the resonant frequency and quality factor at 12.5
◦ C and TCQ
is the temperature coefficient of quality factor. In the linear regime, operating at
resonance, the amplitude is proportional to the quality factor, i.e.
X = X 0 (1 + TCQ(T − T 0 ))
where X 0 is the amplitude of MEMS at T 0 . From the above equations, we get
f T =
f 0
X 0
TCF
TCQ
X
or
f T
f 0
=
TCF
TCQ
X
X 0
(32)
i.e. there exists a linear relationship between f r and X . When nonlinearity is
present the A-f dependence due to duffing-type nonlinearity can be written as
f D
f 0
=
3α/4(X 0 + X )
2
2π f 0
≈
3α
4
X
2
0
2π f 0
+
3α
4π
X
2
0
f 0
X
X 0
(33)
When both the effects are considered
f
f 0
=
f T
f 0
+
f D
f 0
=
3
4
αX
2
0
2π f 0
+
X
X 0
T C F
T C Q
+
3αX
2
0
4π f 0
.
(34)
The effect of temperature variation can be nullified if X 0 is selected for (α < 0) in
such a way that the last term gets cancelled out. Thus, the temperature dependence
of frequency is minimized. Further, electrostatic tuning is also used to suppress
the temperature-frequency drift [36].
G. Chakraborty and N. Jani
compared to the resonant frequency. This scheme is also known as phase feedback
oscillator. Similar noise reduction has been experimentally demonstrated for a 2
DOF-coupled nonlinear oscillator [34].
b. Frequency stability
In timing devices, where stability of the output frequency is of greatest concern, the
change in frequency, caused by increase in amplitude (i.e. amplitude-frequency,
A-f effect), is a problem. However, the nonlinearity can help in achieving temperature compensation in silicon micromechanical resonator [35]. The main challenge
in Si resonator is that Si has a temperature coefficient of frequency (TCF) near
to 30 ppm/
◦ C, which is large compared to that of a quartz oscillator (18 ppm/
◦ C
at 25
◦ C). With increasing temperature, both the resonant frequency and quality
factor increase as follows:
f r = f 0 (1 + TCF(T − T 0 )) ,
Q r = Q 0 (1 + TCQ(T − T 0 )) ,
(31)
where f 0 and Q 0 are the resonant frequency and quality factor at 12.5
◦ C and TCQ
is the temperature coefficient of quality factor. In the linear regime, operating at
resonance, the amplitude is proportional to the quality factor, i.e.
X = X 0 (1 + TCQ(T − T 0 ))
where X 0 is the amplitude of MEMS at T 0 . From the above equations, we get
f T =
f 0
X 0
TCF
TCQ
X
or
f T
f 0
=
TCF
TCQ
X
X 0
(32)
i.e. there exists a linear relationship between f r and X . When nonlinearity is
present the A-f dependence due to duffing-type nonlinearity can be written as
f D
f 0
=
3α/4(X 0 + X )
2
2π f 0
≈
3α
4
X
2
0
2π f 0
+
3α
4π
X
2
0
f 0
X
X 0
(33)
When both the effects are considered
f
f 0
=
f T
f 0
+
f D
f 0
=
3
4
αX
2
0
2π f 0
+
X
X 0
T C F
T C Q
+
3αX
2
0
4π f 0
.
(34)
The effect of temperature variation can be nullified if X 0 is selected for (α < 0) in
such a way that the last term gets cancelled out. Thus, the temperature dependence
of frequency is minimized. Further, electrostatic tuning is also used to suppress
the temperature-frequency drift [36].