72
G. Chakraborty and N. Jani
27]. This dependance of resonant frequency or amplitude limits the performance
of the device where they are required to resonate at a particular frequency, for
example, time device, frequency filters, resonant accelerometers, resonant energy
harvesters, gravimetric sensors, etc.
b. Noise performance
High amplitude of response is preferred in resonant devices for increasing the
signal-to-noise (S/N ) ratio. However, the presence of nonlinear term limits the
amplitude of oscillation for a given excitation level.
To assess the noise performance of the oscillator, a block diagram of the open-loop
system is used as shown in Fig. 6.
In Fig. 6 N 1 (t), N 2 (t) are the noise in excitation and transducer, respectively, and
y(t) is the measured output from the transducer. The maximum energy stored in
the linear oscillator is
E max =
1
2
k 1 X
2
max
While the maximum power that could be extracted from the oscillator is equal to
P sig =
E
2π/ω
=
ω E max
Q
The noise in y(t) is generated by two main sources, namely, the noise generated
from excitation and the thermal noise generated due to motion resistance. With a
properly designed excitation mechanism the contribution of the first term can be
minimized. The thermal noise power can be written approximately as [28]
p
mech
noise = 4k b T
ω 0
2Qω
2
where ω 0 is the centre frequency and δω is the offset from the former. The phase
noise spectrum is obtained as
L (ω) =
p
mech
noise + p N 2
2 p sig
=
2k b T
p sig
ω 0
2Qω
2
+
p N 2
2 p sig
(30)
where p N 2 is the noise in the transducer. In the simplified analysis, the effect of
1/ f noise has not been considered.
It is seen from the above equation that a large value of E max is required for
improving the noise performance of the oscillator. However, the presence of
the nonlinear term limits the value of E max and hence decreases the S/N ratio.
The limitation on amplitude of response due to nonlinearities puts a limit to the
drive current that can be applied resulting in a far-from-carrier-phase noise. Also
the frequency-amplitude dependence converts amplitude noise into phase noise
[9, 29].
G. Chakraborty and N. Jani
27]. This dependance of resonant frequency or amplitude limits the performance
of the device where they are required to resonate at a particular frequency, for
example, time device, frequency filters, resonant accelerometers, resonant energy
harvesters, gravimetric sensors, etc.
b. Noise performance
High amplitude of response is preferred in resonant devices for increasing the
signal-to-noise (S/N ) ratio. However, the presence of nonlinear term limits the
amplitude of oscillation for a given excitation level.
To assess the noise performance of the oscillator, a block diagram of the open-loop
system is used as shown in Fig. 6.
In Fig. 6 N 1 (t), N 2 (t) are the noise in excitation and transducer, respectively, and
y(t) is the measured output from the transducer. The maximum energy stored in
the linear oscillator is
E max =
1
2
k 1 X
2
max
While the maximum power that could be extracted from the oscillator is equal to
P sig =
E
2π/ω
=
ω E max
Q
The noise in y(t) is generated by two main sources, namely, the noise generated
from excitation and the thermal noise generated due to motion resistance. With a
properly designed excitation mechanism the contribution of the first term can be
minimized. The thermal noise power can be written approximately as [28]
p
mech
noise = 4k b T
ω 0
2Qω
2
where ω 0 is the centre frequency and δω is the offset from the former. The phase
noise spectrum is obtained as
L (ω) =
p
mech
noise + p N 2
2 p sig
=
2k b T
p sig
ω 0
2Qω
2
+
p N 2
2 p sig
(30)
where p N 2 is the noise in the transducer. In the simplified analysis, the effect of
1/ f noise has not been considered.
It is seen from the above equation that a large value of E max is required for
improving the noise performance of the oscillator. However, the presence of
the nonlinear term limits the value of E max and hence decreases the S/N ratio.
The limitation on amplitude of response due to nonlinearities puts a limit to the
drive current that can be applied resulting in a far-from-carrier-phase noise. Also
the frequency-amplitude dependence converts amplitude noise into phase noise
[9, 29].