Each - Draper Laboratory

= damping
k = stiffness. Mostly mechanical with modifications
by electrostatic forces
kds = quadrature coupling. The drive axis suspension
force coupling into the sense axes
bds = in-phase damping ‘surfboard’ coupling to sense
axis
x = motion along drive axis (parallel to substrate)
y = motion along sense axis (normal to substrate)
s = Laplace transform of d /dt
Fd = motor drive force applied by the outer combs in
Figure 1
WI = slowly varying input rate
a = drive force coupling to sense axis
Q = quality factor
From (2) and (3), another challenge emerges. The driving
force as well as the drive axis suspension force and drive
axis damping are coupled into the sense axis. With good
design, these forces should be small compared with the
Coriolis Force .
For low-frequency angular rate inputs, the desired output
is the angular rate modulated by the drive frequency. As
shown in the electrical circuit of Figure 3, the proof masses
are the negative input of a high input impedance, highgain
operational amplifier whose input node is at virtual
ground. The feedback resistor is large so that it does not
affect the output at the gyro’s drive frequencies. From (3)
and Figure 3, the preamplifier output is given by (Appendix
B):
where
Vo = output of preamplifier
Vs = bias voltage (plus and minus applied to right and
left sense plates in Figure 1) on sense electrodes
(5 V, example values are given in parentheses)
Vc = coupling (drive feedthrough)
VN = preamplifier input voltage noise (10-8 V/ )
46 Error Sources in In-Plane Silicon Tuning-Fork MEMS Gyroscopes
(4)
Cfb = feedback capacitor about the sense axis preamplifier
(2 pF)
Cs = total of sense capacitors (2 pF)
CN = preamplifier input capacitance to ground (5
pF)
Cc = coupling (undesirable capacitor) to virtual
ground (preamplifier input)
dCs /dy = differential change of sense capacitors with y
motion (2 pF/3 µm)
SC = sum of all capacitors attached to the virtual
ground. Includes strays, working, feedback,
and amplifier capacitors (12 pF).
wd = drive mode undamped natural frequency
(20 kHz x 2 prad/s)
w s = sense mode undamped natural frequency
(22 kHz x 2 prad/s)
x o = amplitude of drive motion (10 µm zero-to-peak)
Fs = cross coupling forces acting along the sense
direction (B-6)
q = phase shift through sense dynamics (B-6)
Proof
Mass
C c Cs/2 C s /2
V c -V s V s
C N
Figure 3. Circuit diagram for sense preamplifier analysis.
In (4), it is assumed that the proof mass motion is driven
so that the displacement is a sinusoidal function of time.
The rate signal, the Coriolis term, is in phase with the
proof mass velocity, i.e., in quadrature with the proof mass
position. For the sample parameters above, the gyro scale
factor at the preamplifier output is 1.3 mV/rad/s. With a
field effect transistor (FET) preamplifier whose input noise
at drive frequency is 10 nV/ , the rate equivalent noise
is 10 deg/h/ . Attaining the theoretical noise limit is a
challenge discussed further in the “Electrical Coupling”
section.
Because of the sense-drive frequency separation and high
sense-axis quality factor, the damping term is omitted in
the denominator of (4); therefore, gain does not depend
on damping. High resonant frequencies are desired to
remove the gyro’s sensitive frequencies from acoustic
noise and vibration and to permit isolators that allow
adequate bandwidth. For a fixed sense-plate bias, higher
sensitivity is achieved by lowering the resonant frequencies
and/or by decreasing the separation between sense
and drive mode. Drive frequencies of 10-25 kHz and
sense-drive mode separations of 5-15% have worked well
for MEMS TFGs. At baseband, the transfer function of
output voltage to rate input has a lightly damped peak
at the frequency separation. Placing the separation at 1-2
kHz allows a 100-Hz bandwidth, which adequately filters
R fb
V N
C fb
–
+
V R
V o
Output
Voltage