CHEN et al.:EM-(M) FORCE FEEDBACK INTERFACES FOR CAPACITIVE MEMS INERTIAL SENSORS: A REVIEW 6479
Fig. 4. Converting a purely electrical M to an unconstrained
EM-M. (a) An unconstrained electrical M structure [25]. (b) The
feedback path to the inner node of the first resonator stage can be converted
to a feedforward path. (c) The feedforward path can be shifted and replaced
by the micromechanical sensor. From [27].
Fig. 5. Positive feedback compensator in a EM- M interface described
in [28] and [29].
Thus, the first two electrical integrators can be replaced with
two mechanical integrators. The resulting structure retains the
same order as the original purely electrical M, and does
not need a compensator for loop stability.
However, the unconstrained M architecture and conven-
tional lead compensation results in stability margins that are
inadequate for high-Q (vacuum packaged) sensing elements.
This problem is due to high-order resonance modes of the
sensing element arising from the electrostatic comb fingers,
which normally are sufficiently damped at atmospheric pres-
sure, but become highly underdamped in vacuum resulting in
a phase lag approaching −360°. Ezekwe and Boser [28], [29]
used a different compensator that utilizes positive feedback
to introduce a phase lead of 180° at all frequencies, which
is sufficient to compensate the phase lag introduced by the
parasitic modes. The block diagram of the EM- M feedback
loop with positive feedback compensator is shown in Fig. 5.
Furthermore, other factors including the time delay between
the position sense and feedback pulse, the sampling
frequency f
s
and whether sensing and feedback use separate
Fig. 6. Cascaded integrators with distributed feedback architecture for a
EM-M [25].
or common electrodes influence the stability and performance
of the EM- M loop. This necessitates a tradeoff between
these design choices.
C. Single Loop Architectures
Several loop structures have been described for implement-
ing a single-loop high-order EM- M; they are trade-offs
between structure complexity and the freedom of choosing the
shape of the signal-transfer-function (STF) and the position of
the noise-transfer-function (NTF) zeros [30]. The most com-
monly used structure is the cascaded integrator with distributed
feedback (CIDF) architecture, as depicted in Fig. 6. It contains
a cascade of several delaying integrators with feedback signals
that are scaled by coefficients a
i
(i = 1, 2 ...)andfedto
each integrator. The distributed feedback architecture bypasses
the mechanical sensing element and therefore contributes
compensating zeros, whose values depend on the feedback
coefficients as well as the parameters of the sensing element.
Furthermore, the feedback voltage of this topology cannot be
adjusted without tuning the values of the coefficients a
i
[30].
Dong [25] analysed the total in-band noise of a typical CIDF
architecture, which is mainly determined by the quantization
noise at the upper end of the signal band. It thus limits the
maximum obtainable signal bandwidth and it is desirable to
further reduce the total in-band quantization noise power.
A single loop high-order EM- M can be constructed by
using several local feedback paths in the loop filter to shape
the quantization noise at the end of the signal band [31].
Fig. 7 (a) and (b) show the architecture of an electronic filter
with feed-forward with local resonators (FFLR) [32], [33], and
distributed feedback loops with local resonators (DFLR) [34],
respectively. The local feedback path g
i
(i = 1, 2 ...) will
create a local resonator, which produces notches to cause fast
decay of the signal magnitude in the stop-band and thus further
suppresses the quantization noise. The architecture in Fig. 7(a)
does not contain signal paths bypassing the sensing element
since it is in series with the filter and the quantizer. The
feedback range is easily adjusted by varying the magnitude
of the feedback pulses.
Normally, a single loop architecture has a low susceptibility
to analog component mismatch [35]. In addition, the quanti-
zation noise is randomized by the high-order loop; therefore,
limit cycle tones are less likely to occur [25].