We describe the experimental procedure used to characterize the circuits of the resonant drive techniques, the characterization of those circuits, and the measured response of the actuators under these excitation regimes.
3.4.1. Electrical Characterization
The experimental setup shown in
Figure 9a was used to measure the natural frequency
of series RLC circuits consisting of an external inductor
L, the circuit parasitic resistance
R, the capacitance of the actuator
, and the circuit parasitic capacitance
. A lumped-element representation of the circuit is shown in
Figure 9b. The measurements were carried out using a Vector Network Analyzer (VNA) (Keysight P5020A). To simplify the measurement process, only one-port S-parameter
measurements were undertaken. This represents the ratio of the reflected wave to the incident wave as a function of frequency at the input port of the circuit. A calibration process was performed prior to measurements to account for the parastics presented by cables and connectors. Calibration was carried out using the short-open-load calibration approach with the Keysight 85521A Mechanical Calibration Kit.
Different inductors were used with the resonant circuit to achieve various resonance frequencies required for the resonant drive techniques discussed in the following sections. Below, the characterization of each inductor is detailed.
For the characterization with a
mH inductor, composed of two series-connected inductors (2.4 mH and 3.2 mH), the power level of the VNA was set to 0 dBm (1 mW). The input signal frequency was swept over the range
MHz with a step frequency of 5 kHz. The intermediate frequency bandwidth (IF-BW) was set to 1 kHz. The amplitude of the
response of the RLC circuit, shown in
Figure 10a in logarithmic (dB) scale, indicates that electrical resonance occurred at
MHz, corresponding to the minimum value of
dB. The electrical quality factor of the RLC circuit was calculated using the half-power bandwidth method from the measured
magnitude, where the bandwidth was determined from a linear scale (rather than dB), yielding
. It was also found that the natural frequencies of the circuits were insensitive to the MEMS actuator in them but varied with the inductor and parasitics in the circuit.
The characterization procedure was repeated for an RLC circuit with a
mH connected in series with the actuator,
Figure 10b. The signal frequency was swept over
MHz, with a step frequency of 1 kHz. The IF-BW was set to 1 kHz, and the data were averaged over five samples. The amplitude of
is represented in a dB scale, while the phase is shown in degrees. The electrical resonance was found at the minimum value of the
dB as
kHz. The corresponding phase angle swung from a minimum of
to a maximum of
. The quality factor of the circuit was calculated as
.
The 950 µH and 675 µH inductors were characterized using the same settings as the 5.6 mH inductor, including the signal frequency sweep range, step size, and IF-BW.
Figure 10c shows the
amplitude and phase response when a 950 µH inductor was connected to the actuator. The electrical resonance was observed at
MHz at
dB. At resonance, the phase angle varied from
to
. The quality factor was determined to be
.
For the 675 µH inductor, the
amplitude and phase response are shown in
Figure 10d. The electrical resonance occurred at a minimum value of
dB corresponding to a frequency of
MHz. The phase angle at resonance ranged from
to
, and the quality factor was calculated as
.
Next, we characterized the resonant circuit with 60 µH and 10 µH inductors. The signal frequency was swept in the range
MHz and the IF-BW was set to 2 kHz. The amplitude and phase of the frequency response of the RLC circuit are shown in
Figure 10e when a 60 µH was connected to the actuator. The electrical resonance was found at the minimum value of
dB to be
MHz, and the corresponding phase angle swung from a minimum of
to a maximum of
. The calculated quality factor in this case was found to be
.
Figure 10f shows the
magnitude and phase of the frequency response of the resonant circuit when a 10 µH inductor is connected with the actuator. The electrical resonance was found at the minimum value of
dB to be
MHz and the corresponding phase angle swung from a minimum of
to a maximum of
. The calculated quality factor in this case was found to be
.
3.4.3. Resonance Matching
The first technique employed was resonance matching. It calls for the use of a single tone signal designed to match the electrical resonance
of the RLC circuit and to result in an electrostatic excitation force with a dominant peak that matches the mechanical resonance
of the actuator. For unbiased signals, the external inductor
L is sized such that
as per Equation (
9). For biased signals, the electrostatic force is described by
and the dominant peak is at
. Therefore, the external inductor
L is sized such that
.
To excite actuator I with an unbiased signal, an
mH inductor was connected in series with the actuator, resulting in a VNA-measured electrical resonance of
kHz,
Figure 10b. It was found that the electrical natural frequency drops in the drive setup, shown in
Figure 11a, due to uncharacterized reactive loading introduced by the function generator into the circuit. Hence, we intentionally set the electrical natural frequency to a value larger than the desired signal frequency
. The signal frequency was swept within the range
kHz in a time window of 100 ms while the voltage amplitude
was held constant.
The frequency response of actuator I is presented in
Figure 12a for four values of the voltage amplitude
. For a voltage amplitude of
V, this excitation scheme achieves a maximum displacement of
nm at resonance, which is more than twice the maximum displacement attained using a conventional voltage amplifier,
Figure 6. The peak of the frequency–response curves was observed to shift to the right. Therefore, the experiment was repeated using a longer time window to investigate whether this shift was due to the hardening of the resonator or an artifact of the sweep transients. The tip velocity of actuator I was measured using the OFV-5000 LDV [
38], and the data were collected by a digital oscilloscope in 4 s long time windows. The actuator velocity,
Figure 12b, demonstrates a similar hardening behavior as the voltage amplitude is increased from
V to 10 V, reaching a peak of
m/s, which confirms the presence of a hardening nonlinearity. Given the relatively small displacements involved, the source of this nonlinearity is probably the electrostatic field rather than the mechanical potential.
The VNA-measured electrical resonance was unchanged at
kHz once the chip carrying actuator II was introduced into the circuit. To excite primary resonance, the signal frequency was swept within the range
kHz in a time window of 100 ms.
Figure 12c shows the frequency response of actuator II tip displacement for four values of the voltage amplitude
. The largest realized displacement for
V was
nm, corresponding to 2.99% of the air gap. The lower peak amplitude in this case is due to a lower quality factor and higher stiffness than those of actuator I. Similar to actuator I, the peak of the frequency response curves was observed to shift to the right, confirming the presence of a hardening nonlinearity.
We deliberately oversized the external inductor to
mH to investigate the impact of mistuning the electrical and mechanical resonances away from
on our actuation technique. The VNA-measured electrical resonance was found to be
kHz, well below half of the mechanical resonance of actuator I at
kHz.
Figure 13a shows the frequency–response curve of actuator I tip displacement under this excitation regime and a voltage amplitude of
V as the signal frequency was swept from
kHz to 741 kHz over a time window of 100 ms. The actuator displacement dropped, as shown in
Figure 13a, by one order of magnitude to
nm, which is much lower than the displacement obtained in the tuned case.
Likewise, we undersized the external inductor to
µH, resulting in a VNA-measured electrical resonance of
MHz, as shown in
Figure 10c, well above half of the mechanical resonance of actuator I.
Figure 13b shows the frequency–response curve of actuator I tip velocity under a voltage amplitude of
V as the signal frequency swept in the range of
kHz over a time window of 30 ms. The tip velocity dropped by two orders of magnitude to
mm/s.
The experimental setup shown in
Figure 11b was adopted to implement biased resonance matching excitation. In this setup, a DC power supply and a bias Tee are introduced to add a bias to the harmonic signal generated by the function generator. For actuator I, two external inductors,
µH and
µH, were connected in series to match the electric and mechanical natural frequencies
. The signal frequency was swept over the range
MHz.
Figure 14a shows the frequency response of the actuator under three levels of bias voltage
and a voltage amplitude of
V. The maximum measured displacement
nm was realized at resonance for a bias voltage of
V.
We note that the introduction of bias into the signal degrades the efficiency of the resonance-matching technique. This can be seen in the fact that adding a bias voltage of V to the same harmonic signal ( V) ends up reducing the realized displacement, in addition to incurring the cost and complexity of biasing the signal and operating in a higher frequency range.
Similarly, we explored the response of actuator II under biased resonance matching. The signal frequency was swept across the range
MHz while connecting an external inductor
µH, resulting in a VNA-measured electrical resonance of
MHz, as shown in
Figure 10d.
Figure 14b shows the frequency response curves of actuator II under three bias voltage levels
and two voltage amplitudes
. The discrepancy in the external inductors required for resonance matching between actuators I and II appears to be a result of unaccounted-for parasitic inductance in the case of actuator I.
The largest achieved displacement for actuator II was nm for a bias voltage of V and a voltage amplitude of V, which is marginally larger than that obtained with the same unbiased signal ( V). This result confirms our conclusion about the effectiveness of the unbiased compared to the biased technique.
We investigated the impact of mistuning on resonance matching with biased signals. Due to the limited drive voltage used in this experiment, a bias Tee was not required, and the experimental setup of
Figure 11a was adopted. First, a significantly oversized external inductor with
µH was introduced to the drive circuit. The VNA-measured electrical resonance was
MHz, as shown in
Figure 10c, mistuning the electrical resonance to the lower side of the mechanical resonance
.
Figure 15a shows the measured frequency–response curve of actuator
tip displacement under resonance matching excitation with the voltage waveform
V as the signal was swept in the frequency range
MHz over a time window of 200 ms. The electrical and mechanical resonances are mismatched, with the former appearing at
MHz and the latter appearing at
MHz.
Since the resonance frequency of actuator I’ is lower than that of actuator II, we used the inductor of the latter
µH as an undersized inductor for the former.
Figure 15b shows the measured frequency–response curve of actuator
tip displacement under the same voltage waveform (
V) as the signal was swept in the frequency range
MHz over a time window of 100 ms. Distinct peaks corresponding to the mismatched electrical and mechanical resonances appear in the frequency response with the electrical resonance at
MHz and the mechanical resonance at
MHz. Note that the VNA-measured electrical resonance was
MHz, as shown in
Figure 10d.
The previous cases demonstrate that optical measurement of the MEMS response can serve as an indicator of the precise location of electrical resonance in the actuation circuit. To accurately match the two resonances , the electrical resonance should be observed optically under a biased signal and adjusted iteratively until it merges with the mechanical resonance into a single peak. Note that when this technique is used to tune the two resonances for an unbiased signal, the two peaks observed in the same vicinity are the electrical resonance and the superharmonic , rather than primary mechanical resonance.
3.4.4. Multi-Frequency Excitation
The multi-frequency excitation technique utilizes the summation of two voltage signals, resulting in an electrostatic force of the form
The frequency of one signal is set to be equal to the electric resonance of the RLC circuit
while the sum or difference of the frequencies of the two signals is set equal to the mechanical resonance
to drive the actuator with one of the last terms on the right-hand side of Equation (
11).
Figure 16 shows the signal for the case of
.
As noted in
Section 3.4.3, it was found that the reactive loading of the function generator in the drive setup, shown in
Figure 11a, causes a drop in the electrical resonance away from that measured using the VNA. To address this, a two-step process was developed to determine the electrical resonance
. First, an estimate of the electrical resonance
was obtained using the VNA and the setup of
Figure 9a to measure the electrical resonance. Then, the drive setup, shown in
Figure 11a, was used to vary the first signal frequency
in the vicinity of
while measuring the response of the actuator. The electrical resonance
was determined as the first signal frequency
at the peak response.
First, we tested the use of a higher electric resonance
in multi-frequency excitation, which corresponds to driving the actuator via the electrostatic force of the frequency difference
. Actuator
was connected in series to an external inductor
µH resulting in an estimated electrical resonance of
MHz; see
Figure 10e. Based on that, the electrical resonance was found to be
MHz using the search routine described above. The frequency of the first signal was set to match the electrical resonance frequency
. The frequency of the second signal was swept down in the range
MHz over 100 ms, so that the difference between the two frequencies was swept up in the vicinity of the mechanical natural frequency
MHz.
Figure 17a shows the frequency–response curves of actuator
tip displacement under five levels of the second signal voltage amplitude
while the amplitude of the amplifier (first) signal is held constant at
V. The maximum displacement realized was
nm, observed at resonance under a voltage amplitude of
V, which is 0.75% of the air gap. The response under this excitation scheme is well regulated as the second signal amplitude is varied.
To test the flexibility of this technique, we tuned the the electrical resonance to a higher value by using a smaller external inductor
µH. As a result, the estimated electrical resonance increased to
MHz; see
Figure 10f. The electrical resonance was found to be
MHz using the drive setup. The first signal frequency was set equal to it
, while the second signal frequency was swept down in the range
MHz over 100 ms.
Figure 17b shows the frequency response of actuator
under a fixed amplifier voltage amplitude
V and seven levels of the second signal voltage amplitude
. The maximum measured displacement with this inductor was
nm for a voltage amplitude of
V, corresponding to 0.45% of the air gap.
The drop in displacement with the increase in the amplifier signal voltage from
V to 7.5 V and the actuation signal voltage from
V to 7.5 V is an interesting finding. It is even more so given that the smaller inductor
in the second case imposes lower losses on the RLC circuit due to a smaller equivalent series resistance (ESR) than the inductor
of the first case. We postulate that the ESRs of both inductors are negligible, as they are in the µH range, compared to the overall resistance of the actuation circuit. On the other hand, the larger inductor
results in a better electrical quality factor than the smaller inductor
given that
thereby making for a better amplifier circuit.
We applied the same excitation technique to actuator II to verify the scheme repeatability. An external inductor µH was connected in series with the actuator, resulting in an electrical resonance of MHz. The frequency of the amplifier signal was set to match the electrical resonance . The frequency of the second signal was swept down in the range MHz over 100 ms so that the difference between the two frequencies was swept up in the vicinity of the mechanical natural frequency MHz.
Figure 18a shows the frequency response of actuator II under a fixed amplifier voltage amplitude
V and five levels of second signal voltage amplitude
. The largest realized displacement with this inductor was
nm for a voltage amplitude of
V, corresponding to 0.435% of the air gap. For comparison, the figure also incorporates a frequency–response curve with a lower amplifier voltage amplitude of
V and an actuation voltage amplitude of
V. It can be seen that the response in the latter case is comparable to that of the higher amplifier voltage under a much smaller actuation voltage of
V, thereby demonstrating the efficacy of the amplifier circuit.
A smaller external inductor
µH was also used in conjunction with actuator II, resulting in an electrical resonance frequency of
MHz. The first signal frequency was set equal to it,
, while the second signal frequency was swept down in the range
over 100 ms.
Figure 18b shows the frequency response of actuator II tip displacement under seven levels of the second signal voltage amplitude while the amplitude of the first amplifier signal was held constant
V. The maximum measured tip displacement was
nm for a voltage amplitude of
V, corresponding to 0.255% of the air gap.
Similar to the case of actuator
, the realized tip displacement using the larger
inductor was greater than that observed with the smaller
inductor, thus confirming our earlier hypothesis.
Figure 18b also incorporates a frequency–response curve with a lower amplifier voltage amplitude of
V and an actuation voltage amplitude of
V. It can be seen that the response in this case is comparable to that of the higher amplifier voltage under a smaller actuation voltage of
V, which confirms the efficacy of the amplifier circuit.
Next, we tested the use of a lower electric resonance
to drive the actuator via the electrostatic force of the frequency sum
. Actuator
was connected in series to a large inductor, made of two inductors
mH and
mH connected in series, resulting in an estimated electrical resonance of
kHz,
Figure 10a. The electrical resonance was found to be
kHz using the drive setup. The first signal frequency was set equal to the electrical resonance
while the second signal frequency was swept up in the range
kHz over 100 ms, so that the frequency sum was swept past the mechanical resonance
in the range
MHz.
The frequency–response curves of actuator
tip displacement are shown in
Figure 19 for four values of the second signal voltage amplitude
while the amplifier voltage amplitude was held constant at
V. The largest displacement in this case was
nm, realized at resonance for a voltage amplitude of
V, corresponding to 0.235% of the air gap.
3.4.5. Amplitude Modulation
In this excitation scheme, we utilize an amplitude-modulated signal, resulting in an electrostatic force of the form
The drive signal, shown in
Figure 20, is the product of an RF carrier signal with a frequency
and a baseband signal with a frequency
. Given that
is, by definition, much larger than
, the actuator acts as a low-pass filter, thereby attenuating the higher frequency forces at
,
, and
. To guarantee this, the RF carrier frequency
is typically set in the range of 3–100
[
23]. On the other hand, the actuator follows the electrostatic force generated by the envelope of the signal. Provided that we set
, the force at
, which is proportional to
as per Equation (
12), can be used as the dominant excitation force. The ratio between the two amplitudes
is the signal modulation index.
The RLC circuit was tuned to set the electrical resonance frequency higher than that of the actuator
using a small external inductor
µH connected in series with actuator
. The resulting electrical resonance was measured as
MHz using the technique described in
Section 3.4.4. The carrier signal frequency was set to match the electrical resonance
, while the frequency of the baseband signal was swept up past the mechanical resonance
in the range
MHz over 100 ms.
Figure 21a shows the frequency response of actuator
tip displacement under six levels of the second voltage amplitude
, while the amplitude of the first (amplifier) signal is held constant at
V. The maximum displacement realized was
nm observed at resonance
MHz under a second voltage amplitude of
V (a modulation index of
), which is 0.99% of the air gap.
To test the flexibility of this technique, we tuned the electrical resonance to a higher value by using a smaller external inductor
µH. As a result, the electrical resonance increased to
MHz.
Figure 21b shows the frequency response of actuator
under four levels of the second voltage amplitude
while the amplifier signal amplitude is held constant at
V. The maximum measured displacement was
nm for a voltage amplitude of
V, corresponding to 0.46% of the air gap.
A third inductor
µH was also used with actuator
to set the electrical resonance to
MHz.
Figure 21c shows the frequency response of actuator I
under seven levels of the second signal voltage amplitude
while the amplifier signal amplitude is held constant at
V. The maximum measured displacement was
nm for a voltage amplitude of
V, corresponding to 0.47% of the air gap. The smaller displacements realized in the last two cases compared to the case of the first case, with the larger
inductor, confirms our hypothesis above that the ESR of inductors at an µH level is negligible compared to the other circuit losses, which allows the larger inductance of
to improve the overall circuit quality factor
.
We applied the same excitation technique to actuator II to verify the scheme repeatability. An external inductor
µH was connected in series with the actuator, resulting in an electrical resonance at
MHz. The difference between the electrical resonance in this case and that in actuator
case is due to inter-inductor variability and variations in the circuit wiring. The baseband signal frequency was swept up in the range
MHz.
Figure 22a shows the frequency–response curves under eight levels of the second voltage amplitude
. The maximum tip displacement achieved was
nm at resonance
MHz for a second voltage amplitude of
V, which is 0.395% of the air gap.
For comparison, the figure also incorporates a frequency–response curve with a lower amplifier voltage amplitude of V and a second voltage amplitude of V. It can be seen that the response in the latter case is comparable to that of the higher amplifier voltage under a much smaller second voltage of V, thereby demonstrating the efficacy of the amplifier circuit.
The smaller
µH inductor was connected in series with actuator II, resulting in an electrical resonance of
MHz.
Figure 22b shows the frequency–response curves of the tip displacement for seven levels of the second voltage amplitude
. The largest measured RMS displacement was
nm at resonance for the second voltage amplitude of
V, corresponding to 0.23% of the air gap. The results also confirm that larger inductors
result in higher amplification (larger
) for µH-sized inductors.
Figure 22b also incorporates a frequency–response curve with a lower amplifier voltage amplitude of
V and a second voltage amplitude of
V. It can be seen that the response in this case falls between those for the smaller voltage amplitudes
V and 2 V and the higher amplifier voltage
V, further confirming the efficacy of the amplifier circuit.
3.4.6. Comparison Among the Actuation Techniques
To evaluate the performance of the three actuation techniques and compare them, we define a magnification factor
that describes the efficiency of electromechanical coupling between the system input, namely the voltage signal applied to the MEMS circuit, and its output, the measured displacement
. To avoid the ambiguity posed by the presence of multiple local maxima in cases where either signal contains more than one frequency, those signals are quantified in terms of their RMS. The magnification factor serves as a figure of merit, allowing us to quantify the relative merits of various actuation schemes in various applications.
The magnification factors realized for actuators I,
, and II are listed in
Table S1,
Table S2, and
Table S3, respectively, in the
Supplementary Materials. Comparison across the various actuation schemes, as shown in
Figure 23, shows that unbiased resonance matching is most effective in achieving the highest magnification factors,
nm/V and
nm/V compared to magnification factors of
nm/V and
nm/V for the biased cases in actuators I and II, respectively. This is expected since unbiased resonance matching amplifies input voltage by
compared an amplification by
only in the biased case.
The performance of multi-frequency and amplitude modulated actuations is inferior to biased resonance matching, but they are similar to each other with a slight advantage to the former. The advantage of biased resonance matching is due to its actuation voltage benefiting from amplification via the of the RLC circuit and the bias voltage , while the latter two realize their amplification only through the of the RLC circuit. In both multi-frequency and amplitude modulated actuations, lower electrical resonance frequencies (larger inductors) were found to realize better magnification factors. Actuator achieved nm/V at lower frequency, MHz, compared to nm/V at higher frequency, MHz. Similarly, actuator II exhibited an improvement from nm/V at higher frequency , MHz, to nm/V at lower frequency, MHz. For amplitude modulation, a similar trend was observed, with actuator achieving nm/V at lower versus nm/V at a higher . Actuator II followed the same pattern, with improving from nm/V at higher to nm/V at lower .
Finally, we observe that the magnification factor of actuator I under voltage amplifier actuation was nm/V compared to nm/V for actuator II under the same actuation scheme. The discrepancy in the magnification factor between the two actuators is probably due to the higher stiffness of actuator II.