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Research Papers

Shaping the Frequency Response of Electromechanical Resonators Using a Signal Interference Control Topology

[+] Author and Article Information
Bryce A. Geesey, Blake A. Wetherton, Nikhil Bajaj

School of Mechanical Engineering;
Birck Nanotechnology Center;
Ray W. Herrick Laboratories,
Purdue University,
West Lafayette, IN 47907

Jeffrey F. Rhoads

School of Mechanical Engineering;
Birck Nanotechnology Center;
Ray W. Herrick Laboratories,
Purdue University,
West Lafayette, IN 47907
e-mail: jfrhoads@purdue.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received June 12, 2016; final manuscript received October 1, 2016; published online January 23, 2017. Assoc. Editor: Dumitru I. Caruntu.

J. Dyn. Sys., Meas., Control 139(3), 031011 (Jan 23, 2017) (9 pages) Paper No: DS-16-1310; doi: 10.1115/1.4034948 History: Received June 12, 2016; Revised October 01, 2016

The recent study of signal interference circuits, which find its origins in earlier work related to active channelized filters, has introduced new methods for shaping the frequency response of electrical systems. This paper seeks to extend this thread of research by investigating the frequency response shaping of electromechanical resonators which are embedded in feedforward, signal interference control architectures. In particular, mathematical models are developed to explore the behavior of linear resonators that are embedded in two prototypical signal interference control topologies, which can exhibit a variety of qualitatively distinct frequency domain behaviors with component-level tuning. Experimental approaches are then used to demonstrate the proposed designs' utility.

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References

Figures

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Fig. 1

Block diagram of the proposed control system with two signal branches, one with a resonator and another with a pure phase-shifter ϕ and gain G. The input U is split equally between the two branches, and the output V is the nonweighted sum of the individual responses x and y.

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Fig. 2

Theoretical phase θ and Q-scaled amplitude D̂ of the nondimensional output V̂, evaluated at resonance (rr=ωr/ωn=1−2ζ2), plotted with respect to the tunable design parameter ϕ. At ϕ=−90 deg, the resonant amplitude is doubled, and at ϕ=90 deg the amplitude is canceled.

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Fig. 3

Various theoretical frequency responses of the nondimensional system output V̂ at frequencies r in a small range near r = 1 for characteristic values of ϕ at 45 deg intervals. Clearly, ϕ=−90 deg and ϕ=90 deg lead to constructive and destructive interference, respectively. Other values of ϕ map the transitions between the two extremes. Resonance occurs in a narrow range of frequencies near r = 1 due to the high-Q assumption, with Q nominally set to 10,000 here.

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Fig. 4

Block diagram of the proposed two resonator control system where each signal branch contains a resonator, and the second branch also includes a pure phase-shifter ϕ and gain G. The input U is split equally between the two branches, y is the response of the second resonator, and the output V is the nonweighted sum of the individual branch responses x and z.

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Fig. 5

Various theoretical frequency responses of the two resonator system output amplitude D̂ scaled by the resonators' Q factor of 10,000. Here, G is chosen to be unity, while ϕ is iterated from 0 deg to 180 deg and the frequency detuning parameter α is adjusted slightly above and below unity. The horizontal axis is the normalized frequency r in a narrow range near r = 1. These results show (a) amplitude doubling, (b) complete amplitude cancellation, (c) bandpass-type responses, and (d) combinations of resonance and cancellation.

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Fig. 6

An example of the types of printed circuit board (PCB) proof-of-concept circuits built in this work. This depiction is the two resonator electromechanical circuit, according to the experimental schematic of Fig. 10.

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Fig. 7

A schematic of the single resonator experimental test circuit, where the input was provided by an Agilent 33250A function generator and the outputs were measured using an Agilent DSO8104A oscilloscope. Power at +5 V, −5 V, and ground was provided by two Agilent E3645A DC supplies. All operational amplifiers were LMH6609 by Texas Instruments, and the quartz crystal resonator was a Kyocera CX3225GB at 50 MHz. The inverting amplifier was switched into the circuit to provide an extra 180 deg phase shift. The power supply pins and decoupling capacitors are not shown.

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Fig. 8

Frequency responses of the experimental and modeled single resonator signal interference output. By adjusting the all-pass variable resistor, several values of ϕ were obtained to tune the circuit behavior. The model curves were generated by approximating system parameters from the experimental results, as shown in Table 2.

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Fig. 9

Three frequency responses of the signal interference test circuit at the constructive interference value of ϕ=−90 deg. The input amplitude was adjusted above and below the nominal 200 mV input in order to test the linearity of this circuit. The gain and phase plots tracked the same response for all three values of input magnitude, so the circuit was operating in a linear response range.

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Fig. 10

A schematic of the two resonator experimental test circuit. The input was provided by an Agilent 33250 A function generator, and the outputs were measured using an Agilent DSO8104A oscilloscope. Power at +5 V, −5 V, and ground was provided by two Agilent E3645A DC supplies. All operational amplifiers were LMH6609 by Texas Instruments, and the quartz crystal resonators were Kyocera CX3225GB at 50 MHz. The inverting amplifier was switched into the circuit to provide an extra 180 deg phase shift. The power supply pins and decoupling capacitors are not shown.

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Fig. 11

Frequency responses of the experimental and modeled two resonator signal interference output with α≈1. By adjusting the all-pass variable resistor, several values of ϕ were obtained to tune the circuit behavior. The model curves were generated by approximating system parameters from the experimental results, as shown in Table 3.

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Fig. 12

Images of the Kyocera CX3225GB 50 MHz bulk-mode quartz resonator (a) with the cover removed and (b) after depositing drops of polystyrene (spots on the electrode)

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Fig. 13

Frequency responses of the experimental and modeled two resonator signal interference output with α≈0.9999. By adjusting the all-pass variable resistor, several values of ϕ were obtained to tune the circuit behavior. The model curves were generated by approximating system parameters from the experimental results, as shown in Table 4.

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Fig. 14

Frequency responses of the experimental and modeled two resonator signal interference output with α≈1.0001. By adjusting the all-pass variable resistor, several values of ϕ were obtained to tune the circuit behavior. The model curves were generated by approximating system parameters from the experimental results, as shown in Table 5.

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