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Voltage-Controlled Synthetic Inductors for Resonant Piezoelectric Shunt Damping: a Comparative Analysis

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applied sciences Article Voltage-Controlled Synthetic Inductors for Resonant Piezoelectric Shunt Damping: A Comparative Analysis Marian Vatavu 1, Vasile Nastasescu 1, Flaviu Turcu 2 and Ioan Burda 2,* 1 Weapons Engineering Systems and Mechatronic Department, Military Technical Academy, 050141 Bucharest, Romania; [email protected] (M.V.); [email protected] (V.N.) 2 Physics Department, Babes-Bolyai University, 400084 Cluj-Napoca, Romania; [email protected] * Correspondence: [email protected] Received: 15 October 2019; Accepted: 5 November 2019; Published: 8 November 2019 Abstract: In this paper, the design, simulations, and experimental results related to new analog circuits for voltage controlled synthetic inductors (VCSI) are presented. The new circuits based on a generalized impedance converter (GIC) are proposed for adaptive resonant piezoelectric shunt damping. The VCSIs are implemented using (1) an analog multiplier and (2) an operational transconductance amplifier (OTA) as voltage-controlled resistor. The simulation and experimental results for the new proposed VCSIs are presented and a comparative analysis follows. The proposed VCSIs work in a stable manner in parallel with negative impedance converters (NIC) to enhance structural damping in resonant piezoelectric resistive-inductive shunt applications. The behavior of the synthetic inductor is identical to a real inductor only in a specific frequency range and this situation can explain the reported spreading performance in the literature for resonant piezoelectric shunt damping. The simulation results are validated by a group of experimental investigations that confirm the improved stability and linearity of the new circuits proposed as VCSIs. Experimental results show that the VCSI based on an analog multiplier have an enhanced linearity in comparison with the OTA version in a limited voltage control range. Keywords: synthetic inductor; voltage control; resonant piezoelectric damping; analog multiplier; operational transconductance amplifier 1. Introduction In structural vibration control, piezoelectric materials [1,2] have been used with regularity for a few decades. The ceramic perovskite (Pb[ZrxTi1 x]O3, 0 x 1) is a piezoelectric material transducer − ≤ ≤ called a PZT patch, which is bonded onto a mechanical structure to obtain a very efficient passive damping technique using an external resonant resistive-inductive circuit known as an RL shunt. Two versions of the parallel and series RL shunt circuits are presented in Yamada et al. [3]. The series and parallel resonant shunts are equivalent in terms of vibration control performance, but the parallel RL shunt is less sensitive to the mistuning of the optimal resistance value. A synthetic inductor is based on a generalized impedance converter (GIC), which is well-known as a gyrator [4]. The gyrator was used instead of a real inductor in the analog circuits era [5] to implement low frequency RC filters. The drawbacks of a real inductor are related to the physical dimensions, residual resistance, and electromagnetic interferences. Synthetic inductors are also used for mechatronic applications [6–8], as in the case of resonant piezoelectric RL shunt vibration damping. The most performant and frequently used circuits to implement synthetic inductors in mechatronic applications are Riordan [9] and Antoniou [10] gyrator circuits. The RL shunt damping technique is very sensitive to resonance frequency mistuning, and adaptive circuits [9,10] were proposed based on an additional piezoelectric sensor and a voltage-controlled synthetic inductor (VCSI). The analog VCSIs presented in the literature Appl. Sci. 2019, 9, 4777; doi:10.3390/app9224777 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 10 Appl.sensor Sci. and2019, 9a, 4777voltage-controlled synthetic inductor (VCSI). The analog VCSIs presented in2 ofthe 10 literature are based on a Junction Field Effect Transistor (JFET) [11] or photoresistive opto-isolator are[12] based used as on a a voltage Junction-controlled Field Effect resistor Transistor (VCR). (JFET) These [11 circuits] or photoresistive controlled by opto-isolator an external [voltage12] used have as a voltage-controlledthe advantages of a resistor wide range (VCR). of Theseresistance circuits values controlled but unfortunately by an external have voltage a nonlinear have the response. advantages The ofvactrol a wide (photoresistive range of resistance opto-isolator) values has but an unfortunatelyextended response have time, a nonlinear a significant response. memory The effect vactrol, and (photoresistiveretains the advantage opto-isolator) of optical has isolation. an extended Considering response time,digital a significantelectronics, memory we can obtain effect, andequivalen retainst thebehavior advantage with ofprevious optical isolation.analog VCSIs Considering versions digital by using electronics, a digital we potentiometer can obtain equivalent [13] as a behaviorVCR. To withavoid previous constrain analoging the VCSIsabovementioned versions by VCS usingI based a digital on analog potentiometer circuits presented [13] as ain VCR. literature, To avoid two constrainingnew approaches the have abovementioned been used to VCSI implement based onVCR analogs based circuits on (1) presented an analog in multiplier literature, and two (2) new an approachesoperational havetransconductance been used to implementamplifier (OTA). VCRs based The best on (1)achievement an analog multiplier of this paper and is (2) related an operational with the transconductancelinear response of amplifier the resonant (OTA). frequency The best achievementto the voltage of control this paper in isresonant related withpiezoelectric the linear RL response shunt ofvibration the resonant damping. frequency to the voltage control in resonant piezoelectric RL shunt vibration damping. This paper is organized as follows:follows: In Section 22,, thethe classicalclassical AntoniouAntoniou [[10]10] synthetic inductor circuit designdesign and and simulation simulation results results are presented.are presented. In Section In Section3, two new3, two circuits new forcircuits a voltage-controlled for a voltage- syntheticcontrolled inductor synthetic are inductor simulated. are The simulated. experimental The results,experimental comparative results, analysis, comparative and mistuning analysis, e ffandect ofmi thestuning VCSI effect non-linearity of the VCSI in RL non shunt-linearity vibration in RL damping shunt vibration are presented damping in Sectionare presented4. Finally, in Section 54. concludesFinally, Section the paper. 5 concludes the paper. 2. Synthetic Inductors Circuit Design In 1948, Bernard D. H. Tellegen [[4]4] proposedproposed aa two-porttwo-port passivepassive electricalelectrical network element as a kind of transformer in which the primary voltage is converted to a secondary current current,, and vice versa versa,, as it is shown in Figure1 1.. HeHe calledcalled thisthis devicedevice aa gyratorgyrator becausebecause ofof aa theoreticaltheoretical analogyanalogy withwith aa flywheelflywheel (gyroscope). Figure 1.1. Gyrator: two two-port-port passive electrical network element. Currently, the gyrator is mostly known as an active electronic circuit [9,10], which in a limited Currently, the gyrator is mostly known as an active electronic circuit [9,10], which in a limited range of voltage, current, and frequency, works as an ideal gyrator based on a transconductance model range of voltage, current, and frequency, works as an ideal gyrator based on a transconductance given by the following equation: model given by the following equation: v v v v j!C C 1= =1 = 1 = 1 = = = , (1) Z = = = = = =j! = j!Leq, (1) i1 g2v2 g2i2Zc g2 g1v1Zc g1 g2 g1 g2 where v is the voltage source, g1 and g2 are transconductances and C is the capacitive load. whereFirstv is, thethe voltagesynthesis source, withg active1 and gelements2 are transconductances of the gyrator andto implementC is the capacitive efficient load. RC filters [5] happenedFirst, many the synthesis years later with as activea result elements of the analog of the circuit’s gyrator innovation to implement period. effi Scientignificant RCfilters progress [5] happenedwas added manyby operational years later amplifier as a result (op of-amp) the analogintegrated circuit’s circuits innovation used in Riordan period. [9] Significant or Antoniou progress’s [10] wasconfiguration added by operationalthat ensures amplifier a stable (op-amp)performance integrated and a high circuits-quality used infactor Riordan of the [9 ]synthetic or Antoniou’s inductor. [10] configurationAfter some significant that ensures applications a stable in performance analog processing and a high-qualityof audio signals, factor a recent of the useful synthetic technolog inductor.ical Aftersupport some was significant offered by applications gyrators in resonant in analog piezoelectric processing ofRL audio shunt signals, vibration a recent damping. useful To technological demonstrate supportthe real performance was offered byof gyratorssynthetic in inductor resonant based piezoelectric on an Antoniou RL shunt op vibration-amp circuit damping., the PSPICE To demonstrate® simulator ® the(version real performance 9.1, OrCAD of®, syntheticCadence Design inductor Systems, based on San an AntoniouJose, CA, op-amp95134, USA circuit,) has the been PSPICE used, simulatoras shown ® (versionin Figure 9.1, 2a. OrCADTo satisfy, the Cadence real experimental Design Systems, condition, San Jose, the CA,
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