A sandwich (Langevin-type) piezoelectric ultrasonic transducer with two separately excited sets of piezoelectric ceramic elements, separated by a middle metal cone, was proposed and analyzed one-dimensionally by Lin and Xu (2008). Its resonance and anti-resonance frequencies depend on the geometrical dimensions and on the electrical boundary conditions (short- or open-circuited) of each ceramic set, making it a candidate frequency-variable or multi-frequency power transducer. Later work extended the cascade concept to three stacks, incorporated losses and load impedances in sibling geometries, studied bolt pre-load effects in conventional single-stack Langevin transducers, and demonstrated more than 15% thermally induced resonance tuning using Nitinol end masses. However, no unified predictive design theory for the two-set tunable sandwich transducer exists that simultaneously includes mechanical and dielectric losses, pre-stress bolt compliance, and external electrical and acoustic loads; establishes the validity limits of the one-dimensional approximation; characterizes input impedance, load characteristics, and electro-acoustic efficiency under realistic loading; and quantifies the tuning span achievable through the device's geometrical dimensions and the electrical boundary conditions of its two ceramic sets while preserving coupling and efficiency, in quantitative agreement with measured data. The problem asks for that theory and for the associated tuning-range-versus-performance limits.

Background

A sandwich, or Langevin, transducer is the standard high-power ultrasonic source: a stack of piezoelectric ceramic rings (typically lead zirconate titanate, PZT, polarized along its thickness so an applied alternating voltage produces longitudinal strain) is clamped between a massive back metal block and a front radiating mass, usually by a pre-stressing bolt that keeps the ceramics under compression. In the classical analysis the transducer is treated as a one-dimensional (1D) longitudinal vibrator: because the radial dimensions are much smaller than the longitudinal ones, each section (metal cylinder or ceramic stack) is described by plane longitudinal waves, and the whole device is represented by an electromechanical equivalent circuit in which the electrical port (voltage, current) couples to the mechanical ports (force, velocity) through the piezoelectric constants. The resonance frequency frf_r (where the input electrical impedance is minimal, current maximal for a voltage drive) and the anti-resonance frequency faf_a (impedance maximum) follow from the impedance expression; the effective electromechanical coupling coefficient keffk_{\mathrm{eff}}, defined by keff2=(fa2−fr2)/fa2k_{\mathrm{eff}}^2 = (f_a^2 - f_r^2)/f_a^2, measures how much electrical energy converts to mechanical energy and is a key figure of merit. The electrical boundary condition of a ceramic set matters: short-circuiting versus open-circuiting its electrodes changes the effective stiffness of that set (piezoelectric stiffening) and hence the spectrum. The frequency-tuning span of the device is the range of its resonance frequency frf_r that is attainable by varying the tuning degrees of freedom — the geometrical dimensions and the electrical boundary conditions (short- or open-circuited) of the two ceramic sets — expressed as a percentage of the nominal resonance frequency. Lin had earlier shown that external electric load impedances shift the performance of ordinary sandwich transducers (Lin 2004). Lin and Xu (2008) analyzed a sandwich transducer with two sets of piezoelectric elements separated by a middle metal cone, each set excited separately: they derived two groups of resonance/anti-resonance frequency equations and showed experimentally that the obtainable frequencies depend on the geometrical dimensions and on the electrical boundary conditions of the two sets — with short-circuiting, the two resonance frequencies coincide but the anti-resonance frequencies differ, and with open-circuiting the reverse holds — so that by choosing geometry and electrical boundary conditions the device can work as a frequency-variable or multi-frequency transducer. The analysis was 1D, lossless, ignored the pre-stressing bolt and any external mechanical (acoustic) load, and computed only resonance frequencies and keffk_{\mathrm{eff}}, not efficiency or load characteristics. Subsequent work kept this framework: the three-stack cascaded extension (Meng and Lin 2019) is still 1D and lossless and attributes its theory-experiment discrepancies to ignored losses and bolt; sibling analyses in the same research line incorporated mechanical and dielectric losses and load impedances for other configurations (length-extensional composite transducers in Lin, Cao and Qiao 2016; radially cascaded spherical transducers in Tang and Lin 2023); bolt clamping was studied for conventional single-stack Langevin transducers (Kim and Lee 2020; Lu et al. 2022); and the resistance-adjusting frequency-variable concept was developed for cylindrical, stack, and cement-based transducers (Wang et al. 2017, 2020; Lan et al. 2023). Very recently, Nitinol (a shape-memory alloy whose elastic modulus changes with temperature-driven phase transformation) end masses demonstrated more than 15% resonance tuning in a Langevin transducer (Hafezi, Liu and Feeney 2026). What has not been done is to combine these ingredients into a predictive and validated design theory for the two-set frequency-variable sandwich transducer itself.

Problem Statement

For the sandwich (Langevin-type) piezoelectric ultrasonic transducer with two sets of piezoelectric elements separated by a middle metal cone, whose resonance and anti-resonance frequencies are tunable through geometrical dimensions and the electrical boundary conditions of the two ceramic sets: establish a predictive design theory, whose impedance and resonance/anti-resonance frequency predictions agree quantitatively with measured data of the kind reported in the source studies, that (1) incorporates mechanical and dielectric losses, the compliance and pre-load of the stressing bolt, and external electrical and mechanical (acoustic) load impedances into the input-electrical-impedance and resonance/anti-resonance frequency equations, so that the measured-theoretical discrepancies (notably in the effective electromechanical coupling coefficient keffk_{\mathrm{eff}}) are quantitatively explained; (2) delimits the validity of the one-dimensional longitudinal approximation by accounting for radial-dimension effects, coupled radial modes, and nonuniform stress in larger-radius designs; (3) predicts the load characteristics, full input electrical impedance behavior under varied boundary conditions, and electro-acoustical efficiency (the ratio of radiated acoustic power to input electrical power) under realistic acoustic loading; and (4) determines, with quantitative design rules, the frequency-tuning span achievable through the transducer's own tuning degrees of freedom — the geometrical dimensions and the electrical boundary conditions of the two ceramic sets — together with the tradeoff between tuning span and acceptable keffk_{\mathrm{eff}} and electro-acoustic efficiency. These four deliverables jointly serve one overarching objective: a complete, predictive engineering-science account of the frequency-variable sandwich transducer, brought into quantitative agreement with measured impedance and frequency data of the kind reported in the source studies. The four items are independently resolvable: an answer that settles only a subset must state explicitly which of items (1)–(4) it resolves, each resolved item counts as settling that deliverable on its own, the unresolved items remain open, and no single ingredient may be presented as resolving the whole objective.

References

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