Online proceedings - EDA Publishing Association

11-13 May 2011, Aix-en-Provence, France
, ,
An Electromechanical Model for clamped-clamped Beam Type Piezoelectric
Transformer
Chi-Shao Chen 1 , Chia-Che Wu 2*
1
Graduate Student
2*
Assistant Professor 1
1,2
Department of Mechanical Engineering, National Chung Hsing University,
250, Kuo Kuang Road, Taichung, Taiwan, 402
Tel: +886-4-22840433 ext 419; Fax: +886-4-22877170;
E-mail: josephwu@dragon.nchu.edu.tw
Abstract- In this paper, an analytical solution of a fixed-fixed
beam type piezoelectric transformer with Euler-Bernoulli beam
assumption is proposed. The electromechanical equations are
first derived for transient motions, and coupled expressions for
the mechanical response and voltage output are obtained. The
resulting equations are further reduced for the case of excitation
around the first resonance frequency. Analyical solutions of
mechanical response, voltage, current, and power outputs are
presented. From analytical model, output voltage depends on
the lengthes of two electrodes, the length of beam, and the
Young’s modulus ratio and thickness ratio between PZT layer
and substrate. The lengthes of input electrodes and output
electrodes should be 0.22 time length of beam to achieve the
largest output when the transformer is excited at first resonance
frequency. The output voltages and the resonance frequencies of
transformers are proportional and inversely proportional to the
lengthes of beams, respectively. The combination of Young’s
modului and thicknesses of PZT layer and substrate change the
position of netural axis and the bending stiffness of beam,
concurrently. However, output voltages of transformers depend
not only on the postion of neutral axes but also on bending
stiffnesses.
I. Introduction
Piezoelectric materials have the piezoelectric effect,
which can convert vibration energy into electrical energy, so
it can be used to make transformers for raising or lowering a
voltage. Piezoelectric transformer (PT) offers many
advantages over the small size, lighter with flat structure,
electromagnetic field immunity, and high transforming ratio.
The idea of a PT was first implemented by Rosen in 1956[1].
It used the effect of couple between electrical and mechanical
energy of piezoelectric materials. Exciting mechanical
vibrations by the part of driver and output voltage can be
induced by the part of generator. Most of the PTs are using
the concept of Rosen-type PT such as uniformly-poled
longitudinal PT [2], stacked disk-type PT [3, 4], and
uniformly-poled disk type PT [5]. M. C. Do et al. used
parallel connection of Rosen-type PTs to increase output
power [6]. T. Inous et al.[7] developed a PT which is a
combination of a longitudinal mode piezoelectric actuator
and a longitudinal mode piezoelectric transducer transverse
in parallel to achieve larger power. However, operating
frequencies of transformers in the literature were usually
from a few kHz to several hundred MHz. External oscillator
and control circuit are required to satisfy the frequency
requirement. However, they will substantially expend the size
and the complexity of transformers.
The PT is not only a mechanical system but also an
electrical system. The electromechanical model approaches
in the recently literature include single degree-of-freedom
(SDOF) models [8], Rayleigh-Ritz method[9], equivalent
circuit method[10], and expansion theory based on the Euler-
Bernoulli beam assumptions [11]. The SDOF modeling
approaches supposes a structure such as a cantilevered beam
as a mass-spring-damper system which is convenient for
coupling the mechanical part and electrical part of
transformer. However, SDOF is just a simple approximation
and it is limited to a single vibration mode. SDOF lacks of
several important information of the system, such as the
dynamic mode shape, the accurate strain or stress distribution
along the beam. Rayleigh-Ritz method is a numerical
approximation technique based on discretization of the
continuous distributed parameter system and it allows
predicting the electromechanical response in higher vibration
modes. The Rayleigh-Ritz method can produce accurate
results with only a small number of terms in the
approximating series, which translates into a discrete model
with a small number of degree of freedom. However, the
Rayleigh-Ritz method can’t use in complex geometry and it’s
not an exact solution. Equivalent circuit model is used to
estimate the electrical characteristics of the PT, such as
voltage ratio between input and output. But, it has no idea
about the mechanical information since all parameters are
transferred into electrical form and some coupled coefficient
must be obtained from the experiments.
Erurk and Inman [11] presented the exact
electromechanical solution of a cantilevered piezoelectric
energy harvester with Euler-Bernoulli beam assumptions.
The electromechanical equations were derived for general
transient motions from expansion series and coupled
expression (not only single vibration mode) for mechanical
response and voltage output were obtained. This method
provides exact solutions of energy harvester. They also used
internal strain rate damping and external air damping to
achieve more accurate model. Backward coupling effect in
the mechanical domain and the contribution from the other
vibration mode were also considered in their model.
In this paper, an analytical solution of a fixed-fixed beam
type piezoelectric transformer with Euler-Bernoulli beam
assumption is proposed. A clamped-clamped beam
transformer consist of a fixed-fixed beam, a layer of
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