Shock and Vibration

Volume 2016 (2016), Article ID 2413578, 9 pages

http://dx.doi.org/10.1155/2016/2413578

## System-Level Coupled Modeling of Piezoelectric Vibration Energy Harvesting Systems by Joint Finite Element and Circuit Analysis

^{1}Science and Technology on Integrated Logistics Support Laboratory, National University of Defense Technology, Changsha 410073, China^{2}Faculty of Engineering and the Environment, University of Southampton, Boldrewood Campus, Southampton SO16 7QF, UK

Received 10 December 2015; Revised 27 January 2016; Accepted 3 February 2016

Academic Editor: Lorenzo Dozio

Copyright © 2016 Congcong Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A practical piezoelectric vibration energy harvesting (PVEH) system is usually composed of two coupled parts: a harvesting structure and an interface circuit. Thus, it is much necessary to build system-level coupled models for analyzing PVEH systems, so that the whole PVEH system can be optimized to obtain a high overall efficiency. In this paper, two classes of coupled models are proposed by joint finite element and circuit analysis. The first one is to integrate the equivalent circuit model of the harvesting structure with the interface circuit and the second one is to integrate the equivalent electrical impedance of the interface circuit into the finite element model of the harvesting structure. Then equivalent circuit model parameters of the harvesting structure are estimated by finite element analysis and the equivalent electrical impedance of the interface circuit is derived by circuit analysis. In the end, simulations are done to validate and compare the proposed two classes of system-level coupled models. The results demonstrate that harvested powers from the two classes of coupled models approximate to theoretic values. Thus, the proposed coupled models can be used for system-level optimizations in engineering applications.

#### 1. Introduction

Nowadays, low power consumption wireless sensor networks (WSNs) are widely used in military and industrial fields, such as equipment operation condition monitoring, bridge health monitoring, and intelligent transportation. As for most applications, WSNs are far from power lines. In particular, wireless sensor nodes may need to be embedded into mechanical structures. Under those cases, it is difficult to power WSNs by wires, so that battery is the most conventional solution for WSNs. However, batteries need to be replaced regularly due to their limited life spans. For WSNs placed in highly dangerous or unreachable areas, it is hard and even impossible to replace batteries. Nowadays, the most suitable solution to extend the life of a WSN is to harvest environmental energy to generate electrical energy. Up to now, many efforts have been made on harvesting energy from environmental vibrations, such as mechanical vibrations cause by operating equipment or in-service bridges. Generally speaking, vibration energy harvesting can be carried out by electrostatic [1], electromagnetic [2], or piezoelectric mechanisms [3, 4]. In particular, piezoelectric mechanism is of particular interest due to high energy density and electromechanical coupling coefficient. Thus, piezoelectric vibration energy harvesting (PVEH) has been widely studied to realize self-powered WSNs [5, 6].

A typical PVEH system usually can be divided into two parts. The first part is the harvesting structure composed of piezoelectric material, elastic base, and electrodes, which determines the electrical energy transformed from the vibration energy. The second part is the electrical part composed of the interface circuit and storage unit, which determines the efficiency of the electrical energy from the harvesting structure into the storage unit such as capacitor. By now, many works have been conducted on those two parts. For the harvesting structure, piezoelectric cantilever beams are widely used due to their tremendous application potential, including unimorph and bimorph type beams. For the interface circuit, standard energy harvesting (SEH) interface circuit composed of a full-wave bridge rectifier is first used. But its efficiency is very low because intrinsic capacitance of a piezoelectric patch is often very large and varies with exciting frequencies, so that it is very difficult for SEH circuits to achieve optimal impedance matching [7]. In order to overcome this drawback, a technique called synchronized switch harvesting on inductor (SSHI) has been proposed to enhance the power output of a PVEH system, including parallel SSHI (P-SSHI) and series SSHI (S-SSHI) circuits [8]. SSHI interface circuits are nonlinear, so it has been testified that SSHI-based circuits can increase the harvested power by several times compared to that of a SEH circuit under the same inputs.

It can be seen that most existing works focus on either the harvesting structure or the interface circuit. In particular, many works on circuits in literature make the excessively simplifying assumption of a purely capacitive model of the piezoelectric circuit. In practice, however, the harvesting structure and the interface circuit are coupled due to inverse piezoelectric effect. Dynamic behaviors of the harvesting structure are affected by the interface circuit. Conversely, the output voltage of the harvesting structure will affect the performance of the interface circuit. From the viewpoint of engineering applications, the overall efficiency of a PVEH system is truly expected, which depends on many contributing factors such as the vibration excitations, the geometries and materials of the harvester structure, and the interface circuit. For this reason, a PVEH system should be considered as a whole for optimization to obtain a high overall efficiency. Thus, it is much necessary to build a system-level coupled model and carry out coupled analysis. Nowadays, one widely used coupled model for a PVEH device is the equivalent electromechanical circuit defined by lumped parameters involving both mechanical and electrical quantities [9–12]. And then a two-port representation can reasonably account for all feedbacks between the mechanical and the electrical parts. Its main difficulty is to estimate rapidly and accurately all model parameters, while direct measurements are time-consuming. In recent years, finite element modeling methods are introduced into the field of PVEH, but it is difficult to directly combine finite element and circuit analysis due to the nonlinearity of interface circuits. In [13] finite element solvers are coupled to SPICE circuit software to simulate electrical responses of a PVEH device. However, it requires transferring data between the finite element solver and SPICE simulator at each iteration, which leads to low calculating efficiency. Hence, there is an urgent need to rapidly and accurately establish a system-level coupled model to analyze and optimize a PVEH system.

In this paper, a typical PVEH system including a bimorph piezoelectric harvesting structure and a parallel SSHI circuit is taken into account. Then joint finite element and circuit analysis is combined to construct system-level coupled models of the PVEH system. The innovation of this paper is to propose two classes of coupled models. The first one is a system-level coupled circuit model, which can be independently built in circuit simulation software. The second one is a system-level coupled finite element model, which can be independently built in finite element software. Both models do not require data transferring during working and experimental tests for parameters estimations. The remainder of this paper is organized as follows: current problems of coupled modeling methods of a typical PVEH system are summarized in Section 2. In Section 3, coupled modeling methodology of the PVEH system is demonstrated based on joint finite element and circuit analysis and two classes of coupled models are proposed. Then the first-class model is investigated in Section 4 and the second-class model is investigated in Section 5. Simulations and experiments are done in Section 6. Finally, Section 7 concludes the whole paper.

#### 2. Problem Statements

In this paper, a typical PVEH system composed of a bimorph cantilever beam and a parallel SSHI circuit is shown in Figure 1. In order to integrate the harvesting structure with the interface circuit, the most conventional way is to build the equivalent circuit model of the harvesting structure. Firstly, the harvesting structure can be modeled as an equivalent mechanical model involving mass (), damping (), spring (), and piezoelectric unit as shown in Figure 2.