Equivalent Circuit Modelling for Unimorph and Bimorph Piezoelectric Energy Harvester

Abstract

Piezoelectricity is a growing area of energy harvesting research. Piezoelectric energy harvester (PEH) generally use cantilever structures in unimorph and bimorph configuration. These harvester sense the mechanical vibration and convert them into usable electric energy. In this work, an analytical model displaying the characteristics of PEH with resonant frequency, tip deflections and output power have been determined. Analytical model is in close agreement with the FEM simulation model having an error of about $5\%$. In order to bridge the gap between FEM simulation and energy harvesting circuitry an equivalent electrical model for the energy harvester is developed. Resonant frequency determined from the equivalent model is 87.12 Hz, whereas from FEM model and analytical model are 90.56 Hz and 93.32 Hz respectively.

Supported by MEITY.

1 Introduction

Wireless sensing devices and portable electronics require periodically maintained external power supply throughout their lifetime. Self powered wireless sensing devices are being developed for low power applications. There are many methods to harvest energy like solar, wind, vibration etc. A promising source of energy is vibration which can be used to harvest power for wireless sensors [1]. There are three ways to harvest vibration energy from the environment to power wireless sensing nodes i.e. electrostatics, capacitive and piezoelectric. Among these methods piezoelectric is the most reliable and robust technique to harvest vibrations. Piezoelectric energy harvesting involves simple structure, low cost, no electromagnetic interference, easiness to fabricate and a high energy density compared to the other two vibration techniques [2]. Vibration energy harvesting has been area of research for the past few years, with a lot of emphasis being given to piezoelectric materials. Modeling of piezoelectric materials has been a focus of research over the past few years [3]. Despite the research a model relating material properties and power output for the harvester in an easy manner is not available. The present research work is aimed on the development of a new model for evaluating the output power of the piezoelectric energy harvester using analogies to electronic circuit theory. The model, initially, relies on equivalent circuit representation of electromechanical transducers. The equivalent circuit representation can model electromechanical transducers (i.e. electrical and mechanical domains), using simple circuit elements, giving an analogy in the differential equations describing both domains, and couple both domains by an ideal electromechanical transformer [4].

SPICE (Simulated Program with Integrated Circuit Emphasis) and FEA (Finite element analysis) have been utilized to model electrical and mechanical properties of a Piezoelectric energy harvester (PEH). Open circuit and short circuit voltage can be estimated with finite analysis. For system with weak electromechanical coupling these results can be directly applied [5]. But when there is strong coupling, mechanical vibrations are effected by backward coupling, which causes inconsistencies in voltage and current values. In order to bridge this gap between SPICE and FEA an Equivalent circuit model (ECM) is developed, which could estimate the performance of PEH accurately.

Table 1. Device parameters for piezoelectric energy harvester

2 Analytical Model

Characteristics of a PEH can be realized with the help of an analytical model. Several coupled and uncoupled lumped and distributed models have been developed in the past few years. Distributed model based on Rayleigh-Ritz and Euler Bernoulli theorem give an appropriate analysis of the harvester under ambient vibration conditions. Ertuk and Inman [5] reviewed many models and addressed the issues in distributed modeling of PEH. A brief description of the distributed parameter model of simple cantilevered unimorph harvester has been provided with calculation of tip deflection as well as generated power output have been derived. The energy harvester in a bimorph configuration can be derived similarly. Table 1 describes the device parameters for PEH.

Fig. 1.

A Piezoelectric unimorph energy harvester [2]

Unimorph Cantilever PEH

A rectangular shaped unimorph cantilever based PEH is shown in Fig. 1. Beam is longitudinally uniform in density. Following assumptions are made in the analytical modeling on the constitutive piezoelectricity relations: (a) Beam assumption for Euler-Bernoulli; (b) negligible air damping for external excitation; (c) proportional damping (i.e., viscous air damping and strain rate damping are assumed to be proportional to bending stiffness and mass per length); and (d) uniform electric field through the piezoelectric thickness [6]. The piezoelectric constitutive equations are given as:

$$\begin{aligned} \sigma =dE+sX \end{aligned}$$

(1)

$$\begin{aligned} D= \epsilon E+ dX \end{aligned}$$

(2)

where s is elasticity modulus and $\sigma$ is stress that is dependent on electric field E and strain X while d is the piezoelectric constant. Electric field and strain determine the dielectric displacement D. $\epsilon$ is absolute permittivity of the piezoelectric material. Assumptions (a) and (b) determine the governing equation of mechanical motion is,

$$\begin{aligned}&\frac{\partial ^2 M(x,t)}{\partial x^2}+c_a\frac{\partial w_{rel}(x,t)}{\partial t}+m\frac{\partial ^2 w_{rel}(x,t)}{\partial t^2}+~~\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad c_sI\frac{\partial ^5 w_{rel}(x,t)}{ \partial t\partial x^4} =-m\frac{\partial ^2 w_{b}(x,t)}{\partial t^2} \end{aligned}$$

(3)

where I is moment of inertia of beam; M(x, t) is the internal bending moment of the beam; $w_{rel}(x t)$ and $w_b(x t)$ are deflection relative to the base motion and base excitation, respectively; $c_a$ & $c_s$ are coefficients for viscous air damping and strain rate damping respectively and m is mass per unit length. The bending moment of the piezoelectric layer [7] is given by:

$$\begin{aligned} D_p=\frac{E_p^2t_p^4+E_{np}^2t_{np}^4+E_pt_{np}(2t_p^2+2t_{np}^2+3t_pt_{np})}{12(E_pt_p+E_{np}t_{np})} \end{aligned}$$

(4)

The natural transverse vibration is written as:

$$\begin{aligned} f_i=\frac{v_i^2}{2\pi \sqrt{2}}(\frac{h}{l_b^2} )\sqrt{\frac{e_T}{\rho }} \end{aligned}$$

(5)

The Bernoulli-Euler equation can be derived in a term related to bending modulus as,

$$\begin{aligned} f_i=\frac{v_i^2}{2\pi l_b^2}\sqrt{\frac{D}{m_w}} \end{aligned}$$

(6)

with $v_i$ for the first three modes are: $v_1=1.8751,$ $v_2=4.6941$ and $v_3=7.857$ The first natural frequency of the unimorph structure is calculated as:

$$\begin{aligned} f_N=\frac{0.1615}{l_b^2}\sqrt{\frac{D_p}{m}} \end{aligned}$$

(7)

The natural frequency of cantilever with proof mass, $f_M$ is:

$$\begin{aligned} f_M=f_N\sqrt{\frac{m_{eff}}{m_{eff}+M_t}} \end{aligned}$$

(8)

$M_t$ is the additional proof mass while the effective mass at the tip of the cantilever structure is $m_{eff}$. Effective mass is given by:

$$\begin{aligned} m_{eff}=0.236\rho _bw_bh_bl_b \end{aligned}$$

(9)

Tip deflection of the unimorph cantilever is given by:

$$\begin{aligned} W_D=\frac{Ymw^2}{k\sqrt{[1-(\frac{\omega }{\omega _n})^2]^2}+(\zeta \frac{\omega }{\omega _n})^2} \end{aligned}$$

(10)

where, $Y=\frac{a}{\omega ^2}$ and $\zeta$ is damping coefficient. The primary focus of the work is on the bimorph energy harvester as it provides better power density then the unimorph energy harvester.

Fig. 2.

A Piezoelectric bimorph energy harvester

Bimorph Cantilever PEH

A bimorph is constructed with a substrate in center sandwiched between two piezoelectric layers as shown in Fig. 2. Thickness of piezoelectric layer is $t_p$ while that for substrate layer is $t_s$. The piezoelectric layers are electrically shorted in series with layers being poled towards the substructure. In order to tune cantilever a proof mass $M_t$ is placed at tip of cantilever (x = L). The parameters with the subscripts s and p refer to substrate and piezoelectric, respectively. Total base movement can be described by:

$$\begin{aligned} w_b(x,t)=xh(t)+g(t) \end{aligned}$$

(11)

Beam motion of a bimorph cantilever for series connected piezoelectric layer is given by:

$$\begin{aligned}&YI\frac{\partial ^4 w_rel(x,t)}{\partial x^4}+c_a\frac{\partial w_{rel}(x,t)}{\partial t}+ k v(t) (\frac{d\delta (x)}{dx}-\frac{d\delta (x-L)}{dx})+~~\nonumber \\&\qquad \quad m\frac{\partial ^2 w_{rel}(x,t)}{\partial t^2}+c_{sI}\frac{\partial ^5 w_{rel}(x,t)}{\partial t\partial x^4 } =-[m+M_t\delta (x-L)\frac{\partial ^2 w_{b}(x,t)}{\partial t^2}] \end{aligned}$$

(12)

$c_{sI}$ and $c_a$ are coefficient for strain rate and viscous air dampening. Piezoelectric coupling term k, m is mass per unit length [8]. v(t) is the voltage over the piezoelectric layers in accordance with Dirac delta function $\delta$. The bending stiffness term YI of the cantilever is given by:

$$\begin{aligned} YI=\frac{2b}{3}[Y_s\frac{t_s^3}{8}+c_{11}^E((t_p+\frac{t_s}{2})^3-\frac{t_s^3}{8})] \end{aligned}$$

(13)

where $c^E_{11}$ is elastic constant at constant E-field while $Y_s$ is Young’s modulus for the substructure. $t_s$ and $t_p$ are thickness of substructure and piezoelectric layers respectively, and width of the cantilever is b.

Air damping constitutes to about $10\%$ of total damping when the harvester is working in air, hence proportional damping exists in the system making the $c_{sI}$ = $c_a$ $= 0$. When harvester is working in higher viscosity fluids, these constants have to be considered [8]. The piezoelectric coupling coefficient k is given by:

Fig. 3.

A Piezoelectric bimorph energy harvester

$$\begin{aligned} k=\frac{d_{31}c_{11}(h_p+h_s)b}{2} \end{aligned}$$

(14)

Mass per unit length m for bimorph is:

$$\begin{aligned} m=b(\rho _sh_s+2\rho _ph_p) \end{aligned}$$

(15)

The resonant frequency for the bimorph cantilever can be calculated using Eqs. (7) and (15). Assuming proportional damping in beam, the response of system can be evaluated by an absolute convergent series of eigen value functions [7]:

$$\begin{aligned} \omega _{rel}^s=\varSigma _{r=1}^{\infty }\phi _r(x)\eta _r(t) \end{aligned}$$

(16)

The mass-normalized eigenfunction $\phi _r(x)$ beam deflection in its $r^{th}$ mode, and modal mechanical function $\eta _r(t)$ describes amplitude of deflection over time. Tip deflection of unimorph cantilever with varying input vibration frequency is depicted in Fig. 3 based on:

$$\begin{aligned} W_D=\frac{Ymw^2}{k\sqrt{[1-(\frac{\omega }{\omega _n})^2]^2}+(2\zeta \frac{\omega }{\omega _n})^2} \end{aligned}$$

(17)

Fig. 4.

(a) The frequency vs the power density, (b) the stress and generated output power for the input vibration frequency

2.1 Output Power

System’s output power depends on amount of charge developed on the piezoelectric layer and due to the applied mechanical stress,

$$\begin{aligned} Q_3=2 \int _{0}^{L}D_3 w_s dx \end{aligned}$$

(18)

By solving (1), (2), (10), (17) and (18) we obtain

$$\begin{aligned} Q_3=2 w_s \int _{0}^{L}(d_{31} \sigma _1 + \epsilon _1^{\sigma } \frac{I_r R}{t_p}) \end{aligned}$$

(19)

Current is defined as the charge per unit area; hence, differentiating and solving (19) gives:

$$\begin{aligned} I_r(f)= \frac{d_{31} E_p t_n w_s (\frac{f}{f_n})^2 m a l_p}{4 EI \sqrt{X+(1-{(\frac{f}{f_n})^2)^2} }} \end{aligned}$$

(20)

The average power dissipated in the load resistor R can be determined:

$$\begin{aligned} P_{avg}(f)= \frac{(I_r(f))^2 R}{2} \end{aligned}$$

(21)

By using (20) in (21) we obtain

$$\begin{aligned} P_{avg}(f)= \frac{d_{31}^2 E_p^2 t_n^2 w_s (a)^2 m^2 (\frac{f^3}{f_n})^2 l_p R}{32 (EI)^2 ((1- (\frac{f}{f_n})^2)^2+{X})} \end{aligned}$$

(22)

where

$$l_p = (4 L^2 + 6 L l_m + 3 l_m^2)^2$$

and

$$X= (2 \zeta \frac{f}{f_n})^2(1+ \frac{2 \epsilon _p^{\sigma } w_s L f R}{t_p})$$

Power density from the analytical model Eq. (22) is compared with the FEM model in Fig. 4a and stress developed on the piezoelectric layer is converted into outout power as shown in Fig. 4b.

3 Equivalent Circuit Modeling

A PEH is generally connected to a resistive load, the power drop across the resistive load can be determined by analytical model and system-level FEA. In practical application non-linear electric components such as rectifier and regulator are included in energy harvesting circuits. These circuits also have storage elements [9]. Energy harvesters having complex geometries cannot be modeled using analytical model and FEA. For these complex structures equivalent circuit modeling is the most appropriate method.

Fig. 5.

Mass spring damper equivalent

3.1 Mechanical Equivalent

A mass spring damper system can be used to model PEH as shown in Fig. 5. Mass of the PEH system is m, $k_s$ is the spring constant and d is damping coefficient. Harmonic movement of frame is y(t) while z(t) is relative motion of seismic mass [10]. Transfer function for the system is given by:

$$\begin{aligned} ma=m\ddot{z}+d\dot{z}+k_sz+F_e \end{aligned}$$

(23)

$$a(t)=\ddot{y}(t)=\hat{a} \sin (\omega t)$$

External force applied on the harvester is ma. Damping force is $F_e$.

$$\begin{aligned} ma=m\ddot{z}+(d_e+d)\dot{z}+k_sz \end{aligned}$$

(24)

Taking laplace transform,

$$\begin{aligned} ms^2y=ms^2z+(d+d_e)sz+k_sz \end{aligned}$$

(25)

The dimensionless electrical & mechanical damping coefficient is given by:

$$\begin{aligned} \zeta _e=\frac{d_e}{2m\omega _n} ~~ and ~~ \zeta _d=\frac{d}{2m\omega _n} \end{aligned}$$

(26)

Transfer function,

$$\begin{aligned} \frac{Z(s)}{y(s)}=\frac{s^2}{s^2+2\omega _n(\zeta _d+\zeta _e)s+\omega _n^2} \end{aligned}$$

(27)

3.2 Electrical Equivalent

Electrical analogy is used to determine the equivalent circuit of the energy harvester. Mechanical force is represented as voltage, while electric current acts as mechanical velocity [11, 12]. From (1) and (2) the electrical equivalent circuit is given in Fig. 6.

Fig. 6.

Electrical equivalent of energy harvester

Electrical circuit parameter can be determined in terms of energy harvester design parameters.

$$\begin{aligned} k_s=\frac{3YI}{L^2} \end{aligned}$$

(28)

$$\begin{aligned} L=m_{eq}=\frac{33}{140}mL+M_t \end{aligned}$$

(29)

where m is the mass per unit length of beam and $M_t$ is mass of proof mass kept at tip of beam. Resistive component values can be determined from Eq. (26). The circuit above only represents the mechanical part of the harvester. Mechanical domain coupling to the electrical domain using electromechanical is established with the help of a transformer having winding ratio 1 : N as shown in Fig. 7.

Fig. 7.

Coupled electromechanical system

Electromechanical coupling coefficient is defined as $k_{31}^2$,

$$\begin{aligned} k_{31}^2=\frac{d_{31}^2}{\AA _{33}^Ts_{11}^E} \end{aligned}$$

(30)

Transformer coupling N is defined in terms of design parameters,

$$\begin{aligned} N=\frac{d_{31}w}{s_{11}^E} \end{aligned}$$

(31)

A simplified model is also derived with RLC circuit components can also be used as an electrical equivalent circuit for PEH as shown in Fig. 8.

Fig. 8.

Simplified electromechanical equivalent circuit

$$\begin{aligned} V_{mc}=\frac{ma}{N} \end{aligned}$$

(32)

$$\begin{aligned} R_{mc}=\frac{d}{N^2} \end{aligned}$$

(33)

$$\begin{aligned} L_{mc}=\frac{m_{eq}}{N^2} \end{aligned}$$

(34)

$$\begin{aligned} C_{mc}=\frac{N^2}{k_s} \end{aligned}$$

(35)

Using the electrical equivalent circuit model resonant frequency of the harvester can be determined as

$$\begin{aligned} \omega _n=\frac{1}{2\pi \sqrt{L_{mc}C_{mc}}} \end{aligned}$$

(36)

for the bimorph energy harvester the resonant frequency using the electrical equivalent circuit is determined as 90.12 Hz.

4 Conclusion

An effective analytical model determining the resonant frequency, tip displacement and output generated power for the PEH has been described. The model matches the simulation result with a maximum error of $5\%$. An electrical equivalent circuit for the energy harvester is also emulated. The circuit consists of Resistor, Capacitor, Inductor with coupling being represented by a transformer. The resonant frequency from equivalent circuit is 87.12 Hz, while those from FEM and analytical model is 90.56 Hz and 93.32 Hz.