# Modeling the performance of a micromachined piezoelectric energy harvester

## Abstract

Piezoelectric energy microgenerators are devices that generate continuously electricity when they are subjected to varying mechanical strain due to e.g. ambient vibrations. This paper presents the mathematical analysis, modelling and validation of a miniaturized piezoelectric energy harvester based on ambient random vibrations. Aluminium nitride as piezoelectric material is arranged between two electrodes. The device design includes a silicon cantilever on which AlN film is deposited and which features a seismic mass at the end of the cantilever. Euler–Bernoulli energy approach and Hamilton’s principle are applied for device modeling and analysis of the operation of the device at various acceleration values. The model shows good agreement with the experimental findings, thus giving confidence into model. Both mechanical and electrical characteristics are considered and compared with the experimental data, and good agreement is obtained. The developed analytical model can be applied for the design of piezoelectric microgenerators with enhanced performance.

### List of symbols

- L
Length of cantilever beam (m)

- B
Width of cantilever beam (m)

- H
Thickness of structural layer (m)

- T
Thickness of piezoelectric layer (m)

- V
Voltage across piezoelectric element (m)

- Z
Coordinate parallel to beam thickness (m)

- X
Coordinate parallel to beam length or axial coordinate (m)

- T
_{k} Kinetic energy (J)

- U
Internal energy (J)

- W
_{e} Electrical work (J)

- W
External work (J)

- ρ
_{s} Density of structural layer (kg m

^{−3})- ρ
_{p} Density of piezoelectric layer (kg m

^{−3})- U
Displacement (M)

- S
Applied strain (None)

- T
Developed stress (Pa)

- E
Electric field (V m

^{−1})- D
Electric displacement (C m

^{−2})- dV
_{s} Differential volume of structural layer (m

^{3})- dV
_{p} Differential volume of piezoelectric layer (m

^{3})- E
Effective piezoelectric coupling coefficient C (m

^{2})- Cs
Elastic modulus of structural layer (Pa)

- c
_{p} Elastic modulus of structural layer (Pa)

- ε
^{S} Clamped permittivity (F/m)

- ε
^{T} Free permittivity (F/m)

- ε
_{0} Permittivity of free space (F/m)

- R
Generalized mechanical coordinate (M)

- Ψ
Bending mode function

*Φ*Voltage distribution function

- F
Discretized force (N)

- Q
Discretized charge (C)

*z*_{t}Coordinate with origin at the neutral axis and direction pointing parallel to the thickness of the beam (M)

*B*_{f}Forcing vector (Kg)

- A
Applied acceleration (m s

^{−2})- M
Effective mass (Kg)

- K
Effective spring constant (N m

^{−1})- Θ
Alternative coupling coefficient (F V

^{−1})- Η
Viscous drag coefficient (N s m

^{−1})- C
_{p} Capacitance of piezoelectric element (F)

- Q
Charge stored (C)

- L
_{0} Length of tip mass (M)

- h
_{0} Thickness of tip mass (M)

- M
_{0} Mass of tip mass (Kg)

- J
_{0} Moment of inertia of tip mass (kg m

^{2})- S
_{0} Static moment of tip mass (kg m)

- M
Mass per length of beam (kg m

^{−1})- EI
The sum of the moments of area of each layer multiplied by their respective elastic moduli (N m

^{2})- Ω
Angular frequency (s

^{−1})- ω
_{n} Resonant angular frequency (s

^{−1})- c, d, e, f
Constants in the bending mode function

- A
_{ij} Matrix elements

- λ
_{N} Mode number (m

^{−1})- A
_{p} Differential cross sectional area of piezoelectric layer (m

^{2})- A
_{s} Differential cross sectional area of structural layer (m

^{2})- z
_{p} Distance between neutral axis of beam and neutral axis of piezoelectric layer (M)

- z
_{s} Distance between neutral axis of beam and neutral axis of structural layer (M)

- D
Piezoelectric coupling coefficient (M V

^{−1})- S
Compliance (GPa

^{−1})- u
_{t,}u_{0} Tip displacement, tip displacement amplitude (M)

- Ψ
_{L} Bending mode function evaluated at x = L

- C
_{e} Filtering Capacitance (F)

- I
Current through device (A)

- F, F
_{0} Force, force amplitude (N)

- Θ
Phase between force and displacement

- V
_{c} Rectified voltage across load (V)

- R
Load resistance (Ω)

- R
Normalized resistance

*k*_{e}Electromechanical coupling coefficient

- Ζ
Mechanical damping ratio

- Ω
Frequency ratio

- P
Power (W)

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