PIEZO STACKS – PHYSICS

Actuators & Sensors

INTRODUCTION

Designing an actuator that utilizes a piezo stack as actuation principles is not trivial. To aid in this this sheet elaborates on the parameters that are of interest for such a design. The behavior a piezo stack can be summarized with the following properties:

  • Geometrical
  • Material
  • Mechanical
  • Electrical

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Geometric properties

A=H\ast B[m^2]

Cross section area

d_s[m]

Layer thickness (typical 60-500 µm)

n=\frac {L}{d_s} [-]

Number of layers

Material Properties

\rho =7.8e3\frac{kg}{m^3}

Density

E=36e9\frac {N}{m^2}

Young’s modulus

\nu =0.34

Poisson ratio

\xi =0.1

Damping constant (typical value = 1750)

-271^oC<T<150^oC

Operating temperature

\varepsilon _{33}=300-4000

Dielectric constant (typical value = 1750)

d_{33}=4e-10\frac {m}{V}

Charge constant

HC=350\frac {J}{kg\ast K}

Spec. heat capacity

TC=1.1\frac {W}{mK}

Spec. thermal conductivity

-20V < U < 120V

Typical voltage

Mechanics

\Delta L_{max}=\frac {L}{1000}

Maximum displacement

C_{axial}=\frac {EA}{L}

Axial stiffness

D=2\xi \sqrt{C_{axial}m}

Damping

f_0=\frac {1}{2\pi }\sqrt{\frac {C_{axial}}{m}

Resonance frequency

t_{min}=\frac {1}{3f_0}

Minimum rise time

\varphi =2\arctan (\frac {f}{f_0})

Phase lag

F_{blocking}=\Delta l_{max}\ast C_{axial}

Blocking force

\sigma _{dynamic}=15MPa

Preload for dynamic use

\sigma _{static}=30 MPa

Preload for static use

4-20%

Hysteresis

{\Delta}x_{creep}(t)=x\ast 0.01\ast \log (\frac {t}{0.1})

Creep @ t [s]

F_{pre tension}=\frac {1}{2}F_{blocking}

Pre tension force (matching push-pull force)

Geometric characteristics of a Piezo stack.

Electronics

x=d_{33}\ast L\ast \frac {U}{d_s}

Displacement @ U [V]

 \varepsilon _0=8.9e-12\frac {F}{m}

Permittivity of free space

C_S=n\ast \varepsilon _0\ast \varepsilon _{33}\ast \frac {A}{d_s}

Small signal capacitance (typical for U < 100 V)

C_L=1.7\ast C_S

Large signal capacitance (typical for U > 100 V)

P=\frac {1}{2}\ast f\ast C\ast U^2

Average Polarization Power

E=\frac {1}{2}\ast C\ast U^2

Polarization energy

Not generic for all frequency ranges, just for indication:

\tan \delta _U=0.015+0.016\sqrt{\frac {U}{[V]}}

Loss factor @ U [V]

\tan \delta _T=0.28+0.17\ast \ln (\frac {T}{[K]})

Loss factor @ T [K]

tan\delta =\tan \delta _U+\tan \delta _T

Loss factor

P_{heat}=P\ast tan\delta

Heat generation

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