# 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 of a piezo stack can be summarized with the following properties:

• Geometrical
• Material
• Mechanical
• Electrical

#### Geometric properties

$A=H\cdot B\left[m^2\right]$

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
(estimate)

$-{271}^oC<T<{150}^oC$

Operating temperature

$\varepsilon_{33}=300-4000$

Dielectric constant (typical value = 1750)

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

Charge constant

$HC=350\frac{J}{kg\cdot K}$

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

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

Phase lag

$F_{blocking}=\Delta l_{max}\cdot C_{axial}$

Blocking force

$\sigma_{dynamic}=15\ MPa$

$\sigma_{static}=30\ MPa$

$4-20 \%$

Hysteresis

$\Delta x_{creep}\left(t\right)=x\cdot0.01\cdot\log{\left(\frac{t}{0.1}\right)}$

Creep @ t [s]

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

Pre tension force (matching push-pull force)

#### Electronics

$x=d_{33}\cdot L\cdot\frac{U}{d_s}$

Displacement @ U [V]

$\varepsilon_0=8.9e-12\frac{F}{m}$

Permittivity of free space

$C_S=n\cdot\varepsilon_0\cdot\varepsilon_{33}\cdot\frac{A}{d_s}$

Small signal capacitance (typical for U < 100 V)

$C_L=1.7\cdot C_S$

Large signal capacitance (typical for U > 100 V)

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

Average Polarization Power

$E=\frac{1}{2}\cdot C\cdot U^2$

Polarization energy

Not generic for all frequency ranges, just for indication:

$\tan{\delta_U}=0.015+0.016\sqrt{\frac{U}{\left[V\right]}}$

Loss factor @ U [V]

$\tan{\delta_T}=0.28+0.17\cdot\ln{\left(\frac{T}{\left[K\right]}\right)}$

Loss factor @ T [K]

$\tan{\delta}=\tan{\delta_U}+\tan{\delta_T}$

Heat generation

$P_{heat}=P\cdot\tan{\delta}$

Heat generation

## Tech Support

Please submit a message and we will come back to you on short notice.