# Piezo Stack

Model electrical and force characteristics of piezoelectric stacked actuator

## Library

Translational Actuators

## Description

The Piezo Stack block represents the electrical and force characteristics of a piezoelectric stacked actuator using the following equations:

`$\begin{array}{l}S={s}^{E}T+{d}^{\text{'}}E\\ D=dT+{\epsilon }^{T}E\end{array}$`

where

• S is the strain tensor.

• T is the stress tensor.

• E is the electric field vector.

• D is the electric displacement vector.

• sE is the elastic compliance matrix when subjected to a constant electric field.

• d is the piezoelectric constant matrix.

• εT is the permittivity measured at a constant stress.

 Note:   The block models one-dimensional lumped parameter behavior, so S, T, E and D are all scalar values.

You can specify the block parameters that determine static force using either datasheet parameters or material properties, as determined by the value of the Parameterization parameter on the Static Force tab of the block dialog box.

The Dynamic Forces tab of the block dialog box lets you include optional effective mass and mechanical damping effects.

• If you specify a nonzero value for the Effective mass parameter or a finite value for the Resonant frequency at constant field parameter, the block attaches a lumped mass to the mechanical R port. When you specify a finite resonant frequency, the block calculates the effective mass to achieve the correct resonant frequency.

• If you specify a nonzero value for the Damping parameter or a finite value for the Mechanical quality factor parameter, the block adds a damping term across the R and C mechanical ports. When you specify a mechanical quality factor, Qm, the block calculates the damping from this parameter value as$\sqrt{Mk}/{Q}_{m}$, where k is the short-circuit device stiffness, or equivalently the stiffness at constant field.

A positive voltage across the electrical + to – ports creates a positive displacement acting from the mechanical C to R ports.

## Basic Assumptions and Limitations

The model does not include hysteresis effects.

## Dialog Box and Parameters

### Static Force Tab

Parameterization

Select one of the following methods for static force parameterization:

• `Specify from a datasheet` — Provide datasheet parameters that the block converts to static force values. This is the default method.

• `Specify material properties` — Provide material properties that the block converts to static force values.

Stack area

Cross-sectional area of the stack. The default value is `100` mm2.

Stack length

Stack length when no load and no electrical potential are applied. This parameter is only visible when you select ```Specify from a datasheet``` for the Parameterization parameter. The default value is `36` mm.

No-load displacement at V0 volts

Unconstrained displacement of the stack when a voltage of V0 volts is applied. This parameter is only visible when you select ```Specify from a datasheet``` for the Parameterization parameter. The default value is `0.038` mm.

Blocking force at V0 volts

Force the stack produces when a voltage of V0 volts is applied and the stack is physically prevented from expanding. This parameter is only visible when you select `Specify from a datasheet` for the Parameterization parameter. The default value is `3.8e+03` N.

Test voltage V0

Voltage used to determine the no-load displacement and blocking force. This parameter is only visible when you select ```Specify from a datasheet``` for the Parameterization parameter. The default value is `120` V.

Capacitance

This parameter is only visible when you select ```Specify from a datasheet``` for the Parameterization parameter. The default value is `13` uF.

Piezo layer thickness

Thickness of each layer in the piezo stack. This parameter is only visible when you select `Specify material properties` for the Parameterization parameter. The default value is `0.3` mm.

Number of layers

Number of layers in the piezo stack. This parameter is only visible when you select `Specify material properties` for the Parameterization parameter. The default value is `50`.

Piezoelectric charge constant

Mechanical strain per unit electric field applied. This parameter is only visible when you select `Specify material properties` for the Parameterization parameter. The default value is `5e-10` m/V.

Dielectric constant

Permittivity or dielectric displacement per unit electric field measured at constant stress. This parameter is only visible when you select `Specify material properties` for the Parameterization parameter. The default value is `2.124e-08` F/m.

Elastic compliance

Strain produced in a piezoelectric material per unit of stress applied. This parameter is only visible when you select ```Specify material properties``` for the Parameterization parameter. The default value is `1.9e-11` m2/N.

### Dynamic Forces Tab

Parameterization

Select one of the following methods for dynamic force parameterization:

• `Specify from a datasheet` — Provide datasheet parameters that the block converts to dynamic force values. This is the default method.

• `Specify material properties` — Provide material properties that the block converts to dynamic force values.

Resonant frequency at constant field

Frequency at which the actuator naturally resonates if mechanically perturbed with the electrical ports shorted. This parameter is only visible when you select `Specify from a datasheet` for the Parameterization parameter. The default value is `Inf` kHz.

Mechanical quality factor

Factor that affects the damping across the R and C mechanical ports. This parameter is only visible when you select ```Specify from a datasheet``` for the Parameterization parameter. The default value is `Inf`.

Damping

Translational damping term. This parameter is only visible when you select `Specify material properties` for the Parameterization parameter. The default value is `0` N/(m/s).

Effective mass

Mass that approximates the distributed dynamics of the device and causes the stack to resonate at the correct frequency when attached to the mechanical R port. This mass is usually about one third of the actual stack mass. This parameter is only visible when you select ```Specify material properties``` for the Parameterization parameter. The default value is `0` g.

### Initial Conditions Tab

Initial stack deflection

Stack deflection at time zero. If you have an external Ideal Translational Motion Sensor block attached across the Piezo Stack block, you must use the same initial deflection parameter for both blocks. The default value is `0` mm.

Initial voltage

Stack voltage at time zero. The default value is `0` V.

## Ports

The block has the following ports:

`+`

Positive electrical port

`-`

Negative electrical port

`C`

Mechanical translational conserving port

`R`

Mechanical translational conserving port