digital-brain/content/zettels/piezoelectric_actuators.md

+++ title = "Piezoelectric Actuators" author = ["Thomas Dehaeze"] draft = false +++

Tags

[Actuators]({{< relref "actuators" >}})

Piezoelectric Stack Actuators

Manufacturers

Manufacturers	Links	Country
Cedrat	link	France
PI	link	USA
Piezo System	link	Germany
Noliac	link	Denmark
Thorlabs	link	USA
PiezoDrive	link	Australia
Mechano Transformer	link	Japan
CoreMorrow	link	China
PiezoData	link	China
Queensgate	link	UK
Matsusada Precision	link	Japan

Model

A model of a multi-layer monolithic piezoelectric stack actuator is described in (Fleming 2010) ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).

Basically, it can be represented by a spring \(k_a\) with the force source \(F_a\) in parallel.

The relation between the applied voltage \(V_a\) to the generated force \(F_a\) is: \[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \] with:

• \(d_{33}\) is the piezoelectric strain constant [m/V]
• \(n\) is the number of layers
• \(k_a\) is the actuator stiffness [N/m]

Mechanically Amplified Piezoelectric actuators

The Amplified Piezo Actuators principle is presented in (Claeyssen et al. 2007):

The displacement amplification effect is related in a first approximation to the ratio of the shell long axis length to the short axis height. The flatter is the actuator, the higher is the amplification.

A model of an amplified piezoelectric actuator is described in (Lucinskis and Mangeot 2016).

{{< figure src="/ox-hugo/ling16_topology_piezo_mechanism_types.png" caption="Figure 1: Topology of several types of compliant mechanisms <sup id="d9e8b33774f1e65d16bd79114db8ac64"><a class="reference-link" href="#ling16_enhan_mathem_model_displ_amplif" title="Mingxiang Ling, Junyi Cao, Minghua Zeng, Jing Lin, &amp; Daniel J Inman, Enhanced Mathematical Modeling of the Displacement Amplification Ratio for Piezoelectric Compliant Mechanisms, {Smart Materials and Structures}, v(7), 075022 (2016).">(Mingxiang Ling {\it et al.}, 2016)" >}}

Manufacturers	Links	Country
Cedrat	link	France
PiezoDrive	link	Australia
Dynamic-Structures	link	USA
Thorlabs	link	USA
Noliac	link	Denmark
Mechano Transformer	link, link, link	Japan
CoreMorrow	link	China
PiezoData	link	China

Specifications

Typical Specifications

Typical specifications of piezoelectric stack actuators are usually in terms of:

• Displacement/ Travel range \([\mu m]\)
• Blocked force \([N]\)
• Stiffness \([N/\mu m]\)
• Resolution \([nm]\)
• Length \([mm]\)
• Electrical Capacitance \([nF]\)

Displacement and Length

The maximum displacement specified is the displacement of the actuator when the maximum voltage is applied without any load.

Typical maximum strain of Piezoelectric Stack Actuators is \(0.1%\). The free displacement \(\Delta L_{f}\) is then related to the length \(L\) of piezoelectric stack by:

\begin{equation} \Delta L_f \approx \frac{L}{1000} \end{equation}

A “free” actuator — one that experiences no resistance to movement — will produce its maximum displacement, often referred to as “free stroke,” and generate zero force.

Note that this maximum displacement is only attainable at DC. For dynamical applications, the electrical capacitance of the piezoelectric actuator is an important factor (see bellow).

Blocked Force

The blocked force \(F_b\) is measured by first applying the maximum voltage to the piezoelectric stack without any load. Thus, the piezoelectric stack experiences its maximum displacement.

A force is then applied to return the actuator to its original length. This force is measured and recorded as the blocking force.

The blocking force is also the maximum force that can produce the piezoelectric stack in contact with an infinitely stiff environment.

When an actuator is blocked from moving, it will produce its maximum force, which is referred to as the blocked, or blocking, force.

Stiffness

The stiffness of the actuator is the ratio of the blocking force to the free stroke:

\begin{equation} k_p = \frac{F_b}{\Delta L_f} \end{equation}

with:

• \(k_p\): stiffness of the piezo actuator
• \(F_b\): blocking force
• \(\Delta L_f\): free stroke

Resolution

The resolution is limited by the noise in the [Voltage Amplifier]({{< relref "voltage_amplifier" >}}).

Typical [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}}) of voltage amplifiers is \(100dB = 10^{5}\). Thus, for a piezoelectric stack with a displacement \(L\), the resolution will be

\begin{equation} r \approx \frac{L}{10^5} \end{equation}

For a piezoelectric stack with a displacement of \(100,[\mu m]\), the resolution will be \(\approx 1,[nm]\).

Electrical Capacitance

The electrical capacitance may limit the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure 2). This is due to the fact that voltage amplifier has a limitation on the deliverable current.

[Voltage Amplifier]({{< relref "voltage_amplifier" >}}) with high maximum output current should be used if either high bandwidth is wanted or piezoelectric stacks with high capacitance are to be used.

{{< figure src="/ox-hugo/piezoelectric_capacitance_voltage_max.png" caption="Figure 2: Maximum sin-wave amplitude as a function of frequency for several piezoelectric capacitance" >}}

Piezoelectric actuator experiencing a mass load

When the piezoelectric actuator is supporting a payload, it will experience a static deflection due to its finite stiffness \(\Delta l_n = \frac{mg}{k_p}\), but its stroke will remain unchanged (Figure 3).

{{< figure src="/ox-hugo/piezoelectric_mass_load.png" caption="Figure 3: Motion of a piezoelectric stack actuator under external constant force" >}}

Piezoelectric actuator in contact with a spring load

Then the piezoelectric actuator is in contact with a spring load \(k_e\), its maximum stroke \(\Delta L\) is less than its free stroke \(\Delta L_f\) (Figure 4):

\begin{equation} \Delta L = \Delta L_f \frac{k_p}{k_p + k_e} \end{equation}

{{< figure src="/ox-hugo/piezoelectric_spring_load.png" caption="Figure 4: Motion of a piezoelectric stack actuator in contact with a stiff environment" >}}

For piezo actuators, force and displacement are inversely related (Figure 5). Maximum, or blocked, force (\(F_b\)) occurs when there is no displacement. Likewise, at maximum displacement, or free stroke, (\(\Delta L_f\)) no force is generated. When an external load is applied, the stiffness of the load (\(k_e\)) determines the displacement (\(\Delta L_A\)) and force (\(\Delta F_A\)) that can be produced.

{{< figure src="/ox-hugo/piezoelectric_force_displ_relation.png" caption="Figure 5: Relation between the maximum force and displacement" >}}

Bibliography

Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static & Dynamic Applications.” Ferroelectrics 351 (1):3–14. https://doi.org/10.1080/00150190701351865.

Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” IEEE/ASME Transactions on Mechatronics 15 (3):433–47. https://doi.org/10.1109/tmech.2009.2028422.

Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”

Backlinks

• [Actuators]({{< relref "actuators" >}})
• [Voltage Amplifier]({{< relref "voltage_amplifier" >}})