A model of a multi-layer monolithic piezoelectric stack actuator is described in (<ahref="#citeproc_bib_item_2">Fleming 2010</a>) ([Notes]({{<relref"fleming10_nanop_system_with_force_feedb.md">}})).
Typical topology of mechanically amplified piezoelectric actuators are displayed in Figure [1](#figure--fig:ling16-topology-piezo-mechanism-types) (from (<ahref="#citeproc_bib_item_3">Ling et al. 2016</a>)).
{{<figuresrc="/ox-hugo/ling16_topology_piezo_mechanism_types.png"caption="<span class=\"figure-number\">Figure 1: </span>Topology of several types of compliant mechanisms">}}
> 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.
The electrical capacitance may limit the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure [2](#figure--fig:piezoelectric-capacitance-voltage-max)).
[Voltage Amplifier]({{< relref "voltage_amplifier.md" >}}) with high maximum output current should be used if either high bandwidth is wanted or piezoelectric stacks with high capacitance are to be used.
{{<figuresrc="/ox-hugo/piezoelectric_capacitance_voltage_max.png"caption="<span class=\"figure-number\">Figure 2: </span>Maximum sin-wave amplitude as a function of frequency for several piezoelectric capacitance">}}
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--fig:piezoelectric-mass-load)).
{{<figuresrc="/ox-hugo/piezoelectric_mass_load.png"caption="<span class=\"figure-number\">Figure 3: </span>Motion of a piezoelectric stack actuator under external constant force">}}
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](#figure--fig:piezoelectric-spring-load)):
{{<figuresrc="/ox-hugo/piezoelectric_spring_load.png"caption="<span class=\"figure-number\">Figure 4: </span>Motion of a piezoelectric stack actuator in contact with a stiff environment">}}
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.
{{<figuresrc="/ox-hugo/piezoelectric_force_displ_relation.png"caption="<span class=\"figure-number\">Figure 5: </span>Relation between the maximum force and displacement">}}
The stiffness of the piezoelectric stack varies a little bit whether it is open-circuited or short-circuited (<ahref="#citeproc_bib_item_4">Liu et al. 2007</a>).
This this experiment: <https://research.tdehaeze.xyz/test-bench-force-sensor/>.
Therefore, if the piezoelectric actuator is driven by a charge amplifier (i.e. high input impedance), the stiffness will be a little bit higher than if it is driven with a voltage amplifier (i.e. small input impedance).
Piezoelectric actuators can be driven either using a voltage to charge converter or a [Voltage Amplifier]({{< relref "voltage_amplifier.md" >}}).
Limitations of the electronics is discussed in [Design, modeling and control of nanopositioning systems]({{< relref "fleming14_desig_model_contr_nanop_system.md" >}}).
<divclass="csl-entry"><aid="citeproc_bib_item_1"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static & Dynamic Applications.” <i>Ferroelectrics</i> 351 (1): 3–14. doi:<ahref="https://doi.org/10.1080/00150190701351865">10.1080/00150190701351865</a>.</div>
<divclass="csl-entry"><aid="citeproc_bib_item_2"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 15 (3): 433–47. doi:<ahref="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
<divclass="csl-entry"><aid="citeproc_bib_item_3"></a>Ling, Mingxiang, Junyi Cao, Minghua Zeng, Jing Lin, and Daniel J Inman. 2016. “Enhanced Mathematical Modeling of the Displacement Amplification Ratio for Piezoelectric Compliant Mechanisms.” <i>Smart Materials and Structures</i> 25 (7): 075022. doi:<ahref="https://doi.org/10.1088/0964-1726/25/7/075022">10.1088/0964-1726/25/7/075022</a>.</div>
<divclass="csl-entry"><aid="citeproc_bib_item_4"></a>Liu, W. Q., Z. H. Feng, R. B. Liu, and J. Zhang. 2007. “The Influence of Preamplifiers on the Piezoelectric Sensor’s Dynamic Property.” <i>Review of Scientific Instruments</i> 78 (12): 125107. doi:<ahref="https://doi.org/10.1063/1.2825404">10.1063/1.2825404</a>.</div>
<divclass="csl-entry"><aid="citeproc_bib_item_5"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”</div>
