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content/book/fleming14_desig_model_contr_nanop_system.md
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@ -10,7 +10,7 @@ Tags
: [Piezoelectric Actuators](piezoelectric_actuators.md), [Flexible Joints](flexible_joints.md) : [Piezoelectric Actuators](piezoelectric_actuators.md), [Flexible Joints](flexible_joints.md)
Reference Reference
: ([Fleming and Leang 2014](#orgc8028e0)) : ([Fleming and Leang 2014](#org3a2500f))
Author(s) Author(s)
: Fleming, A. J., & Leang, K. K. : Fleming, A. J., & Leang, K. K.
@ -783,11 +783,11 @@ Year
### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models} ### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
<a id="orgded1d91"></a> <a id="org1b7a832"></a>
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}} {{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
Consider the electrical circuit shown in Figure [1](#orgded1d91) where a voltage source is connected to a piezoelectric actuator. Consider the electrical circuit shown in Figure [1](#org1b7a832) where a voltage source is connected to a piezoelectric actuator.
The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\). The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively. The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
@ -911,4 +911,4 @@ The bandwidth limitations of standard piezoelectric drives were identified as:
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgc8028e0"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>. <a id="org3a2500f"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.

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content/zettels/piezoelectric_actuators.md
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Tags Tags
: [Actuators]({{< relref "actuators" >}}), [Voltage Amplifier]({{< relref "voltage_amplifier" >}}) : [Actuators](actuators.md), [Voltage Amplifier](voltage_amplifier.md)
## Piezoelectric Stack Actuators {#piezoelectric-stack-actuators} ## Piezoelectric Stack Actuators {#piezoelectric-stack-actuators}
@ -32,7 +32,7 @@ Tags
### Model {#model} ### Model {#model}
A model of a multi-layer monolithic piezoelectric stack actuator is described in ([Fleming 2010](#org4089875)) ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})). A model of a multi-layer monolithic piezoelectric stack actuator is described in ([Fleming 2010](#orgc916f93)) ([Notes](fleming10_nanop_system_with_force_feedb.md)).
Basically, it can be represented by a spring \\(k\_a\\) with the force source \\(F\_a\\) in parallel. Basically, it can be represented by a spring \\(k\_a\\) with the force source \\(F\_a\\) in parallel.
@ -56,14 +56,14 @@ Some manufacturers propose "raw" plate actuators that can be used as actuator /
## Mechanically Amplified Piezoelectric actuators {#mechanically-amplified-piezoelectric-actuators} ## Mechanically Amplified Piezoelectric actuators {#mechanically-amplified-piezoelectric-actuators}
The Amplified Piezo Actuators principle is presented in ([Claeyssen et al. 2007](#orge4dbf99)): The Amplified Piezo Actuators principle is presented in ([Claeyssen et al. 2007](#orgaaabf8d)):
> 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 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. > The flatter is the actuator, the higher is the amplification.
A model of an amplified piezoelectric actuator is described in ([Lucinskis and Mangeot 2016](#orga7e7177)). A model of an amplified piezoelectric actuator is described in ([Lucinskis and Mangeot 2016](#org8ca201e)).
<a id="org22709f8"></a> <a id="org5d92181"></a>
{{< figure src="/ox-hugo/ling16_topology_piezo_mechanism_types.png" caption="Figure 1: Topology of several types of compliant mechanisms <sup id=\"d9e8b33774f1e65d16bd79114db8ac64\"><a 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).\">ling16_enhan_mathem_model_displ_amplif</a></sup>" >}} {{< figure src="/ox-hugo/ling16_topology_piezo_mechanism_types.png" caption="Figure 1: Topology of several types of compliant mechanisms <sup id=\"d9e8b33774f1e65d16bd79114db8ac64\"><a 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).\">ling16_enhan_mathem_model_displ_amplif</a></sup>" >}}
@ -141,9 +141,9 @@ with:
### Resolution {#resolution} ### Resolution {#resolution}
The resolution is limited by the noise in the [Voltage Amplifier]({{< relref "voltage_amplifier" >}}). The resolution is limited by the noise in the [Voltage Amplifier](voltage_amplifier.md).
Typical [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}}) of voltage amplifiers is \\(100dB = 10^{5}\\). Typical [Signal to Noise Ratio](signal_to_noise_ratio.md) of voltage amplifiers is \\(100dB = 10^{5}\\).
Thus, for a piezoelectric stack with a displacement \\(L\\), the resolution will be Thus, for a piezoelectric stack with a displacement \\(L\\), the resolution will be
\begin{equation} \begin{equation}
@ -155,58 +155,58 @@ For a piezoelectric stack with a displacement of \\(100\,[\mu m]\\), the resolut
### Electrical Capacitance {#electrical-capacitance} ### Electrical Capacitance {#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](#org38927da)). The electrical capacitance may limit the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure [2](#org2c60a2d)).
This is due to the fact that voltage amplifier has a limitation on the deliverable current. 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. [Voltage Amplifier](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.
<a id="org38927da"></a> <a id="org2c60a2d"></a>
{{< 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" >}} {{< 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 {#piezoelectric-actuator-experiencing-a-mass-load} ## Piezoelectric actuator experiencing a mass load {#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](#org35604e1)). 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](#org7af4476)).
<a id="org35604e1"></a> <a id="org7af4476"></a>
{{< figure src="/ox-hugo/piezoelectric_mass_load.png" caption="Figure 3: Motion of a piezoelectric stack actuator under external constant force" >}} {{< 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 {#piezoelectric-actuator-in-contact-with-a-spring-load} ## Piezoelectric actuator in contact with a spring load {#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](#org2f55c26)): 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](#org97370ea)):
\begin{equation} \begin{equation}
\Delta L = \Delta L\_f \frac{k\_p}{k\_p + k\_e} \Delta L = \Delta L\_f \frac{k\_p}{k\_p + k\_e}
\end{equation} \end{equation}
<a id="org2f55c26"></a> <a id="org97370ea"></a>
{{< figure src="/ox-hugo/piezoelectric_spring_load.png" caption="Figure 4: Motion of a piezoelectric stack actuator in contact with a stiff environment" >}} {{< 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](#orgf384614)). For piezo actuators, force and displacement are inversely related (Figure [5](#org8c01425)).
Maximum, or blocked, force (\\(F\_b\\)) occurs when there is no displacement. 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. 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. 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.
<a id="orgf384614"></a> <a id="org8c01425"></a>
{{< figure src="/ox-hugo/piezoelectric_force_displ_relation.png" caption="Figure 5: Relation between the maximum force and displacement" >}} {{< figure src="/ox-hugo/piezoelectric_force_displ_relation.png" caption="Figure 5: Relation between the maximum force and displacement" >}}
## Driving Electronics {#driving-electronics} ## Driving Electronics {#driving-electronics}
Piezoelectric actuators can be driven either using a voltage to charge converter or a [Voltage Amplifier]({{< relref "voltage_amplifier" >}}). Piezoelectric actuators can be driven either using a voltage to charge converter or a [Voltage Amplifier](voltage_amplifier.md).
Limitations of the electronics is discussed in the book [Design, modeling and control of nanopositioning systems]({{< relref "fleming14_desig_model_contr_nanop_system#electrical-considerations" >}}). Limitations of the electronics is discussed in [Design, modeling and control of nanopositioning systems](fleming14_desig_model_contr_nanop_system.md).
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orge4dbf99"></a>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>. <a id="orgaaabf8d"></a>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>.
<a id="org4089875"></a>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>. <a id="orgc916f93"></a>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>.
<a id="orga7e7177"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.” <a id="org8ca201e"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”