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content/zettels/piezoelectric_actuators.md
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@@ -64,7 +64,7 @@ The Amplified Piezo Actuators principle is presented in (<a href="#citeproc_bib_
A model of an amplified piezoelectric actuator is described in (<a href="#citeproc_bib_item_5">Lucinskis and Mangeot 2016</a>).
Typical topology of mechanically amplified piezoelectric actuators are displayed in Figure [1](#figure--fig:ling16-topology-piezo-mechanism-types) (from (<a href="#citeproc_bib_item_3">Ling et al. 2016</a>)).
Typical topology of mechanically amplified piezoelectric actuators are displayed in [Figure 1](#figure--fig:ling16-topology-piezo-mechanism-types) (from (<a href="#citeproc_bib_item_3">Ling et al. 2016</a>)).
<a id="figure--fig:ling16-topology-piezo-mechanism-types"></a>
@@ -158,28 +158,28 @@ For a piezoelectric stack with a displacement of \\(100\\,[\mu m]\\), the resolu
### 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](#figure--fig:piezoelectric-capacitance-voltage-max)).
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)).
This is due to the fact that voltage amplifier has a limitation on the deliverable current.
[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.
<a id="figure--fig:piezoelectric-capacitance-voltage-max"></a>
{{< figure src="/ox-hugo/piezoelectric_capacitance_voltage_max.png" caption="<span class=\"figure-number\">Figure 1: </span>Maximum sin-wave amplitude as a function of frequency for several piezoelectric capacitance" >}}
{{< figure src="/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" >}}
## 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 [1](#figure--fig:piezoelectric-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--fig:piezoelectric-mass-load)).
<a id="figure--fig:piezoelectric-mass-load"></a>
{{< figure src="/ox-hugo/piezoelectric_mass_load.png" caption="<span class=\"figure-number\">Figure 1: </span>Motion of a piezoelectric stack actuator under external constant force" >}}
{{< figure src="/ox-hugo/piezoelectric_mass_load.png" caption="<span class=\"figure-number\">Figure 3: </span>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}
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 [1](#figure--fig:piezoelectric-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](#figure--fig:piezoelectric-spring-load)):
\begin{equation}
\Delta L = \Delta L\_f \frac{k\_p}{k\_p + k\_e}
@@ -187,16 +187,16 @@ Then the piezoelectric actuator is in contact with a spring load \\(k\_e\\), its
<a id="figure--fig:piezoelectric-spring-load"></a>
{{< figure src="/ox-hugo/piezoelectric_spring_load.png" caption="<span class=\"figure-number\">Figure 1: </span>Motion of a piezoelectric stack actuator in contact with a stiff environment" >}}
{{< figure src="/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" >}}
For piezo actuators, force and displacement are inversely related (Figure [1](#figure--fig:piezoelectric-force-displ-relation)).
For piezo actuators, force and displacement are inversely related ([Figure 5](#figure--fig:piezoelectric-force-displ-relation)).
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.
<a id="figure--fig:piezoelectric-force-displ-relation"></a>
{{< figure src="/ox-hugo/piezoelectric_force_displ_relation.png" caption="<span class=\"figure-number\">Figure 1: </span>Relation between the maximum force and displacement" >}}
{{< figure src="/ox-hugo/piezoelectric_force_displ_relation.png" caption="<span class=\"figure-number\">Figure 5: </span>Relation between the maximum force and displacement" >}}
## Piezoelectric stiffness - Electrical Boundaries {#piezoelectric-stiffness-electrical-boundaries}