@@ -33,11 +33,11 @@ Let's suppose that the ADC is ideal and the only noise comes from the quantizati
Interestingly, the noise amplitude is uniformly distributed.
The quantization noise can take a value between \\(\pm q/2\\), and the probability density function is constant in this range (i.e., it’s a uniform distribution).
Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 <e<q/2\\)(Fig. [1](#org4bd731c)).
Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 <e<q/2\\)(Fig. [1](#figure--fig:probability-density-function-adc)).
{{<figuresrc="/ox-hugo/probability_density_function_adc.png"caption="Figure 1: Probability density function \\(p(e)\\) of the ADC error \\(e\\)">}}
{{<figuresrc="/ox-hugo/probability_density_function_adc.png"caption="<span class=\"figure-number\">Figure 1: </span>Probability density function \\(p(e)\\) of the ADC error \\(e\\)">}}
Now, we can calculate the time average power of the quantization noise as
@@ -59,13 +59,12 @@ Thus, the two-sided PSD (from \\(\frac{-f\_s}{2}\\) to \\(\frac{f\_s}{2}\\)), we
\end{equation}
<divclass="important">
<div></div>
Finally, the Power Spectral Density of the quantization noise of an ADC is equal to:
\begin{equation}
\begin{aligned}
\Gamma &= \frac{q^2}{12 f\_s} \\\\\\
\Gamma &= \frac{q^2}{12 f\_s} \\\\
&= \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 f\_s} \text{ in } \left[ \frac{V^2}{Hz} \right]
\end{aligned}
\end{equation}
@@ -73,7 +72,6 @@ Finally, the Power Spectral Density of the quantization noise of an ADC is equal
</div>
<divclass="exampl">
<div></div>
Let's take a 18bits ADC with a range of +/-10V and a sample frequency of 10kHz.
@@ -108,6 +108,24 @@ Which is much more efficient that the single stage decimation.
</div>
There are two **practical issues** to consider for two-stage decimation:
- First, if the dual-filter system is required to have a pass-band peak-peak ripple of \\(R\\) dB, then both filters must be designed to have a pass-band peak-peak ripple of no greater than \\(R/2\\) dB.
- Second, the number of multiplications needed to compute each \\(x\_{\text{new}}(m)\\) output sample is much larger than \\(N\_\text{total}\\) because we must compute so many \\(\text{LPF}\_1\\) and \\(\text{LPF}\_2\\) output samples destined to be discarded.
In order to cope with the second issue, an efficient decimation filter implementation scheme called _polyphase decomposition_ can be used.
<summary>The advantages of two stage decimation, over single-stage decimation are:
<ulclass="org-ul">
<li>an overall reduction in computation workload</li>
<li>reduced signal and filter coefficient data storage</li>
<li>simpler filter designs</li>
<li>a decrease in the ill effects of finite binary-work-length filter coefficients</li>
</ul>
These advantages become more pronounced as the overall desired decimation factor \(M\) becomes larger.</summary>
## Self-Sensing for perfect collocation {#self-sensing-for-perfect-collocation}
This can be done with a [Voice Coil Actuator]({{< relref "voice_coil_actuators.md" >}}) (see <verma20_perfec_colloc_using_self_sensin_elect_actuat>) or with a [Piezoelectric Actuator]({{< relref "piezoelectric_actuators.md" >}}) (see <jansen19_activ_dampin_dynam_struc_using>).
| [Thorlabs](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup%5Fid=8700) | USA |
| [Thorlabs](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=8700) | USA |
| [PiezoDrive](https://www.piezodrive.com/actuators/) | Australia |
| [Mechano Transformer](http://www.mechano-transformer.com/en/products/10.html) | Japan |
| [CoreMorrow](http://www.coremorrow.com/en/pro-9-1.html) | China |
@@ -33,7 +33,7 @@ Tags
### Model {#model}
A model of a multi-layer monolithic piezoelectric stack actuator is described in ([Fleming 2010](#orgd563065)) ([Notes]({{<relref "fleming10_nanop_system_with_force_feedb.md#" >}})).
A model of a multi-layer monolithic piezoelectric stack actuator is described in <fleming10_nanop_system_with_force_feedb> ([Notes]({{<relref "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.
@@ -57,25 +57,25 @@ Some manufacturers propose "raw" plate actuators that can be used as actuator /
{{<figuresrc="/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, \& 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>">}}
{{<figuresrc="/ox-hugo/ling16_topology_piezo_mechanism_types.png"caption="<span class=\"figure-number\">Figure 1: </span>Topology of several types of compliant mechanisms <ling16_enhan_mathem_model_displ_amplif>">}}
The electrical capacitance may limit the maximum voltage that can be used to drive the piezoelectric actuator as a function of frequency (Figure [2](#orgca6870e)).
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.
[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="Figure 2: Maximum sin-wave amplitude as a function of frequency for several piezoelectric capacitance">}}
{{<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">}}
## 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](#orge05f5e6)).
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)).
<aid="orge05f5e6"></a>
<aid="figure--fig:piezoelectric-mass-load"></a>
{{<figuresrc="/ox-hugo/piezoelectric_mass_load.png"caption="Figure 3: Motion of a piezoelectric stack actuator under external constant force">}}
{{<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">}}
## 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](#orgfcd374f)):
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}
\end{equation}
<aid="orgfcd374f"></a>
<aid="figure--fig:piezoelectric-spring-load"></a>
{{<figuresrc="/ox-hugo/piezoelectric_spring_load.png"caption="Figure 4: Motion of a piezoelectric stack actuator in contact with a stiff environment">}}
{{<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">}}
For piezo actuators, force and displacement are inversely related (Figure [5](#orgada6c4c)).
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.
{{<figuresrc="/ox-hugo/piezoelectric_force_displ_relation.png"caption="Figure 5: Relation between the maximum force and displacement">}}
{{<figuresrc="/ox-hugo/piezoelectric_force_displ_relation.png"caption="<span class=\"figure-number\">Figure 5: </span>Relation between the maximum force and displacement">}}
## Driving Electronics {#driving-electronics}
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#" >}}).
## Bibliography {#bibliography}
<aid="orgb463c4c"></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>.
<aid="orgd563065"></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>.
<aid="org2bf81f0"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”
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" >}}).
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