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content/zettels/analog_to_digital_converters.md
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title = "Analog to Digital Converters" title = "Analog to Digital Converters"
author = ["Thomas Dehaeze"] author = ["Dehaeze Thomas"]
keywords = ["electronics"] keywords = ["electronics"]
draft = false draft = false
category = "equipment" category = "equipment"
+++ +++
Tags Tags
: [Electronics]({{<relref "electronics.md#" >}}) : [Electronics]({{< relref "electronics.md" >}})
## Types of Analog to Digital Converters {#types-of-analog-to-digital-converters} ## Types of Analog to Digital Converters {#types-of-analog-to-digital-converters}
<https://dewesoft.com/daq/types-of-adc-converters> <https://dewesoft.com/daq/types-of-adc-converters>
- Delta Sigma ([Baker 2011](#orgbdb61af)) - Delta Sigma <baker11_how_delta_sigma_adcs_work_part>
- Successive Approximation - Successive Approximation
@ -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. 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). 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)).
<a id="org4bd731c"></a> <a id="figure--fig:probability-density-function-adc"></a>
{{< figure src="/ox-hugo/probability_density_function_adc.png" caption="Figure 1: Probability density function \\(p(e)\\) of the ADC error \\(e\\)" >}} {{< figure src="/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 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} \end{equation}
<div class="important"> <div class="important">
<div></div>
Finally, the Power Spectral Density of the quantization noise of an ADC is equal to: Finally, the Power Spectral Density of the quantization noise of an ADC is equal to:
\begin{equation} \begin{equation}
\begin{aligned} \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] &= \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 f\_s} \text{ in } \left[ \frac{V^2}{Hz} \right]
\end{aligned} \end{aligned}
\end{equation} \end{equation}
@ -73,7 +72,6 @@ Finally, the Power Spectral Density of the quantization noise of an ADC is equal
</div> </div>
<div class="exampl"> <div class="exampl">
<div></div>
Let's take a 18bits ADC with a range of +/-10V and a sample frequency of 10kHz. Let's take a 18bits ADC with a range of +/-10V and a sample frequency of 10kHz.
@ -87,7 +85,4 @@ The quantization is:
{{< youtube b9lxtOJj3yU >}} {{< youtube b9lxtOJj3yU >}}
## Oversampling {#oversampling}
## Bibliography {#bibliography}
<a id="orgbdb61af"></a>Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” _Analog Applications_ 7.

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content/zettels/decimation.md
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@ -108,6 +108,24 @@ Which is much more efficient that the single stage decimation.
</div> </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:
<ul class="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>
## References: {#references} ## References: {#references}

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content/zettels/integral_force_feedback.md
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title = "Integral Force Feedback" title = "Integral Force Feedback"
author = ["Thomas Dehaeze"] author = ["Dehaeze Thomas"]
draft = false draft = false
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Tags Tags
: [Active Damping]({{< relref "active_damping" >}}) : [Active Damping]({{< relref "active_damping.md" >}}), [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}), [Force Sensors]({{< relref "force_sensors.md" >}})
## 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>).

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content/zettels/piezoelectric_actuators.md
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title = "Piezoelectric Actuators" title = "Piezoelectric Actuators"
author = ["Thomas Dehaeze"] author = ["Dehaeze Thomas"]
draft = false draft = false
category = "equipment" category = "equipment"
+++ +++
Tags Tags
: [Actuators]({{<relref "actuators.md#" >}}), [Voltage Amplifier]({{<relref "voltage_amplifier.md#" >}}) : [Actuators]({{< relref "actuators.md" >}}), [Voltage Amplifier]({{< relref "voltage_amplifier.md" >}})
## Piezoelectric Stack Actuators {#piezoelectric-stack-actuators} ## Piezoelectric Stack Actuators {#piezoelectric-stack-actuators}
@ -18,9 +18,9 @@ Tags
|----------------------------------------------------------------------------------------------------------------------|-----------| |----------------------------------------------------------------------------------------------------------------------|-----------|
| [Cedrat](http://www.cedrat-technologies.com/) | France | | [Cedrat](http://www.cedrat-technologies.com/) | France |
| [PI](https://www.physikinstrumente.com/en/) | USA | | [PI](https://www.physikinstrumente.com/en/) | USA |
| [Piezo System](https://www.piezosystem.com/products/piezo%5Factuators/stacktypeactuators/) | Germany | | [Piezo System](https://www.piezosystem.com/products/piezo_actuators/stacktypeactuators/) | Germany |
| [Noliac](http://www.noliac.com/products/actuators/plate-stacks/) | Denmark | | [Noliac](http://www.noliac.com/products/actuators/plate-stacks/) | Denmark |
| [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 | | [PiezoDrive](https://www.piezodrive.com/actuators/) | Australia |
| [Mechano Transformer](http://www.mechano-transformer.com/en/products/10.html) | Japan | | [Mechano Transformer](http://www.mechano-transformer.com/en/products/10.html) | Japan |
| [CoreMorrow](http://www.coremorrow.com/en/pro-9-1.html) | China | | [CoreMorrow](http://www.coremorrow.com/en/pro-9-1.html) | China |
@ -33,7 +33,7 @@ Tags
### Model {#model} ### 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. 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 /
## 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](#orgb463c4c)): The Amplified Piezo Actuators principle is presented in <claeyssen07_amplif_piezoel_actuat>:
> 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](#org2bf81f0)). A model of an amplified piezoelectric actuator is described in <lucinskis16_dynam_charac>.
<a id="org77a46eb"></a> <a id="figure--fig:ling16-topology-piezo-mechanism-types"></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="<span class=\"figure-number\">Figure 1: </span>Topology of several types of compliant mechanisms <ling16_enhan_mathem_model_displ_amplif>" >}}
| Manufacturers | Country | | Manufacturers | Country |
|----------------------------------------------------------------------------------------------------|-----------| |----------------------------------------------------------------------------------------------------|-----------|
| [Cedrat](https://www.cedrat-technologies.com/en/products/actuators/amplified-piezo-actuators.html) | France | | [Cedrat](https://www.cedrat-technologies.com/en/products/actuators/amplified-piezo-actuators.html) | France |
| [PiezoDrive](https://www.piezodrive.com/actuators/ap-series-amplified-piezoelectric-actuators/) | Australia | | [PiezoDrive](https://www.piezodrive.com/actuators/ap-series-amplified-piezoelectric-actuators/) | Australia |
| [Dynamic-Structures](https://www.dynamic-structures.com/category/piezo-actuators-stages) | USA | | [Dynamic-Structures](https://www.dynamic-structures.com/category/piezo-actuators-stages) | USA |
| [Thorlabs](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup%5Fid=8700) | USA | | [Thorlabs](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=8700) | USA |
| [Noliac](http://www.noliac.com/products/actuators/amplified-actuators/) | Denmark | | [Noliac](http://www.noliac.com/products/actuators/amplified-actuators/) | Denmark |
| [Mechano Transformer](http://www.mechano-transformer.com/en/products/01a%5Factuator%5F5.html) | Japan | | [Mechano Transformer](http://www.mechano-transformer.com/en/products/01a_actuator_5.html) | Japan |
| [CoreMorrow](http://www.coremorrow.com/en/pro-13-1.html) | China | | [CoreMorrow](http://www.coremorrow.com/en/pro-13-1.html) | China |
| [PiezoData](https://www.piezodata.com/piezoelectric-actuator-amplifier/) | China | | [PiezoData](https://www.piezodata.com/piezoelectric-actuator-amplifier/) | China |
@ -99,7 +99,7 @@ Typical specifications of piezoelectric stack actuators are usually in terms of:
The maximum displacement specified is the displacement of the actuator when the maximum voltage is applied without any load. 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\%\\). 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: The free displacement \\(\Delta L\_{f}\\) is then related to the length \\(L\\) of piezoelectric stack by:
\begin{equation} \begin{equation}
@ -142,72 +142,62 @@ with:
### Resolution {#resolution} ### Resolution {#resolution}
The resolution is limited by the noise in the [Voltage Amplifier]({{<relref "voltage_amplifier.md#" >}}). The resolution is limited by the noise in the [Voltage Amplifier]({{< relref "voltage_amplifier.md" >}}).
Typical [Signal to Noise Ratio]({{<relref "signal_to_noise_ratio.md#" >}}) of voltage amplifiers is \\(100dB = 10^{5}\\). Typical [Signal to Noise Ratio]({{< relref "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}
r \approx \frac{L}{10^5} r \approx \frac{L}{10^5}
\end{equation} \end{equation}
For a piezoelectric stack with a displacement of \\(100\,[\mu m]\\), the resolution will be \\(\approx 1\,[nm]\\). For a piezoelectric stack with a displacement of \\(100\\,[\mu m]\\), the resolution will be \\(\approx 1\\,[nm]\\).
### 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](#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. 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.
<a id="orgca6870e"></a> <a id="figure--fig:piezoelectric-capacitance-voltage-max"></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="<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} ## 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)).
<a id="orge05f5e6"></a> <a id="figure--fig:piezoelectric-mass-load"></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="<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} ## 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} \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="orgfcd374f"></a> <a id="figure--fig:piezoelectric-spring-load"></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="<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. 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="orgada6c4c"></a> <a id="figure--fig:piezoelectric-force-displ-relation"></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="<span class=\"figure-number\">Figure 5: </span>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.md#" >}}). 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#" >}}). Limitations of the electronics is discussed in [Design, modeling and control of nanopositioning systems]({{< relref "fleming14_desig_model_contr_nanop_system.md" >}}).
## Bibliography {#bibliography}
<a id="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>.
<a id="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>.
<a id="org2bf81f0"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”

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content/zettels/stepper_motor.md
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For a two phase stepper motor, there are (typically) 200 steps per revolution. For a two phase stepper motor, there are (typically) 200 steps per revolution.
Errors with a period of 200 period/revolution can be expected. Errors with a period of 200 period/revolution can be expected.
## Manufacturers {#manufacturers}
| Manufacturers | Country |
|--------------------------------------------------------------------------|----------------|
| [AML](https://arunmicro.com/) | United Kingdom |
| [Sanyo](https://www.sanyodenki.com/catalogs/servo/stepping_systems.html) | |

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