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content/book/hatch00_vibrat_matlab_ansys.md
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@ -4,7 +4,7 @@ author = ["Thomas Dehaeze"]
draft = false draft = false
+++ +++
### Backlinks {#backlinks} Backlinks:
- [Finite Element Model]({{< relref "finite_element_model" >}}) - [Finite Element Model]({{< relref "finite_element_model" >}})
@ -12,7 +12,7 @@ Tags
: [Finite Element Model]({{< relref "finite_element_model" >}}) : [Finite Element Model]({{< relref "finite_element_model" >}})
Reference Reference
: ([Hatch 2000](#orgf974cac)) : ([Hatch 2000](#org1dc1f0a))
Author(s) Author(s)
: Hatch, M. R. : Hatch, M. R.
@ -25,14 +25,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction} ## Introduction {#introduction}
<a id="org30f30a5"></a> <a id="org7ccf52c"></a>
The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models. The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models.
### Modal Analysis {#modal-analysis} ### Modal Analysis {#modal-analysis}
The diagram in Figure [1](#org47ba802) shows the methodology for analyzing a lightly damped structure using normal modes. The diagram in Figure [1](#org72fd3ea) shows the methodology for analyzing a lightly damped structure using normal modes.
<div class="important"> <div class="important">
<div></div> <div></div>
@ -50,7 +50,7 @@ The steps are:
</div> </div>
<a id="org47ba802"></a> <a id="org72fd3ea"></a>
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}} {{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}}
@ -62,7 +62,7 @@ Because finite element models usually have a very large number of states, an imp
<div class="important"> <div class="important">
<div></div> <div></div>
Figure [2](#org5605afc) shows such process, the steps are: Figure [2](#org6c056ac) shows such process, the steps are:
- start with the finite element model - start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model) - compute the eigenvalues and eigenvectors (as many as dof in the model)
@ -75,14 +75,14 @@ Figure [2](#org5605afc) shows such process, the steps are:
</div> </div>
<a id="org5605afc"></a> <a id="org6c056ac"></a>
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}} {{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}
### Notations {#notations} ### Notations {#notations}
Tables [3](#org8a3a401), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document. Tables [3](#org7b2a048), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
<a id="table--tab:notations-modes-nodes"></a> <a id="table--tab:notations-modes-nodes"></a>
<div class="table-caption"> <div class="table-caption">
@ -131,22 +131,22 @@ Tables [3](#org8a3a401), [2](#table--tab:notations-eigen-vectors-values) and [3]
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems} ## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
<a id="org76a3c3a"></a> <a id="orgb5e5e43"></a>
The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency. The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency.
We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**. We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**.
Figure [3](#org8a3a401) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs. Figure [3](#org7b2a048) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\). The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
<a id="org8a3a401"></a> <a id="org7b2a048"></a>
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}} {{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
<div class="important"> <div class="important">
<div></div> <div></div>
([Miu 1993](#orga24e99d)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function. ([Miu 1993](#org946c04d)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
The resonances of the "overhanging appendages" of the constrained system create the zeros. The resonances of the "overhanging appendages" of the constrained system create the zeros.
@ -155,12 +155,12 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis} ## State Space Analysis {#state-space-analysis}
<a id="org0ebaa11"></a> <a id="org0d77f45"></a>
## Modal Analysis {#modal-analysis} ## Modal Analysis {#modal-analysis}
<a id="orge0d0333"></a> <a id="org58639cc"></a>
Lightly damped structures are typically analyzed with the "normal mode" method described in this section. Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
@ -200,9 +200,9 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and
#### Equation of Motion {#equation-of-motion} #### Equation of Motion {#equation-of-motion}
Let's consider the model shown in Figure [4](#org9ab4d6c) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\). Let's consider the model shown in Figure [4](#org04e0e48) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
<a id="org9ab4d6c"></a> <a id="org04e0e48"></a>
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}} {{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}}
@ -301,17 +301,17 @@ One then find:
\end{bmatrix} \end{bmatrix}
\end{equation} \end{equation}
Virtual interpretation of the eigenvectors are shown in Figures [5](#orga0c09be), [6](#orgcbdfa37) and [7](#org3296dae). Virtual interpretation of the eigenvectors are shown in Figures [5](#orgb26ad8f), [6](#orgdbad135) and [7](#org8c56bc5).
<a id="orga0c09be"></a> <a id="orgb26ad8f"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}} {{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
<a id="orgcbdfa37"></a> <a id="orgdbad135"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}} {{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}
<a id="org3296dae"></a> <a id="org8c56bc5"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}} {{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}}
@ -350,9 +350,9 @@ There are many options for change of basis, but we will show that **when eigenve
The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems. The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems.
The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate. The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate.
This procedure is schematically shown in Figure [8](#orgc4bf3bf). This procedure is schematically shown in Figure [8](#org442424c).
<a id="orgc4bf3bf"></a> <a id="org442424c"></a>
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}} {{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}
@ -700,7 +700,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
## Frequency Response: Modal Form {#frequency-response-modal-form} ## Frequency Response: Modal Form {#frequency-response-modal-form}
<a id="orgce4e676"></a> <a id="org065931e"></a>
The procedure to obtain the frequency response from a modal form is as follow: The procedure to obtain the frequency response from a modal form is as follow:
@ -708,9 +708,9 @@ The procedure to obtain the frequency response from a modal form is as follow:
- use Laplace transform to obtain the transfer functions in principal coordinates - use Laplace transform to obtain the transfer functions in principal coordinates
- back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident - back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident
This will be applied to the model shown in Figure [9](#org2f073ed). This will be applied to the model shown in Figure [9](#orgd12bfd1).
<a id="org2f073ed"></a> <a id="orgd12bfd1"></a>
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}} {{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}
@ -892,9 +892,9 @@ Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows tha
</div> </div>
Figure [10](#orgd3fa126) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\). Figure [10](#orga3fac1c) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
<a id="orgd3fa126"></a> <a id="orga3fac1c"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}} {{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@ -903,16 +903,16 @@ The zeros for SISO transfer functions are the roots of the numerator, however, f
## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model} ## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
<a id="orgeeb2d23"></a> <a id="orgbd2c762"></a>
### Introduction {#introduction} ### Introduction {#introduction}
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org2b2ab45). In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orgf951884).
A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output. A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics. The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
<a id="org2b2ab45"></a> <a id="orgf951884"></a>
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}} {{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}}
@ -991,7 +991,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model} ## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
<a id="orgecdee84"></a> <a id="orgd9c4438"></a>
### Model Description {#model-description} ### Model Description {#model-description}
@ -1005,25 +1005,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model} ## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
<a id="orgaa7ea31"></a> <a id="orgdbad836"></a>
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#orgea9abf7)). In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#orgd9ad21b)).
### Actuator Description {#actuator-description} ### Actuator Description {#actuator-description}
<a id="orgea9abf7"></a> <a id="orgd9ad21b"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}} {{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}}
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#org2735634)). The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#orgf3713e6)).
<a id="org2735634"></a> <a id="orgf3713e6"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}} {{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required. For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
Figure [13](#org2735634) shows the nodes used for the reduced Matlab model. Figure [13](#orgf3713e6) shows the nodes used for the reduced Matlab model.
The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force. The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
The arrows at the nodes indicate the direction of forces. The arrows at the nodes indicate the direction of forces.
@ -1083,7 +1083,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction} ## Balanced Reduction {#balanced-reduction}
<a id="org5cec71c"></a> <a id="org49a054f"></a>
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods. In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
@ -1198,14 +1198,14 @@ The **states to be kept are the states with the largest diagonal terms**.
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model} ## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
<a id="org0f5f4fe"></a> <a id="org7289209"></a>
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#orgd698d3c)). In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#orgc80e17f)).
### Actuator Description {#actuator-description} ### Actuator Description {#actuator-description}
<a id="orgd698d3c"></a> <a id="orgc80e17f"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}} {{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
@ -1227,9 +1227,9 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description} ### Ansys Model Description {#ansys-model-description}
In Figure [15](#orgafe5660) are shown the principal nodes used for the model. In Figure [15](#org459464b) are shown the principal nodes used for the model.
<a id="orgafe5660"></a> <a id="org459464b"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}} {{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
@ -1348,11 +1348,11 @@ And we note:
G = zn * Gp; G = zn * Gp;
``` ```
<a id="org0d94307"></a> <a id="org7b5581a"></a>
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}} {{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
<a id="org34d9546"></a> <a id="org8b11f4c"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}} {{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@ -1450,13 +1450,13 @@ G_f = ss(A, B, C, D);
### Simple mode truncation {#simple-mode-truncation} ### Simple mode truncation {#simple-mode-truncation}
Let's plot the frequency of the modes (Figure [18](#orgc4a9679)). Let's plot the frequency of the modes (Figure [18](#org58cec9b)).
<a id="orgc4a9679"></a> <a id="org58cec9b"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
<a id="org6924b78"></a> <a id="org0dbe1d4"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
@ -1525,7 +1525,7 @@ dc_gain = abs(xn(i_input, :).*xn(i_output, :))./(2*pi*f0).^2;
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend'); [dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
``` ```
<a id="org118ba52"></a> <a id="org9a62a29"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
@ -1869,7 +1869,7 @@ wo = gram(G_m, 'o');
And we plot the diagonal terms And we plot the diagonal terms
<a id="orgc90e1af"></a> <a id="org59c65d8"></a>
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}} {{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
@ -1887,7 +1887,7 @@ We use `balreal` to rank oscillatory states.
[G_b, G, T, Ti] = balreal(G_m); [G_b, G, T, Ti] = balreal(G_m);
``` ```
<a id="org5169fd1"></a> <a id="org5730ca2"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
@ -2132,6 +2132,6 @@ pos_frames = pos([1, i_input, i_output], :);
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgf974cac"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press. <a id="org1dc1f0a"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="orga24e99d"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York. <a id="org946c04d"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.

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+++
title = "Analog to Digital Converters"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Electronics]({{< relref "electronics" >}})
## Power Spectral Density of the Quantization Noise {#power-spectral-density-of-the-quantization-noise}
This analysis is taken from [here](https://www.allaboutcircuits.com/technical-articles/quantization-nois-amplitude-quantization-error-analog-to-digital-converters/).
Let's note:
- \\(q = \frac{\Delta V}{2^n}\\) the quantization in [V] (the corresponding value in [V] of the least significant bit)
- \\(\Delta V\\) is the full range of the ADC in [V]
- \\(n\\) is the number of ADC's bits
- \\(f\_s\\) is the sample frequency in [Hz]
Let's suppose that the ADC is ideal and the only noise comes from the quantization error.
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](#org5158d30)).
<a id="org5158d30"></a>
{{< figure src="/ox-hugo/probability_density_function_adc.png" caption="Figure 1: Probability density function \\(p(e)\\) of the ADC error \\(e\\)" >}}
Now, we can calculate the time average power of the quantization noise as
\begin{equation}
P\_q = \int\_{-q/2}^{q/2} e^2 p(e) de = \frac{q^2}{12}
\end{equation}
The other important parameter of a noise source is the power spectral density (PSD), which indicates how the noise power spreads in different frequency bands.
To find the power spectral density, we need to calculate the Fourier transform of the autocorrelation function of the noise.
Assuming that the noise samples are not correlated with one another, we can approximate the autocorrelation function with a delta function in the time domain.
Since the Fourier transform of a delta function is equal to one, the **power spectral density will be frequency independent**.
Therefore, the quantization noise is white noise with total power equal to \\(P\_q = \frac{q^2}{12}\\).
Thus, the two-sided PSD (from \\(\frac{-f\_s}{2}\\) to \\(\frac{f\_s}{2}\\)), we should divide the noise power \\(P\_q\\) by \\(f\_s\\):
\begin{equation}
\int\_{-f\_s/2}^{f\_s/2} \Gamma(f) d f = f\_s \Gamma = \frac{q^2}{12}
\end{equation}
<div class="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} \\\\\\
&= \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 f\_s} \text{ in } \left[ \frac{V^2}{Hz} \right]
\end{aligned}
\end{equation}
</div>
<div class="examp">
<div></div>
Let's take a 18bits ADC with a range of +/-10V and a sample frequency of 10kHz.
The quantization is:
\\[ q = \frac{20}{2^{18}} = 0.000076 \ [V] = 76 \ [\mu V] \\]
\\[ \Gamma\_Q = \frac{q^2}{12 f\_N} = 4.85 \cdot 10^{-14} \quad [V^2/Hz] \\]
</div>
<./biblio/references.bib>

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title = "Digital to Analog Converters"
author = ["Thomas Dehaeze"]
draft = false
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Tags
: [Electronics]({{< relref "electronics" >}})
<./biblio/references.bib>

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title = "Flexures"
author = ["Thomas Dehaeze"]
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Tags
: [Flexible Joints]({{< relref "flexible_joints" >}})
## Material Used {#material-used}
## Materials {#materials}
- ([Smith 2000](#org0c6025e))
- ([Lobontiu 2002](#org42ce68f))
- ([Henein 2003](#org59d412b))
- ([Cosandier 2017](#org637114f))
## Bibliography {#bibliography}
<a id="org637114f"></a>Cosandier, Florent. 2017. _Flexure Mechanism Design_. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press.
<a id="org59d412b"></a>Henein, Simon. 2003. _Conception Des Guidages Flexibles_. Lausanne, Suisse: Presses polytechniques et universitaires romandes.
<a id="org42ce68f"></a>Lobontiu, Nicolae. 2002. _Compliant Mechanisms: Design of Flexure Hinges_. CRC press.
<a id="org0c6025e"></a>Smith, Stuart T. 2000. _Flexures: Elements of Elastic Mechanisms_. Crc Press.

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## Manufacturers {#manufacturers} ## Manufacturers {#manufacturers}
| Manufacturers | Links | | Manufacturers | Links | Country |
|---------------|---------------------------------------------------------------------------------------------------------------| |---------------|---------------------------------------------------------------------------------------------------------------|----------|
| PCB | [link](https://www.pcb.com/sensors-for-test-measurement/impact-hammers-electrodynamic-shakers/impact-hammers) | | PCB | [link](https://www.pcb.com/sensors-for-test-measurement/impact-hammers-electrodynamic-shakers/impact-hammers) | USA |
| DJB | [link](https://www.djbinstruments.com/products/instrumentation/impact-hammers) | | DJB | [link](https://www.djbinstruments.com/products/instrumentation/impact-hammers) | UK |
| Dewesoft | [link](https://dewesoft.com/fr/products/interfaces-and-sensors/accelerometers-and-modal-hammers) | Slovenia |
<./biblio/references.bib> <./biblio/references.bib>

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draft = false draft = false
+++ +++
### Backlinks {#backlinks} Backlinks:
- [A review of nanometer resolution position sensors: operation and performance]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}}) - [A review of nanometer resolution position sensors: operation and performance]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}})
- [Measurement technologies for precision positioning]({{< relref "gao15_measur_techn_precis_posit" >}}) - [Measurement technologies for precision positioning]({{< relref "gao15_measur_techn_precis_posit" >}})
- [Inertial Sensors]({{< relref "inertial_sensors" >}})
- [Sensors]({{< relref "sensors" >}}) - [Sensors]({{< relref "sensors" >}})
- [Collocated Control]({{< relref "collocated_control" >}}) - [Collocated Control]({{< relref "collocated_control" >}})
- [Inertial Sensors]({{< relref "inertial_sensors" >}})
Tags Tags
: [Inertial Sensors]({{< relref "inertial_sensors" >}}), [Force Sensors]({{< relref "force_sensors" >}}), [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Signal Conditioner]({{< relref "signal_conditioner" >}}), [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}}) : [Inertial Sensors]({{< relref "inertial_sensors" >}}), [Force Sensors]({{< relref "force_sensors" >}}), [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Signal Conditioner]({{< relref "signal_conditioner" >}}), [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}})
@ -18,7 +18,7 @@ Tags
## Reviews of Relative Position Sensors {#reviews-of-relative-position-sensors} ## Reviews of Relative Position Sensors {#reviews-of-relative-position-sensors}
- Fleming, A. J., A review of nanometer resolution position sensors: operation and performance ([Fleming 2013](#orgdd1b6d5)) ([Notes]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}})) - Fleming, A. J., A review of nanometer resolution position sensors: operation and performance ([Fleming 2013](#org0e7fb0d)) ([Notes]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}}))
<a id="table--tab:characteristics-relative-sensor"></a> <a id="table--tab:characteristics-relative-sensor"></a>
<div class="table-caption"> <div class="table-caption">
@ -116,9 +116,9 @@ Description:
| Renishaw | 0.2 | 1 | 6 | 1 | | Renishaw | 0.2 | 1 | 6 | 1 |
| Picoscale | 0.2 | 2 | 2 | 1 | | Picoscale | 0.2 | 2 | 2 | 1 |
([Jang and Kim 2017](#orgbcf1569)) ([Jang and Kim 2017](#orga6fb604))
<a id="orgf2b5520"></a> <a id="org22624ed"></a>
{{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 1: Expected precision of interferometer as a function of measured distance" >}} {{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 1: Expected precision of interferometer as a function of measured distance" >}}
@ -130,10 +130,11 @@ Description:
| Heidenhain | [link](https://www.heidenhain.com/en%5FUS/products/linear-encoders/) | Germany | | Heidenhain | [link](https://www.heidenhain.com/en%5FUS/products/linear-encoders/) | Germany |
| MicroE Systems | [link](https://www.celeramotion.com/microe/products/linear-encoders/) | USA | | MicroE Systems | [link](https://www.celeramotion.com/microe/products/linear-encoders/) | USA |
| Renishaw | [link](https://www.renishaw.com/en/browse-encoder-range--6440) | UK | | Renishaw | [link](https://www.renishaw.com/en/browse-encoder-range--6440) | UK |
| Celera Motion | [link](https://www.celeramotion.com/microe/) | USA |
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgdd1b6d5"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” _Sensors and Actuators a: Physical_ 190 (nil):106–26. <https://doi.org/10.1016/j.sna.2012.10.016>. <a id="org0e7fb0d"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” _Sensors and Actuators a: Physical_ 190 (nil):106–26. <https://doi.org/10.1016/j.sna.2012.10.016>.
<a id="orgbcf1569"></a>Jang, Yoon-Soo, and Seung-Woo Kim. 2017. “Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: A Review.” _International Journal of Precision Engineering and Manufacturing_ 18 (12):1881–90. <https://doi.org/10.1007/s12541-017-0217-y>. <a id="orga6fb604"></a>Jang, Yoon-Soo, and Seung-Woo Kim. 2017. “Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: A Review.” _International Journal of Precision Engineering and Manufacturing_ 18 (12):1881–90. <https://doi.org/10.1007/s12541-017-0217-y>.

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title = "Rotation Stage"
author = ["Thomas Dehaeze"]
draft = false
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Tags
: [Slip Rings]({{< relref "slip_rings" >}})
## Manufacturers {#manufacturers}
| Manufacturers | Links | Country |
|-------------------|-------------------------------------------|---------|
| Huber | [link](https://www.xhuber.com/en/) | Germany |
| LAB Motion System | [link](http://www.leuvenairbearings.com/) | Belgium |
<./biblio/references.bib>

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<https://www.bksv.com/en/products/shakers-and-exciters/LDS-shaker-systems/permanent-magnet-shakers/V201> <https://www.bksv.com/en/products/shakers-and-exciters/LDS-shaker-systems/permanent-magnet-shakers/V201>
| Manufacturers | Links | | Manufacturers | Links | Country |
|--------------------|----------------------------------------------------------------------------------| |--------------------|----------------------------------------------------------------------------------|-----------|
| Labsen | [link](http://labsentec.com.au/category/products/vibrationshock/) | | Labsen | [link](http://labsentec.com.au/category/products/vibrationshock/) | Australia |
| The Modal Shop | [link](http://www.modalshop.com/excitation/Electrodynamic-Exciter-Family?ID=243) | | The Modal Shop | [link](http://www.modalshop.com/excitation/Electrodynamic-Exciter-Family?ID=243) | USA |
| Deweshop | [link](https://dewesoft.com/fr/products/interfaces-and-sensors/shakers) | | Deweshop | [link](https://dewesoft.com/fr/products/interfaces-and-sensors/shakers) | Slovenia |
| Bruel and Kjaer | [link](https://www.bksv.com/en/products/shakers-and-exciters/LDS-shaker-systems) | | Bruel and Kjaer | [link](https://www.bksv.com/en/products/shakers-and-exciters/LDS-shaker-systems) | Denmark |
| YMC | [link](http://www.chinaymc.com/product/showproduct.php?id=78&lang=en) | | YMC | [link](http://www.chinaymc.com/product/showproduct.php?id=78&lang=en) | China |
| BKSV | [link](https://www.bksv.com/en/products/shakers-and-exciters) | | Vibration Research | [link](https://vibrationresearch.com/shakers/) | USA |
| Vibration Research | [link](https://vibrationresearch.com/shakers/) | | Sentek Dynamics | [link](https://www.sentekdynamics.com/) | USA |
| Sentek Dynamics | [link](https://www.sentekdynamics.com/) |
<./biblio/references.bib> <./biblio/references.bib>

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### Backlinks {#backlinks} Backlinks:
- [Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments]({{< relref "tang18_decen_vibrat_contr_voice_coil" >}}) - [Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments]({{< relref "tang18_decen_vibrat_contr_voice_coil" >}})
- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}}) - [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})
@ -32,40 +32,59 @@ Tags
: :
## Manufacturers {#manufacturers}
| Manufacturers | Links | Country |
|---------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|---------|
| PI | [link](https://www.physikinstrumente.com/en/products/parallel-kinematic-hexapods/) | Germany |
| Newport | [link](https://www.newport.com/search/?q1=hexapod%3Arelevance%3Acompatibility%3AMETRIC%3AisObsolete%3Afalse%3A-excludeCountries%3AFR%3AnpCategory%3Ahexapods&ajax&text=hexapod) | USA |
| Symetrie | [link](https://symetrie.fr/en/hexapods-en/positioning-hexapods/) | France |
## Stewart Platforms at ESRF {#stewart-platforms-at-esrf}
| Beamline | Manufacturer | Comments |
|----------|--------------|-----------------------------------|
| ID11 | Symetrie | Small, Piezo based |
| ID31 | Symetrie | Large Stroke, Encoders, DC motors |
| ID01 | PI | |
| ID16a | ESRF | Piezo (PI) |
## Flexure Jointed Stewart Platforms {#flexure-jointed-stewart-platforms} ## Flexure Jointed Stewart Platforms {#flexure-jointed-stewart-platforms}
Papers by J.E. McInroy: Papers by J.E. McInroy:
- ([O’Brien et al. 1998](#orgaa46d57)) - ([O’Brien et al. 1998](#org71c69cc))
- ([McInroy, O’Brien, and Neat 1999](#org378c866)) - ([McInroy, O’Brien, and Neat 1999](#orgd9fe3c1))
- ([McInroy 1999](#org3334ff2)) - ([McInroy 1999](#org82cce67))
- ([McInroy and Hamann 2000](#orgbb67e4d)) - ([McInroy and Hamann 2000](#orgc17f973))
- ([Chen and McInroy 2000](#org37a21cf)) - ([Chen and McInroy 2000](#org21fffc9))
- ([McInroy 2002](#org8af76b7)) - ([McInroy 2002](#org2f95611))
- ([Li, Hamann, and McInroy 2001](#orgd55cfdb)) - ([Li, Hamann, and McInroy 2001](#org247940b))
- ([Lin and McInroy 2003](#orged11f1d)) - ([Lin and McInroy 2003](#org39928ef))
- ([Jafari and McInroy 2003](#org3d4fb3c)) - ([Jafari and McInroy 2003](#org1e7c00b))
- ([Chen and McInroy 2004](#orgda0daba)) - ([Chen and McInroy 2004](#orgc6995cb))
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgda0daba"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>. <a id="orgc6995cb"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>.
<a id="org37a21cf"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>. <a id="org21fffc9"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
<a id="org3d4fb3c"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>. <a id="org1e7c00b"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>.
<a id="orged11f1d"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>. <a id="org39928ef"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>.
<a id="orgd55cfdb"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>. <a id="org247940b"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>.
<a id="org3334ff2"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>. <a id="org82cce67"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
<a id="org8af76b7"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>. <a id="org2f95611"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
<a id="orgbb67e4d"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>. <a id="orgc17f973"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>.
<a id="org378c866"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>. <a id="orgd9fe3c1"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>.
<a id="orgaa46d57"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>. <a id="org71c69cc"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>.

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+++ +++
## Backlinks {#backlinks} ### Backlinks {#backlinks}
- [Element and system design for active and passive vibration isolation]({{< relref "zuo04_elemen_system_desig_activ_passiv_vibrat_isolat" >}}) - [Element and system design for active and passive vibration isolation]({{< relref "zuo04_elemen_system_desig_activ_passiv_vibrat_isolat" >}})
- [A six-axis single-stage active vibration isolator based on stewart platform]({{< relref "preumont07_six_axis_singl_stage_activ" >}}) - [A six-axis single-stage active vibration isolator based on stewart platform]({{< relref "preumont07_six_axis_singl_stage_activ" >}})
@ -31,4 +31,11 @@ draft = false
Tags Tags
: :
## Vibration Isolation Tables {#vibration-isolation-tables}
| Manufacturer | links |
|--------------|-----------------------------------------------------------|
| TMC | [link](https://www.techmfg.com/products/stacis/stacisiii) |
<./biblio/references.bib> <./biblio/references.bib>

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draft = false draft = false
+++ +++
### Backlinks {#backlinks} Backlinks:
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}) - [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
@ -15,13 +15,32 @@ Tags
## Voltage Amplifiers to drive Capacitive Load {#voltage-amplifiers-to-drive-capacitive-load} ## Voltage Amplifiers to drive Capacitive Load {#voltage-amplifiers-to-drive-capacitive-load}
### Manufacturers {#manufacturers}
| Manufacturers | Links | Country |
|---------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------|-------------|
| Piezo Drive | [link](https://www.piezodrive.com/drivers/) | Australia |
| Thorlabs | [link](https://www.thorlabs.com/navigation.cfm?guide%5FID=2085) | USA |
| PI | [link](https://www.pi-usa.us/en/products/controllers-drivers-motion-control-software/piezo-drivers-controllers-power-supplies-high-voltage-amplifiers/) | USA |
| Micromega Dynamics | | Belgium |
| Lab Systems | [link](https://www.lab-systems.com/products/amplifier/amplifier.html) | Isreal |
| Falco System | [link](https://www.falco-systems.com/products.html) | Netherlands |
| Piezomechanics | [link](https://www.piezomechanik.com/products/) | Germany |
| Cedrat Technologies | [link](https://www.cedrat-technologies.com/en/products/piezo-controllers/electronic-amplifier-boards.html) | France |
| Trek | [link](https://www.trekinc.com/products/HV%5FAmp.asp) | USA |
| Madcitylabs | [link](http://www.madcitylabs.com/piezoactuators.html) | USA |
| Piezosystem | [link](https://www.piezosystem.com/products/controller/) | Germany |
| Matsusada Precision | [link](https://www.matsusada.com/product/pz/) | Japan |
| Mechano Transformer | [link](http://www.mechano-transformer.com/en/products/08.html) | Japan |
### Limitation in Current {#limitation-in-current} ### Limitation in Current {#limitation-in-current}
The piezoelectric stack can be represented as a capacitance. The piezoelectric stack can be represented as a capacitance.
Let's take a capacitance driven by a voltage amplifier (Figure [1](#org7969f96)). Let's take a capacitance driven by a voltage amplifier (Figure [1](#orgf2b344c)).
<a id="org7969f96"></a> <a id="orgf2b344c"></a>
{{< figure src="/ox-hugo/voltage_amplifier_capacitance.png" caption="Figure 1: Piezoelectric actuator model with a voltage source" >}} {{< figure src="/ox-hugo/voltage_amplifier_capacitance.png" caption="Figure 1: Piezoelectric actuator model with a voltage source" >}}
@ -41,7 +60,7 @@ Thus, for a specified maximum current \\(I\_\text{max}\\), the "power bandwidth"
- Above \\(\omega\_{0, \text{max}}\\), the maximum current \\(I\_\text{max}\\) is reached and the maximum voltage that can be applied decreases with frequency: - Above \\(\omega\_{0, \text{max}}\\), the maximum current \\(I\_\text{max}\\) is reached and the maximum voltage that can be applied decreases with frequency:
\\[ U\_\text{max} = \frac{I\_\text{max}}{\omega C} \\] \\[ U\_\text{max} = \frac{I\_\text{max}}{\omega C} \\]
The maximum voltage as a function of frequency is shown in Figure [2](#org310483b). The maximum voltage as a function of frequency is shown in Figure [2](#org1190638).
```matlab ```matlab
Vpkp = 170; % [V] Vpkp = 170; % [V]
@ -55,7 +74,7 @@ C = 1e-6; % [F]
56.172 56.172
``` ```
<a id="org310483b"></a> <a id="org1190638"></a>
{{< figure src="/ox-hugo/voltage_amplifier_max_V_piezo.png" caption="Figure 2: Maximum voltage as a function of the frequency for \\(C = 1 \mu F\\), \\(I\_\text{max} = 30mA\\) and \\(V\_{pkp} = 170 V\\)" >}} {{< figure src="/ox-hugo/voltage_amplifier_max_V_piezo.png" caption="Figure 2: Maximum voltage as a function of the frequency for \\(C = 1 \mu F\\), \\(I\_\text{max} = 30mA\\) and \\(V\_{pkp} = 170 V\\)" >}}
@ -69,7 +88,7 @@ If driven at \\(\Delta U = 100V\\), \\(C = 1 \mu F\\) and \\(I\_\text{max} = 1 A
### Bandwidth limitation (small signals) {#bandwidth-limitation--small-signals} ### Bandwidth limitation (small signals) {#bandwidth-limitation--small-signals}
This is takken from Chapter 14 of ([Fleming and Leang 2014](#orga9ea9d3)). This is takken from Chapter 14 of ([Fleming and Leang 2014](#org2e80fee)).
```matlab ```matlab
L = 250e-9; % Cable inductance [H] L = 250e-9; % Cable inductance [H]
@ -92,23 +111,7 @@ Specifications are usually:
The bandwidth can be estimated from the Maximum Current and the Capacitance of the Piezoelectric Actuator. The bandwidth can be estimated from the Maximum Current and the Capacitance of the Piezoelectric Actuator.
| Manufacturers | Links | Country |
|---------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------|-------------|
| Piezo Drive | [link](https://www.piezodrive.com/drivers/) | Australia |
| Thorlabs | [link](https://www.thorlabs.com/navigation.cfm?guide%5FID=2085) | USA |
| PI | [link](https://www.pi-usa.us/en/products/controllers-drivers-motion-control-software/piezo-drivers-controllers-power-supplies-high-voltage-amplifiers/) | USA |
| Micromega Dynamics | | Belgium |
| Lab Systems | [link](https://www.lab-systems.com/products/amplifier/amplifier.html) | Isreal |
| Falco System | [link](https://www.falco-systems.com/products.html) | Netherlands |
| Piezomechanics | [link](https://www.piezomechanik.com/products/) | Germany |
| Cedrat Technologies | [link](https://www.cedrat-technologies.com/en/products/piezo-controllers/electronic-amplifier-boards.html) | France |
| Trek | [link](https://www.trekinc.com/products/HV%5FAmp.asp) | USA |
| Madcitylabs | [link](http://www.madcitylabs.com/piezoactuators.html) | USA |
| Piezosystem | [link](https://www.piezosystem.com/products/controller/) | Germany |
| Matsusada Precision | [link](https://www.matsusada.com/product/pz/) | Japan |
| Mechano Transformer | [link](http://www.mechano-transformer.com/en/products/08.html) | Japan |
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orga9ea9d3"></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="org2e80fee"></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>.