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content/book/lyons11_under_digit_signal_proces.md
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Tags Tags
: [IRR and FIR Filters]({{<relref "irr_and_fir_filters.md#" >}}), [Digital Filters]({{<relref "digital_filters.md#" >}}) : [IRR and FIR Filters]({{< relref "irr_and_fir_filters.md" >}}), [Digital Filters]({{< relref "digital_filters.md" >}})
Reference Reference
: <lyons11_under_digit_signal_proces> : (<a href="#citeproc_bib_item_1">Lyons 2011</a>)
Author(s) Author(s)
: Lyons, R. : Lyons, R.
@ -43,13 +43,13 @@ Year
### Analyzing Linear Time-Invariant Systems {#analyzing-linear-time-invariant-systems} ### Analyzing Linear Time-Invariant Systems {#analyzing-linear-time-invariant-systems}
<a id="orgcbbc38b"></a> <a id="figure--fig:lyons11-lti-impulse-response"></a>
{{< figure src="/ox-hugo/lyons11_lti_impulse_response.png" caption="Figure 1: LTI system unit impulse response sequences. (a) system block diagram. (b) impulse input sequence \\(x(n)\\) and impulse reponse output sequence \\(y(n)\\)." >}} {{< figure src="/ox-hugo/lyons11_lti_impulse_response.png" caption="<span class=\"figure-number\">Figure 1: </span>LTI system unit impulse response sequences. (a) system block diagram. (b) impulse input sequence \\(x(n)\\) and impulse reponse output sequence \\(y(n)\\)." >}}
<a id="org50f1362"></a> <a id="figure--fig:lyons11-moving-average"></a>
{{< figure src="/ox-hugo/lyons11_moving_average.png" caption="Figure 2: Analyzing a moving average filter. (a) averager block diagram; (b) impulse input and impulse response; (c) averager frequency magnitude reponse." >}} {{< figure src="/ox-hugo/lyons11_moving_average.png" caption="<span class=\"figure-number\">Figure 2: </span>Analyzing a moving average filter. (a) averager block diagram; (b) impulse input and impulse response; (c) averager frequency magnitude reponse." >}}
## Periodic Sampling {#periodic-sampling} ## Periodic Sampling {#periodic-sampling}
@ -57,20 +57,20 @@ Year
### Aliasing: Signal Ambiguity In The Frequency Domain {#aliasing-signal-ambiguity-in-the-frequency-domain} ### Aliasing: Signal Ambiguity In The Frequency Domain {#aliasing-signal-ambiguity-in-the-frequency-domain}
<a id="org0ff6bb3"></a> <a id="figure--fig:lyons11-frequency-ambiguity"></a>
{{< figure src="/ox-hugo/lyons11_frequency_ambiguity.png" caption="Figure 3: Frequency ambiguity; (a) discrete time sequence of values; (b) two different sinewaves that pass through the points of discete sequence" >}} {{< figure src="/ox-hugo/lyons11_frequency_ambiguity.png" caption="<span class=\"figure-number\">Figure 3: </span>Frequency ambiguity; (a) discrete time sequence of values; (b) two different sinewaves that pass through the points of discete sequence" >}}
### Sampling Lowpass Signals {#sampling-lowpass-signals} ### Sampling Lowpass Signals {#sampling-lowpass-signals}
<a id="org38fdf07"></a> <a id="figure--fig:lyons11-noise-spectral-replication"></a>
{{< figure src="/ox-hugo/lyons11_noise_spectral_replication.png" caption="Figure 4: Spectral replications; (a) original continuous signal plus noise spectrum; (b) discrete spectrum with noise contaminating the signal of interest" >}} {{< figure src="/ox-hugo/lyons11_noise_spectral_replication.png" caption="<span class=\"figure-number\">Figure 4: </span>Spectral replications; (a) original continuous signal plus noise spectrum; (b) discrete spectrum with noise contaminating the signal of interest" >}}
<a id="org5e8c824"></a> <a id="figure--fig:lyons11-lowpass-sampling"></a>
{{< figure src="/ox-hugo/lyons11_lowpass_sampling.png" caption="Figure 5: Low pass analog filtering prior to sampling at a rate of \\(f\_s\\) Hz." >}} {{< figure src="/ox-hugo/lyons11_lowpass_sampling.png" caption="<span class=\"figure-number\">Figure 5: </span>Low pass analog filtering prior to sampling at a rate of \\(f\_s\\) Hz." >}}
## The Discrete Fourier Transform {#the-discrete-fourier-transform} ## The Discrete Fourier Transform {#the-discrete-fourier-transform}
@ -531,96 +531,8 @@ X(m) = \sum\_{n = 0}^{N-1} x(n) e^{-j2 \pi n m /N}
### 13.51 A Stable Goertzel Algorithm {#13-dot-51-a-stable-goertzel-algorithm} ### 13.51 A Stable Goertzel Algorithm {#13-dot-51-a-stable-goertzel-algorithm}
## A: The Arithmetic Of Complex Numbers {#a-the-arithmetic-of-complex-numbers} ## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
### A.1 Graphical Representation Of Real And Complex Numbers {#a-dot-1-graphical-representation-of-real-and-complex-numbers} <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Lyons, Richard. 2011. <i>Understanding Digital Signal Processing</i>. Upper Saddle River, NJ: Prentice Hall.</div>
</div>
### A.2 Arithmetic Representation Of Complex Numbers {#a-dot-2-arithmetic-representation-of-complex-numbers}
### A.3 Arithmetic Operations Of Complex Numbers {#a-dot-3-arithmetic-operations-of-complex-numbers}
### A.4 Some Practical Implications Of Using Complex Numbers {#a-dot-4-some-practical-implications-of-using-complex-numbers}
## B: Closed Form Of A Geometric Series {#b-closed-form-of-a-geometric-series}
## C: Time Reversal And The Dft {#c-time-reversal-and-the-dft}
## D: Mean,Variance, And Standard Deviation {#d-mean-variance-and-standard-deviation}
### D.1 Statistical Measures {#d-dot-1-statistical-measures}
### D.2 Statistics Of Short Sequences {#d-dot-2-statistics-of-short-sequences}
### D.3 Statistics Of Summed Sequences {#d-dot-3-statistics-of-summed-sequences}
### D.4 Standard Deviation (Rms) Of A Continuous Sinewave {#d-dot-4-standard-deviation--rms--of-a-continuous-sinewave}
### D.5 Estimating Signal-To-Noise Ratios {#d-dot-5-estimating-signal-to-noise-ratios}
### D.6 The Mean And Variance Of Random Functions {#d-dot-6-the-mean-and-variance-of-random-functions}
### D.7 The Normal Probability Density Function {#d-dot-7-the-normal-probability-density-function}
## E: Decibels (Db And Dbm) {#e-decibels--db-and-dbm}
### E.1 Using Logarithms To Determine Relative Signal Power {#e-dot-1-using-logarithms-to-determine-relative-signal-power}
### E.2 Some Useful Decibel Numbers {#e-dot-2-some-useful-decibel-numbers}
### E.3 Absolute Power Using Decibels {#e-dot-3-absolute-power-using-decibels}
## F: Digital Filter Terminology {#f-digital-filter-terminology}
## G: Frequency Sampling Filter Derivations {#g-frequency-sampling-filter-derivations}
### G.1 Frequency Response Of A Comb Filter {#g-dot-1-frequency-response-of-a-comb-filter}
### G.2 Single Complex Fsf Frequency Response {#g-dot-2-single-complex-fsf-frequency-response}
### G.3 Multisection Complex Fsf Phase {#g-dot-3-multisection-complex-fsf-phase}
### G.4 Multisection Complex Fsf Frequency Response {#g-dot-4-multisection-complex-fsf-frequency-response}
### G.5 Real Fsf Transfer Function {#g-dot-5-real-fsf-transfer-function}
### G.6 Type-Iv Fsf Frequency Response {#g-dot-6-type-iv-fsf-frequency-response}
## H: Frequency Sampling Filter Design Tables {#h-frequency-sampling-filter-design-tables}
## I: Computing Chebyshev Window Sequences {#i-computing-chebyshev-window-sequences}
### I.1 Chebyshev Windows For Fir Filter Design {#i-dot-1-chebyshev-windows-for-fir-filter-design}
### I.2 Chebyshev Windows For Spectrum Analysis {#i-dot-2-chebyshev-windows-for-spectrum-analysis}
<./biblio/references.bib>

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content/book/smith99_scien_engin_guide_digit_signal.md
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title = "The scientist and engineer's guide to digital signal processing - second edition" title = "The scientist and engineer's guide to digital signal processing - second edition"
author = ["Thomas Dehaeze"] author = ["Dehaeze Thomas"]
keywords = ["Signal Processing"] keywords = ["Signal Processing"]
draft = true draft = true
+++ +++
Tags Tags
: [Digital Signal Processing]({{< relref "digital_signal_processing" >}}) : [Digital Signal Processing]({{< relref "digital_signal_processing.md" >}})
Reference Reference
: ([Smith 1999](#org023917a)) : (<a href="#citeproc_bib_item_1">Smith 1999</a>)
Author(s) Author(s)
: Smith, S. W. : Smith, S. W.
@ -18,7 +18,8 @@ Year
: 1999 : 1999
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org023917a"></a>Smith, Steven W. 1999. _The Scientist and Engineer’s Guide to Digital Signal Processing - Second Edition_. California Technical Publishing. <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Smith, Steven W. 1999. <i>The Scientist and Engineer’s Guide to Digital Signal Processing - Second Edition</i>. California Technical Publishing.</div>
</div>

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title = "Active damping of vibrations in high-precision motion systems" title = "Active damping of vibrations in high-precision motion systems"
author = ["Thomas Dehaeze"] author = ["Dehaeze Thomas"]
draft = false draft = false
ref_author = "Babakhani, B." ref_author = "Babakhani, B."
ref_year = 2012 ref_year = 2012
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Tags Tags
: [Active Damping]({{<relref "active_damping.md#" >}}) : [Active Damping]({{< relref "active_damping.md" >}})
Reference Reference
: ([Babakhani 2012](#orgf88b4c7)) : (<a href="#citeproc_bib_item_1">Babakhani 2012</a>)
Author(s) Author(s)
: Babakhani, B. : Babakhani, B.
@ -19,7 +19,8 @@ Year
: 2012 : 2012
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgf88b4c7"></a>Babakhani, Bayan. 2012. “Active Damping of Vibrations in High-Precision Motion Systems.” University of Twente. <https://doi.org/10.3990/1.9789036534642>. <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Babakhani, Bayan. 2012. “Active Damping of Vibrations in High-Precision Motion Systems.” University of Twente. doi:<a href="https://doi.org/10.3990/1.9789036534642">10.3990/1.9789036534642</a>.</div>
</div>

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Tags Tags
: :
Links to specific actuators:
## Short Stroke Actuators {#short-stroke-actuators}
For short stroke and very high dynamic applications, mainly two types of actuators can be used:
- [Voice Coil Actuators]({{< relref "voice_coil_actuators.md" >}}) - [Voice Coil Actuators]({{< relref "voice_coil_actuators.md" >}})
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}) - [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}})
## Long Stroke Actuators {#long-stroke-actuators}
- Linear motors
- Piezoelectric Walking Drive ([PI](https://www.physikinstrumente.com/en/expertise/technology/piezoelectric-drives/piezowalk-piezo-motors/))
Rotational drives can be combined with ball-screw mechanisms for long (infinite) axial motion:
- Brush-less DC Motor. See (<a href="#citeproc_bib_item_2">Yedamale 2003</a>) and this [working principle](https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html).
- [Stepper Motor]({{< relref "stepper_motor.md" >}})
## How to choose the correct actuator for my application? {#how-to-choose-the-correct-actuator-for-my-application} ## How to choose the correct actuator for my application? {#how-to-choose-the-correct-actuator-for-my-application}
For vibration isolation: For vibration isolation:
@ -20,16 +34,6 @@ For vibration isolation:
- In (<a href="#citeproc_bib_item_1">Ito and Schitter 2016</a>), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat.md" >}})) - In (<a href="#citeproc_bib_item_1">Ito and Schitter 2016</a>), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat.md" >}}))
## Brush-less DC Motor {#brush-less-dc-motor}
- (<a href="#citeproc_bib_item_2">Yedamale 2003</a>)
<https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html>
## [Stepper Motor]({{< relref "stepper_motor.md" >}}) {#stepper-motor--stepper-motor-dot-md}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body"> <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">

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title = "Continuous Transfer Functions"
author = ["Dehaeze Thomas"]
draft = false
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Tags
:
## Typical Transfer Functions {#typical-transfer-functions}

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title = "Digital Filters" title = "Digital Filters"
author = ["Thomas Dehaeze"] author = ["Dehaeze Thomas"]
draft = false draft = false
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@ -9,4 +9,8 @@ Tags
A nice open access book on digital filter is accessible here: <https://ccrma.stanford.edu/~jos/filters/filters.html> A nice open access book on digital filter is accessible here: <https://ccrma.stanford.edu/~jos/filters/filters.html>
<./biblio/references.bib>
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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title = "Discrete Transfer Functions"
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Digital Filters]({{< relref "digital_filters.md" >}})
## Typical Transfer functions {#typical-transfer-functions}
### Delay {#delay}
### First Order Low Pass {#first-order-low-pass}
### First Order High Pass {#first-order-high-pass}
### Integrator {#integrator}
### Derivator {#derivator}
### Second Order Low Pass {#second-order-low-pass}
### PID {#pid}
### Notch {#notch}
### Moving Average {#moving-average}

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Tags Tags
: :
TODOS:
- [X] Electronics circuits containing input voltage, output voltage, R L and C components
- [ ] Bode plot of the filter from input voltage to output voltage
- [ ] Equation of the transfer functions with nice parameters (\\(\omega\_c\\), \\(\xi\\))
## First Order Low Pass Filter {#first-order-low-pass-filter} ## First Order Low Pass Filter {#first-order-low-pass-filter}
<a id="figure--fig:elec-passive-first-order-low-pass-filter"></a> <a id="figure--fig:analog-filt-first-order-lpf"></a>
{{< figure src="/ox-hugo/elec_passive_first_order_low_pass_filter.png" caption="<span class=\"figure-number\">Figure 1: </span>First Order Low Pass Filter using an RC circuit" >}} {{< figure src="/ox-hugo/analog_filt_first_order_lpf.png" caption="<span class=\"figure-number\">Figure 1: </span>First Order Low Pass Filter using an RC circuit" >}}
\begin{equation}
\boxed{\frac{V\_o}{V\_i}(s) = \frac{1}{1 + \frac{s}{\omega\_0}}, \quad \omega\_0 = \frac{1}{RC}}
\end{equation}
\begin{equation}
Z\_f(s) = \frac{V\_i}{i\_i}(s) = \frac{1 + RC \cdot s}{C \cdot s}
\end{equation}
```matlab
%% First Order Low Pass Filter
R = 1e3; % [Ohm]
C = 1e-6; % [F]
```
```matlab
G = 1/(1 + R*C*s); % Filter Transfer Function
Z = (1 + R*C*s)/(C*s); % Filter Impedance
```
<a id="figure--fig:analog-filt-first-order-lpf-characteristics"></a>
{{< figure src="/ox-hugo/analog_filt_first_order_lpf_characteristics.png" caption="<span class=\"figure-number\">Figure 2: </span>First Order Low Pass Filter - Filter transfer function and filter impedance" >}}
## First Order High Pass Filter {#first-order-high-pass-filter} ## First Order High Pass Filter {#first-order-high-pass-filter}
<a id="figure--fig:elec-passive-first-order-high-pass-filter"></a> <a id="figure--fig:analog-filt-first-order-hpf"></a>
{{< figure src="/ox-hugo/elec_passive_first_order_high_pass_filter.png" caption="<span class=\"figure-number\">Figure 2: </span>First Order High Pass Filter using an RC circuit" >}} {{< figure src="/ox-hugo/analog_filt_first_order_hpf.png" caption="<span class=\"figure-number\">Figure 3: </span>First Order High Pass Filter using an RC circuit" >}}
\begin{equation}
\boxed{\frac{V\_o}{V\_i}(s) = \frac{\frac{s}{\omega\_0}}{1 + \frac{s}{\omega\_0}}, \quad \omega\_0 = \frac{1}{RC}}
\end{equation}
\begin{equation}
Z\_f(s) = \frac{V\_i}{i\_i}(s) = \frac{1 + RC \cdot s}{C \cdot s}
\end{equation}
```matlab
%% First Order High Pass Filter
R = 1e3; % [Ohm]
C = 1e-6; % [F]
```
```matlab
G = R*C*s/(1 + R*C*s); % Filter Transfer Function
Z = (1 + R*C*s)/(C*s); % Filter Impedance
```
<a id="figure--fig:analog-filt-first-order-hpf-characteristics"></a>
{{< figure src="/ox-hugo/analog_filt_first_order_hpf_characteristics.png" caption="<span class=\"figure-number\">Figure 4: </span>First Order High Pass Filter - Filter transfer function and filter impedance" >}}
## Second Order Low Pass Filter {#second-order-low-pass-filter} ## Second Order Low Pass Filter {#second-order-low-pass-filter}
<a id="figure--fig:elec-passive-second-order-low-pass-filter"></a> <a id="figure--fig:analog-filt-second-order-lpf"></a>
{{< figure src="/ox-hugo/elec_passive_second_order_low_pass_filter.png" caption="<span class=\"figure-number\">Figure 3: </span>Second Order Low Pass Filter using an RLC circuit" >}} {{< figure src="/ox-hugo/analog_filt_second_order_lpf.png" caption="<span class=\"figure-number\">Figure 5: </span>Second Order Low Pass Filter using an RLC circuit" >}}
\begin{equation}
\boxed{\frac{V\_o}{V\_i}(s) = \frac{1}{1 + 2 \xi \frac{s}{\omega\_0} + \frac{s^2}{\omega\_0^2}}, \quad \omega\_0 = \frac{1}{\sqrt{LC}},\quad \xi = \frac{1}{2R\sqrt{LC}}}
\end{equation}
\begin{equation}
Z\_f(s) = \frac{V\_i}{i\_i}(s) = L \cdot s + \frac{R}{1 + RC \cdot s}
\end{equation}
```matlab
%% Second Order Low Pass Filter
R = 1e2; % [Ohm]
C = 1e-5; % [F]
L = 1e-2; % [H]
```
```matlab
G = 1/(1 + L/R*s + C*L*s^2); % Filter Transfer Function
Z = L*s + (R)/(1 + R*C*s); % Filter Impedance
```
<a id="figure--fig:analog-filt-second-order-lpf-characteristics"></a>
{{< figure src="/ox-hugo/analog_filt_second_order_lpf_characteristics.png" caption="<span class=\"figure-number\">Figure 6: </span>Second Order Low Pass Filter - Filter transfer function and filter impedance" >}}
## Second Order High Pass Filter {#second-order-high-pass-filter} ## Second Order High Pass Filter {#second-order-high-pass-filter}
<a id="figure--fig:elec-passive-second-order-high-pass-filter"></a> <a id="figure--fig:analog-filt-second-order-hpf"></a>
{{< figure src="/ox-hugo/elec_passive_second_order_high_pass_filter.png" caption="<span class=\"figure-number\">Figure 4: </span>Second Order High Pass Filter using an RLC circuit" >}} {{< figure src="/ox-hugo/analog_filt_second_order_hpf.png" caption="<span class=\"figure-number\">Figure 7: </span>Second Order High Pass Filter using an RLC circuit" >}}
## Bibliography {#bibliography} ## Bibliography {#bibliography}

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+++
title = "Laplace Transform"
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Continuous Transfer Functions]({{< relref "continuous_transfer_functions.md" >}})
## Definition {#definition}
Laplace transform:
\begin{equation}
\boxed{F(s) = \mathcal{L}{f(t)} = \int\_0^\infty e^{-st} f(t) dt}
\end{equation}
## Laplace transform of signals {#laplace-transform-of-signals}
<https://en.wikipedia.org/wiki/List_of_Laplace_transforms>
## Laplace transform of functions {#laplace-transform-of-functions}
<https://en.wikibooks.org/wiki/Signals_and_Systems/Table_of_Laplace_Transforms>
## Solving Linear Differential equations {#solving-linear-differential-equations}
### Mechanical System {#mechanical-system}
### Electrical System {#electrical-system}

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content/zettels/piezoelectric_actuators.md
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@ -62,11 +62,13 @@ The Amplified Piezo Actuators principle is presented in (<a href="#citeproc_bib_
> 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 (<a href="#citeproc_bib_item_3">Lucinskis and Mangeot 2016</a>). A model of an amplified piezoelectric actuator is described in (<a href="#citeproc_bib_item_4">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>)).
<a id="figure--fig:ling16-topology-piezo-mechanism-types"></a> <a id="figure--fig:ling16-topology-piezo-mechanism-types"></a>
{{< 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>" >}} {{< 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" >}}
| Manufacturers | Country | | Manufacturers | Country |
|----------------------------------------------------------------------------------------------------|-----------| |----------------------------------------------------------------------------------------------------|-----------|
@ -208,5 +210,6 @@ Limitations of the electronics is discussed in [Design, modeling and control of
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body"> <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static &#38; Dynamic Applications.” <i>Ferroelectrics</i> 351 (1): 3–14. doi:<a href="https://doi.org/10.1080/00150190701351865">10.1080/00150190701351865</a>.</div> <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static &#38; Dynamic Applications.” <i>Ferroelectrics</i> 351 (1): 3–14. doi:<a href="https://doi.org/10.1080/00150190701351865">10.1080/00150190701351865</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div> <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
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</div> </div>

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: :
## 2 phase VS 5 phase stepper motor {#2-phase-vs-5-phase-stepper-motor} ## Errors between steps (micro-stepping) {#errors-between-steps--micro-stepping}
<https://www.orientalmotor.com/stepper-motors/technology/2-phase-vs-5-phase-stepper-motors.html> For a two phase stepper motor, there are (typically) **200 steps per revolution** (i.e. 1.8% per step).
Between each step, even when using some micro-stepping, there are some position errors that are due to non-perfect magnetic and electromagnetic fields.
## Errors {#errors} The period of this error is corresponding to 200 period/revolution.
For a two phase stepper motor, there are (typically) 200 steps per revolution. Then scanning, this can lead to high frequency vibrations.
Errors with a period of 200 period/revolution can be expected.
This is what is typically limiting the accuracy of the stepper motor (usually specified in between 3% and 5% of the step increment).
This is approximately corresponding to **1mrad**.
<div class="exampl">
Consider a stepper motor with 200 steps by turn attached to a ball-screw with a pitch of 1mm per turn.
A rotation of 1 turn per second will induce vibrations at 200Hz with an amplitude of \\(1\\,\mu m\\).
</div>
Note that this error is not a pure sine, it also has some harmonics with corresponding periods of 1/100 revolution and 1/50 revolution.
This error should repeat every turn and can be calibrated provided it is repeatable over time.
One way to reduce these errors is to use a ball-screw mechanism with a smaller pitch.
The price to pay is smaller velocity.
## Manufacturers {#manufacturers} ## Manufacturers {#manufacturers}
@ -25,3 +42,14 @@ Errors with a period of 200 period/revolution can be expected.
|--------------------------------------------------------------------------|----------------| |--------------------------------------------------------------------------|----------------|
| [AML](https://arunmicro.com/) | United Kingdom | | [AML](https://arunmicro.com/) | United Kingdom |
| [Sanyo](https://www.sanyodenki.com/catalogs/servo/stepping_systems.html) | | | [Sanyo](https://www.sanyodenki.com/catalogs/servo/stepping_systems.html) | |
## 2 phase VS 5 phase stepper motor {#2-phase-vs-5-phase-stepper-motor}
<https://www.orientalmotor.com/stepper-motors/technology/2-phase-vs-5-phase-stepper-motors.html>
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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title = "Z-Transform"
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Discrete Transfer Functions]({{< relref "discrete_transfer_functions.md" >}})
## From Continuous to Discrete transfer function {#from-continuous-to-discrete-transfer-function}
A continuous transfer function (i.e. in the Laplace domain) can easily be converted to the discrete time domain (i.e. in the z-domain) using the `c2d` Matlab command ([doc](https://fr.mathworks.com/help/control/ref/lti.c2d.html;jsessionid=206bd0fc8950c5f8cb6b568d7393#mw_53fc4689-2099-41d0-93b3-de1e51a174c1)).
Several methods can be used, each with some advantages and drawbacks, see [this document](https://fr.mathworks.com/help/control/ug/continuous-discrete-conversion-methods.html).

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