+++
title = "Design, modeling and control of nanopositioning systems"
author = ["Dehaeze Thomas"]
description = "Talks about various topics related to nano-positioning systems."
keywords = ["Control", "Metrology", "Flexible Joints"]
draft = false
+++

Tags
: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})

Reference
: (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>)

Author(s)
: Fleming, A. J., &amp; Leang, K. K.

Year
: 2014


## Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}

<a id="figure--fig:fleming14-amplifier-model"></a>

{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="<span class='figure-number'>Figure 1: </span>A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}

Consider the electrical circuit shown in [Figure 1](#figure--fig:fleming14-amplifier-model) where a voltage source is connected to a piezoelectric actuator.
The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.

<div class="exampl">

Typical inductance of standard RG-58 coaxial cable is \\(250 nH/m\\).
Typical value of \\(R\_s\\) is between \\(10\\) and \\(100 \Omega\\).

</div>

When considering the effects of both output impedance and cable inductance, the transfer function from source voltage \\(V\_s\\) to load voltage \\(V\_L\\) is second-order low pass filter:

\begin{equation}
  \frac{V\_L(s)}{V\_s(s)} = \frac{1}{\frac{s^2}{\omega\_r^2} + 2 \zeta \frac{s}{\omega\_r} + 1}
\end{equation}

with:

-   \\(\omega\_r = \frac{1}{\sqrt{L C\_p}}\\)
-   \\(\zeta = \frac{R\_s \sqrt{L C\_p}}{2 L}\\)


## Amplifier small-signal Bandwidth {#amplifier-small-signal-bandwidth}

The most obvious bandwidth limitation is the small-signal bandwidth of the amplifier.

If the inductance \\(L\\) is neglected, the transfer function from source voltage \\(V\_s\\) to load voltage \\(V\_L\\) forms a first order filter with a cut-off frequency

\begin{equation}
  \omega\_c = \frac{1}{R\_s C\_p}
\end{equation}

This is thus highly dependent of the load.

The high capacitive impedance nature of piezoelectric loads introduces phase-lag into the feedback path.
A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-off frequency of the pole formed by the amplifier output impedance \\(R\_s\\) and load capacitance \\(C\_p\\) (see [Table 1](#table--tab:piezo-limitation-Rs) for values).

<a id="table--tab:piezo-limitation-Rs"></a>
<div class="table-caption">
  <span class="table-number"><a href="#table--tab:piezo-limitation-Rs">Table 1</a>:</span>
  Bandwidth limitation due to \(R_s\)
</div>

|              | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
|--------------|-------------|-----------|------------|
| Rs = 1 Ohm   | 1.6 MHz     | 160 kHz   | 16 kHz     |
| Rs = 10 Ohm  | 160 kHz     | 16 kHz    | 1.6 kHz    |
| Rs = 100 Ohm | 16 kHz      | 1.6 kHz   | 160 Hz     |

The inductance \\(L\\) does also play a role in the amplifier bandwidth as it changes the resonance frequency.
Ideally, low inductance cables should be used.
It is however usually quite high compare to \\(\omega\_c\\) as shown in [Table 2](#table--tab:piezo-limitation-L).

<a id="table--tab:piezo-limitation-L"></a>
<div class="table-caption">
  <span class="table-number"><a href="#table--tab:piezo-limitation-L">Table 2</a>:</span>
  Bandwidth limitation due to \(R_s\)
</div>

|             | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
|-------------|-------------|-----------|------------|
| L = 25 nH   | 3.2 MHz     | 1 MHz     | 320 kHz    |
| L = 250 nH  | 1 MHz       | 320 kHz   | 100 kHz    |
| L = 2500 nH | 320 kHz     | 100 kHz   | 32 kHz     |


## Amplifier maximum slew rate {#amplifier-maximum-slew-rate}

Further bandwidth restrictions are imposed by the maximum **slew rate** of the amplifier.
This is the maximum rate at which the output voltage can change and is usually expressed in \\(V/\mu s\\).

For sinusoidal signals, the amplifiers slew rate must exceed:
\\[ SR\_{\text{sin}} > V\_{p-p} \pi f \\]
where \\(V\_{p-p}\\) is the peak to peak voltage and \\(f\\) is the frequency.

<div class="exampl">

If a 300kHz sine wave is to be reproduced with an amplitude of 10V, the required slew rate is \\(\approx 20 V/\mu s\\).

</div>

When dealing with capacitive loads, **the current limit is usually exceed well before the slew rate limit**.


## Current and Power Limitations {#current-and-power-limitations}

When driving the actuator off-resonance, the current delivered to a piezoelectric actuator is approximately:
\\[ I\_L(s) = V\_L(s) C\_p s \\]

For sinusoidal signals, the maximum positive and negative current is equal to:
\\[ I\_L^\text{max} = V\_{p-p} \pi f C\_p \\]

<a id="table--tab:piezo-required-current"></a>
<div class="table-caption">
  <span class="table-number"><a href="#table--tab:piezo-required-current">Table 3</a>:</span>
  Minimum current requirements for a 10V sinusoid
</div>

|             | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
|-------------|-------------|-----------|------------|
| f = 30 Hz   | 0.19 mA     | 1.9 mA    | 19 mA      |
| f = 3 kHz   | 19 mA       | 190 mA    | 1.9 A      |
| f = 300 kHz | 1.9 A       | 19 A      | 190 A      |


## Summary {#summary}

The bandwidth limitations  of standard piezoelectric drives were identified as:

-   High output impedance
-   The presence of a ple in the voltage-feedback loop due to output impedance and load capacitance
-   Insufficient current capacity due to power dissipation
-   High cable and connector inductance


## Bibliography {#bibliography}

<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>Fleming, A. J., and K. K. Leang. 2014. <i>Design, Modeling and Control of Nanopositioning Systems</i>. Advances in Industrial Control. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-06617-2">10.1007/978-3-319-06617-2</a>.</div>
</div>