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+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
+
+
+## Introduction to Nanotechnology {#introduction-to-nanotechnology}
+
+
+## Introduction to Nanopositioning {#introduction-to-nanopositioning}
+
+
+## Scanning Probe Microscopy {#scanning-probe-microscopy}
+
+
+## Challenges with Nanopositioning Systems {#challenges-with-nanopositioning-systems}
+
+
+### Hysteresis {#hysteresis}
+
+
+### Creep {#creep}
+
+
+### Thermal Drift {#thermal-drift}
+
+
+### Mechanical Resonance {#mechanical-resonance}
+
+
+## Control of Nanopositioning Systems {#control-of-nanopositioning-systems}
+
+
+### Feedback Control {#feedback-control}
+
+
+### Feedforward Control {#feedforward-control}
+
+
+## Book Summary {#book-summary}
+
+
+### Assumed Knowledge {#assumed-knowledge}
+
+
+### Content Summary {#content-summary}
+
+
+## References {#references}
+
+
+## The Piezoelectric Effect {#the-piezoelectric-effect}
+
+
+## Piezoelectric Compositions {#piezoelectric-compositions}
+
+
+## Manufacturing Piezoelectric Ceramics {#manufacturing-piezoelectric-ceramics}
+
+
+## Piezoelectric Transducers {#piezoelectric-transducers}
+
+
+## Application Considerations {#application-considerations}
+
+
+## Response of Piezoelectric Actuators {#response-of-piezoelectric-actuators}
+
+
+## Modeling Creep and Vibration in Piezoelectric Actuators {#modeling-creep-and-vibration-in-piezoelectric-actuators}
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Piezoelectric Tube Nanopositioners {#piezoelectric-tube-nanopositioners}
+
+
+### 63mm Piezoelectric Tube {#63mm-piezoelectric-tube}
+
+
+### 40mm Piezoelectric Tube Nanopositioner {#40mm-piezoelectric-tube-nanopositioner}
+
+
+## Piezoelectric Stack Nanopositioners {#piezoelectric-stack-nanopositioners}
+
+
+### Phyisk Instrumente P-734 Nanopositioner {#phyisk-instrumente-p-734-nanopositioner}
+
+
+### Phyisk Instrumente P-733.3DD Nanopositioner {#phyisk-instrumente-p-733-dot-3dd-nanopositioner}
+
+
+### Vertical Nanopositioners {#vertical-nanopositioners}
+
+
+### Rotational Nanopositioners {#rotational-nanopositioners}
+
+
+### Low Temperature and UHV Nanopositioners {#low-temperature-and-uhv-nanopositioners}
+
+
+### Tilting Nanopositioners {#tilting-nanopositioners}
+
+
+### Optical Objective Nanopositioners {#optical-objective-nanopositioners}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Operating Environment {#operating-environment}
+
+
+## Methods for Actuation {#methods-for-actuation}
+
+
+## Flexure Hinges {#flexure-hinges}
+
+
+### Introduction {#introduction}
+
+
+### Types of Flexures {#types-of-flexures}
+
+
+### Flexure Hinge Compliance Equations {#flexure-hinge-compliance-equations}
+
+
+### Stiff Out-of-Plane Flexure Designs {#stiff-out-of-plane-flexure-designs}
+
+
+### Failure Considerations {#failure-considerations}
+
+
+### Finite Element Approach for Flexure Design {#finite-element-approach-for-flexure-design}
+
+
+## Material Considerations {#material-considerations}
+
+
+### Materials for Flexure and Platform Design {#materials-for-flexure-and-platform-design}
+
+
+### Thermal Stability of Materials {#thermal-stability-of-materials}
+
+
+## Manufacturing Techniques {#manufacturing-techniques}
+
+
+## Design Example: A High-Speed Serial-Kinematic Nanopositioner {#design-example-a-high-speed-serial-kinematic-nanopositioner}
+
+
+### State-of-the-Art Designs {#state-of-the-art-designs}
+
+
+### Tradeoffs and Limitations in Speed {#tradeoffs-and-limitations-in-speed}
+
+
+### Serial- Versus Parallel-Kinematic Configurations {#serial-versus-parallel-kinematic-configurations}
+
+
+### Piezoactuator Considerations {#piezoactuator-considerations}
+
+
+### Preloading Piezo-Stack Actuators {#preloading-piezo-stack-actuators}
+
+
+### Flexure Design for Lateral Positioning {#flexure-design-for-lateral-positioning}
+
+
+### Design of Vertical Stage {#design-of-vertical-stage}
+
+
+### Fabrication and Assembly {#fabrication-and-assembly}
+
+
+### Drive Electronics {#drive-electronics}
+
+\*\*\*\*0 Experimental Results
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Sensor Characteristics {#sensor-characteristics}
+
+
+### Calibration and Nonlinearity {#calibration-and-nonlinearity}
+
+
+### Drift and Stability {#drift-and-stability}
+
+
+### Bandwidth {#bandwidth}
+
+
+### Noise {#noise}
+
+
+### Resolution {#resolution}
+
+
+### Combining Errors {#combining-errors}
+
+
+### Metrological Traceability {#metrological-traceability}
+
+
+## Nanometer Position Sensors {#nanometer-position-sensors}
+
+
+### Resistive Strain Sensors {#resistive-strain-sensors}
+
+
+### Piezoresistive Strain Sensors {#piezoresistive-strain-sensors}
+
+
+### Piezoelectric Strain Sensors {#piezoelectric-strain-sensors}
+
+
+### Capacitive Sensors {#capacitive-sensors}
+
+
+### MEMs Capacitive and Thermal Sensors {#mems-capacitive-and-thermal-sensors}
+
+
+### Eddy-Current Sensors {#eddy-current-sensors}
+
+
+### Linear Variable Displacement Transformers {#linear-variable-displacement-transformers}
+
+
+### Laser Interferometers {#laser-interferometers}
+
+
+### Linear Encoders {#linear-encoders}
+
+
+## Comparison and Summary {#comparison-and-summary}
+
+
+## Outlook and Future Requirements {#outlook-and-future-requirements}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Shunt Circuit Modeling {#shunt-circuit-modeling}
+
+
+### Open-Loop {#open-loop}
+
+
+### Shunt Damping {#shunt-damping}
+
+
+## Implementation {#implementation}
+
+
+## Experimental Results {#experimental-results}
+
+
+### Tube Dynamics {#tube-dynamics}
+
+
+### Amplifier Performance {#amplifier-performance}
+
+
+### Shunt Damping Performance {#shunt-damping-performance}
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Experimental Setup {#experimental-setup}
+
+
+## PI Control {#pi-control}
+
+
+## PI Control with Notch Filters {#pi-control-with-notch-filters}
+
+
+## PI Control with IRC Damping {#pi-control-with-irc-damping}
+
+
+## Performance Comparison {#performance-comparison}
+
+
+## Noise and Resolution {#noise-and-resolution}
+
+
+## Analog Implementation {#analog-implementation}
+
+
+## Application to AFM Imaging {#application-to-afm-imaging}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Modeling {#modeling}
+
+
+### Actuator Dynamics {#actuator-dynamics}
+
+
+### Sensor Dynamics {#sensor-dynamics}
+
+
+### Sensor Noise {#sensor-noise}
+
+
+### Mechanical Dynamics {#mechanical-dynamics}
+
+
+### System Properties {#system-properties}
+
+
+### Example System {#example-system}
+
+
+## Damping Control {#damping-control}
+
+
+## Tracking Control {#tracking-control}
+
+
+### Relationship Between Force and Displacement {#relationship-between-force-and-displacement}
+
+
+### Integral Displacement Feedback {#integral-displacement-feedback}
+
+
+### Direct Tracking Control {#direct-tracking-control}
+
+
+### Dual Sensor Feedback {#dual-sensor-feedback}
+
+
+### Low Frequency Bypass {#low-frequency-bypass}
+
+
+### Feedforward Inputs {#feedforward-inputs}
+
+
+### Higher-Order Modes {#higher-order-modes}
+
+
+## Experimental Results {#experimental-results}
+
+
+### Experimental Nanopositioner {#experimental-nanopositioner}
+
+
+### Actuators and Force Sensors {#actuators-and-force-sensors}
+
+
+### Control Design {#control-design}
+
+
+### Noise Performance {#noise-performance}
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Why Feedforward? {#why-feedforward}
+
+
+## Modeling for Feedforward Control {#modeling-for-feedforward-control}
+
+
+## Feedforward Control of Dynamics and Hysteresis {#feedforward-control-of-dynamics-and-hysteresis}
+
+
+### Simple DC-Gain Feedforward Control {#simple-dc-gain-feedforward-control}
+
+
+### An Inversion-Based Feedforward Approach  for Linear Dynamics {#an-inversion-based-feedforward-approach-for-linear-dynamics}
+
+
+### Frequency-Weighted Inversion: The Optimal Inverse {#frequency-weighted-inversion-the-optimal-inverse}
+
+
+### Application to AFM Imaging {#application-to-afm-imaging}
+
+
+## Feedforward and Feedback Control {#feedforward-and-feedback-control}
+
+
+### Application to AFM Imaging {#application-to-afm-imaging}
+
+
+## Iterative Feedforward Control {#iterative-feedforward-control}
+
+
+### The ILC Problem {#the-ilc-problem}
+
+
+### Model-Based ILC {#model-based-ilc}
+
+
+### Nonlinear ILC {#nonlinear-ilc}
+
+
+### Conclusions {#conclusions}
+
+
+## References {#references}
+
+
+## 10.1 Introduction {#10-dot-1-introduction}
+
+
+### 10.1.1 Background {#10-dot-1-dot-1-background}
+
+
+### 10.1.2 The Optimal Periodic Input {#10-dot-1-dot-2-the-optimal-periodic-input}
+
+
+## 10.2 Signal Optimization {#10-dot-2-signal-optimization}
+
+
+## 10.3 Frequency Domain Cost Functions {#10-dot-3-frequency-domain-cost-functions}
+
+
+### 10.3.1 Background: Discrete Fourier Series {#10-dot-3-dot-1-background-discrete-fourier-series}
+
+
+### 10.3.2 Minimizing Signal Power {#10-dot-3-dot-2-minimizing-signal-power}
+
+
+### 10.3.3 Minimizing Frequency Weighted Power {#10-dot-3-dot-3-minimizing-frequency-weighted-power}
+
+
+### 10.3.4 Minimizing Velocity and Acceleration {#10-dot-3-dot-4-minimizing-velocity-and-acceleration}
+
+
+### 10.3.5 Single-Sided Frequency Domain Calculations {#10-dot-3-dot-5-single-sided-frequency-domain-calculations}
+
+
+## 10.4 Time Domain Cost Function {#10-dot-4-time-domain-cost-function}
+
+
+### 10.4.1 Minimum Velocity {#10-dot-4-dot-1-minimum-velocity}
+
+
+### 10.4.2 Minimum Acceleration {#10-dot-4-dot-2-minimum-acceleration}
+
+
+### 10.4.3 Frequency Weighted Objectives {#10-dot-4-dot-3-frequency-weighted-objectives}
+
+
+## 10.5 Application to Scan Generation {#10-dot-5-application-to-scan-generation}
+
+
+### 10.5.1 Choosing β and K {#10-dot-5-dot-1-choosing-β-and-k}
+
+
+### 10.5.2 Improving Feedback and Feedforward Controllers {#10-dot-5-dot-2-improving-feedback-and-feedforward-controllers}
+
+
+## 10.6 Comparison to Other Techniques {#10-dot-6-comparison-to-other-techniques}
+
+
+## 10.7 Experimental Application {#10-dot-7-experimental-application}
+
+
+## 10.8 Chapter Summary {#10-dot-8-chapter-summary}
+
+
+## References {#references}
+
+
+## 11.1 Introduction {#11-dot-1-introduction}
+
+
+## 11.2 Modeling Hysteresis {#11-dot-2-modeling-hysteresis}
+
+
+### 11.2.1 Simple Polynomial Model {#11-dot-2-dot-1-simple-polynomial-model}
+
+
+### 11.2.2 Maxwell Slip Model {#11-dot-2-dot-2-maxwell-slip-model}
+
+
+### 11.2.3 Duhem Model {#11-dot-2-dot-3-duhem-model}
+
+
+### 11.2.4 Preisach Model {#11-dot-2-dot-4-preisach-model}
+
+
+### 11.2.5 Classical Prandlt-Ishlinksii Model {#11-dot-2-dot-5-classical-prandlt-ishlinksii-model}
+
+
+## 11.3 Feedforward Hysteresis Compensation {#11-dot-3-feedforward-hysteresis-compensation}
+
+
+### 11.3.1 Feedforward Control Using the Presiach Model {#11-dot-3-dot-1-feedforward-control-using-the-presiach-model}
+
+
+### 11.3.2 Feedforward Control Using the Prandlt-Ishlinksii Model {#11-dot-3-dot-2-feedforward-control-using-the-prandlt-ishlinksii-model}
+
+
+## 11.4 Chapter Summary {#11-dot-4-chapter-summary}
+
+
+## References {#references}
+
+
+## 12.1 Introduction {#12-dot-1-introduction}
+
+
+## 12.2 Charge Drives {#12-dot-2-charge-drives}
+
+
+## 12.3 Application to Piezoelectric Stack Nanopositioners {#12-dot-3-application-to-piezoelectric-stack-nanopositioners}
+
+
+## 12.4 Application to Piezoelectric Tube Nanopositioners {#12-dot-4-application-to-piezoelectric-tube-nanopositioners}
+
+
+## 12.5 Alternative Electrode Configurations {#12-dot-5-alternative-electrode-configurations}
+
+
+### 12.5.1 Grounded Internal Electrode {#12-dot-5-dot-1-grounded-internal-electrode}
+
+
+### 12.5.2 Quartered Internal Electrode {#12-dot-5-dot-2-quartered-internal-electrode}
+
+
+## 12.6 Charge Versus Voltage {#12-dot-6-charge-versus-voltage}
+
+
+### 12.6.1 Advantages {#12-dot-6-dot-1-advantages}
+
+
+### 12.6.2 Disadvantages {#12-dot-6-dot-2-disadvantages}
+
+
+## 12.7 Impact on Closed-Loop Control {#12-dot-7-impact-on-closed-loop-control}
+
+
+## 12.8 Chapter Summary {#12-dot-8-chapter-summary}
+
+
+## References {#references}
+
+
+## 13.1 Introduction {#13-dot-1-introduction}
+
+
+## 13.2 Review of Random Processes {#13-dot-2-review-of-random-processes}
+
+
+### 13.2.1 Probability Distributions {#13-dot-2-dot-1-probability-distributions}
+
+
+### 13.2.2 Expected Value, Moments, Variance, and RMS {#13-dot-2-dot-2-expected-value-moments-variance-and-rms}
+
+
+### 13.2.3 Gaussian Random Variables {#13-dot-2-dot-3-gaussian-random-variables}
+
+
+### 13.2.4 Continuous Random Processes {#13-dot-2-dot-4-continuous-random-processes}
+
+
+### 13.2.5 Joint Density Functions and Stationarity {#13-dot-2-dot-5-joint-density-functions-and-stationarity}
+
+
+### 13.2.6 Correlation Functions {#13-dot-2-dot-6-correlation-functions}
+
+
+### 13.2.7 Gaussian Random Processes {#13-dot-2-dot-7-gaussian-random-processes}
+
+
+### 13.2.8 Power Spectral Density {#13-dot-2-dot-8-power-spectral-density}
+
+
+### 13.2.9 Filtered Random Processes {#13-dot-2-dot-9-filtered-random-processes}
+
+
+### 13.2.10 White Noise {#13-dot-2-dot-10-white-noise}
+
+
+### 13.2.11 Spectral Density in V/sqrtHz {#13-dot-2-dot-11-spectral-density-in-v-sqrthz}
+
+
+### 13.2.12 Single- and Double-Sided Spectra {#13-dot-2-dot-12-single-and-double-sided-spectra}
+
+
+## 13.3 Resolution and Noise {#13-dot-3-resolution-and-noise}
+
+
+## 13.4 Sources of Nanopositioning Noise {#13-dot-4-sources-of-nanopositioning-noise}
+
+
+### 13.4.1 Sensor Noise {#13-dot-4-dot-1-sensor-noise}
+
+
+### 13.4.2 External Noise {#13-dot-4-dot-2-external-noise}
+
+
+### 13.4.3 Amplifier Noise {#13-dot-4-dot-3-amplifier-noise}
+
+
+## 13.5 Closed-Loop Position Noise {#13-dot-5-closed-loop-position-noise}
+
+
+### 13.5.1 Noise Sensitivity Functions {#13-dot-5-dot-1-noise-sensitivity-functions}
+
+
+### 13.5.2 Closed-Loop Position Noise Spectral Density {#13-dot-5-dot-2-closed-loop-position-noise-spectral-density}
+
+
+### 13.5.3 Closed-Loop Noise Approximations with Integral Control {#13-dot-5-dot-3-closed-loop-noise-approximations-with-integral-control}
+
+
+### 13.5.4 Closed-Loop Position Noise Variance {#13-dot-5-dot-4-closed-loop-position-noise-variance}
+
+
+### 13.5.5 A Note on Units {#13-dot-5-dot-5-a-note-on-units}
+
+
+## 13.6 Simulation Examples {#13-dot-6-simulation-examples}
+
+
+### 13.6.1 Integral Controller Noise Simulation {#13-dot-6-dot-1-integral-controller-noise-simulation}
+
+
+### 13.6.2 Noise Simulation with Inverse Model Controller {#13-dot-6-dot-2-noise-simulation-with-inverse-model-controller}
+
+
+### 13.6.3 Feedback Versus Feedforward Control {#13-dot-6-dot-3-feedback-versus-feedforward-control}
+
+
+## 13.7 Practical Frequency Domain Noise Measurements {#13-dot-7-practical-frequency-domain-noise-measurements}
+
+
+### 13.7.1 Preamplification {#13-dot-7-dot-1-preamplification}
+
+
+### 13.7.2 Spectrum Estimation {#13-dot-7-dot-2-spectrum-estimation}
+
+
+### 13.7.3 Direct Measurement of Position Noise {#13-dot-7-dot-3-direct-measurement-of-position-noise}
+
+
+### 13.7.4 Measurement of the External Disturbance {#13-dot-7-dot-4-measurement-of-the-external-disturbance}
+
+
+## 13.8 Experimental Demonstration {#13-dot-8-experimental-demonstration}
+
+
+## 13.9 Time-Domain Noise Measurements {#13-dot-9-time-domain-noise-measurements}
+
+
+### 13.9.1 Total Integrated Noise {#13-dot-9-dot-1-total-integrated-noise}
+
+
+### 13.9.2 Estimating the Position Noise {#13-dot-9-dot-2-estimating-the-position-noise}
+
+
+### 13.9.3 Practical Considerations {#13-dot-9-dot-3-practical-considerations}
+
+
+### 13.9.4 Experimental Demonstration {#13-dot-9-dot-4-experimental-demonstration}
+
+
+## 13.10 A Simple Method for Measuring the Resolution  of Nanopositioning Systems {#13-dot-10-a-simple-method-for-measuring-the-resolution-of-nanopositioning-systems}
+
+
+## 13.11 Techniques for Improving Resolution {#13-dot-11-techniques-for-improving-resolution}
+
+
+## 13.12 Chapter Summary {#13-dot-12-chapter-summary}
+
+
+## References {#references}
+
+
+## Electrical Considerations {#electrical-considerations}
+
+
+### 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 \xi \frac{s}{\omega\_r} + 1}
+\end{equation}
+
+with:
+
+-   \\(\omega\_r = \frac{1}{\sqrt{L C\_p}}\\)
+-   \\(\xi = \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      |
+
+
+### Chapter Summary {#chapter-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
+
+
+### References {#references}
+
+
+## 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>