hugo/content/book/fleming14_desig_model_contr_nanop_system.md

+++ 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

(Fleming and Leang 2014)

Author(s)

Fleming, A. J., & Leang, K. K.

Year

2014

Amplifier and Piezo electrical models

{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: 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 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.

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

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

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 for values).

Table 1: Bandwidth limitation due to \(R_s\)

	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 2: Bandwidth limitation due to \(R_s\)

	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

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.

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

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

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: \[ I_L^\text{max} = V_{p-p} \pi f C_p \]

Table 3: Minimum current requirements for a 10V sinusoid

	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

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

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Fleming, A. J., and K. K. Leang. 2014. Design, Modeling and Control of Nanopositioning Systems. Advances in Industrial Control. Springer International Publishing. doi:10.1007/978-3-319-06617-2.