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content/book/fleming14_desig_model_contr_nanop_system.md
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Reference
: ([Fleming and Leang 2014](#org2385d08))
: ([Fleming and Leang 2014](#org611ad6b))
Author(s)
: Fleming, A. J., & Leang, K. K.
@@ -816,46 +816,131 @@ Year
### References {#references}
## 14 Electrical Considerations {#14-electrical-considerations}
## Electrical Considerations {#electrical-considerations}
### 14.1 Introduction {#14-dot-1-introduction}
### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
<a id="org393f35b"></a>
{{< 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](#org393f35b) 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="examp">
<div></div>
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}\\)
### 14.2 Bandwidth Limitations {#14-dot-2-bandwidth-limitations}
### 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 |
#### 14.2.1 Passive Bandwidth Limitations {#14-dot-2-dot-1-passive-bandwidth-limitations}
### 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="examp">
<div></div>
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**.
#### 14.2.2 Amplifier Bandwidth {#14-dot-2-dot-2-amplifier-bandwidth}
### 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 |
#### 14.2.3 Current and Power Limitations {#14-dot-2-dot-3-current-and-power-limitations}
### Chapter Summary {#chapter-summary}
The bandwidth limitations of standard piezoelectric drives were identified as:
### 14.3 Dual-Amplifier {#14-dot-3-dual-amplifier}
#### 14.3.1 Circuit Operation {#14-dot-3-dot-1-circuit-operation}
#### 14.3.2 Range Considerations {#14-dot-3-dot-2-range-considerations}
### 14.4 Electrical Design {#14-dot-4-electrical-design}
#### 14.4.1 High-Voltage Stage {#14-dot-4-dot-1-high-voltage-stage}
#### 14.4.2 Low-Voltage Stage {#14-dot-4-dot-2-low-voltage-stage}
#### 14.4.3 Cabling and Interconnects {#14-dot-4-dot-3-cabling-and-interconnects}
### 14.5 Chapter Summary {#14-dot-5-chapter-summary}
- 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}
@@ -863,4 +948,4 @@ Year
## Bibliography {#bibliography}
<a id="org2385d08"></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="org611ad6b"></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>.