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Update Content - 2024-12-17

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Thomas Dehaeze
2024-12-17 15:37:17 +01:00
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content/book/fleming14_desig_model_contr_nanop_system.md
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@@ -732,7 +732,7 @@ Year
{{< 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.
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.
@@ -768,11 +768,11 @@ If the inductance \\(L\\) is neglected, the transfer function from source voltag
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 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>:
<span class="table-number"><a href="#table--tab:piezo-limitation-Rs">Table 1</a>:</span>
Bandwidth limitation due to \(R_s\)
</div>
@@ -784,11 +784,11 @@ A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-of
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).
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>:
<span class="table-number"><a href="#table--tab:piezo-limitation-L">Table 2</a>:</span>
Bandwidth limitation due to \(R_s\)
</div>
@@ -827,7 +827,7 @@ For sinusoidal signals, the maximum positive and negative current is equal to:
<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>:
<span class="table-number"><a href="#table--tab:piezo-required-current">Table 3</a>:</span>
Minimum current requirements for a 10V sinusoid
</div>