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content/article/fleming10_nanop_system_with_force_feedb.md

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+title = "Nanopositioning system with force feedback for high-performance tracking and vibration control"
+author = ["Thomas Dehaeze"]
+draft = false
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+
+Tags
+: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Force Sensors]({{< relref "force_sensors" >}})
+
+Reference
+: <sup id="c823f68dd2a72b9667a61b3c046b4731"><a class="reference-link" href="#fleming10_nanop_system_with_force_feedb" title="Fleming, Nanopositioning System With Force Feedback for  High-Performance Tracking and Vibration Control, {IEEE/ASME Transactions on Mechatronics}, v(3), 433-447 (2010).">(Fleming, 2010)</a></sup>
+
+Author(s)
+: Fleming, A.
+
+Year
+: 2010
+
+Summary:
+
+-   The noise generated by a piezoelectric force sensor is much less than a capacitive sensor
+-   Dynamical model of a piezoelectric stack actuator and piezoelectric force sensor
+-   Noise of a piezoelectric force sensor
+-   IFF with a piezoelectric stack actuator and piezoelectric force sensor
+-   A force sensor is used as a displacement sensor below the frequency of the first zero
+-   Sensor fusion architecture with a capacitive sensor and a force sensor and using complementary filters
+-   Virtual sensor fusion architecture (called low-frequency bypass)
+-   Analog implementation of the control strategies to avoid quantization noise, finite resolution and sampling delay
+
+
+## Model of a multi-layer monolithic piezoelectric stack actuator {#model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator}
+
+<a id="org3f4c96b"></a>
+
+{{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="Figure 1: Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
+
+The actuator experiences an internal stress in response to an applied voltage.
+This stress is represented by the voltage dependent force \\(F\_a\\) and is related to free displacement by
+\\[ \Delta L = \frac{F\_a}{k\_a} \\]
+
+-   \\(\Delta L\\) is the change in actuator length in [m]
+-   \\(k\_a\\) is the actuator stiffness in [N/m]
+
+The developed force \\(F\_a\\) is related to the applied voltage by:
+\\[ \Delta L = d\_{33} n V\_a \\]
+
+-   \\(d\_{33}\\) is the piezoelectric strain constant in [m/V]
+-   \\(n\\) is the number of layers
+-   \\(V\_a\\) is the applied voltage in [V]
+
+Combining the two equations, we obtain:
+\\[ F\_a = d\_{33} n k\_a V\_a \\]
+
+The ratio of the developed force to applied voltage is \\(d\_{33} n k\_a\\) in [N/V].
+We denote this constant by \\(g\_a\\) and:
+\\[ F\_a = g\_a V\_a, \quad g\_a = d\_{33} n k\_a \\]
+
+
+## Dynamics of a piezoelectric force sensor {#dynamics-of-a-piezoelectric-force-sensor}
+
+Piezoelectric force sensors provide a high sensitivity and bandwidth with low noise at high frequencies.
+
+If a **single wafer** of piezoelectric material is sandwiched between the actuator and platform:
+\\[ D = d\_{33} T \\]
+
+-   \\(D\\) is the amount of generated charge per unit area in \\([C/m^2]\\)
+-   \\(T\\) is the stress in \\([N/m^2]\\)
+-   \\(d\_{33}\\) is the piezoelectric strain constant in \\([m/V] = [C/N]\\)
+
+The generated charge is then
+\\[ q = d\_{33} F\_s \\]
+
+If an **n-layer** piezoelectric transducer is used as a force sensor, the generated charge is then:
+\\[ q = n d\_{33} F\_s \\]
+
+---
+
+We can use a **charge amplifier** to measure the force \\(F\_s\\).
+
+{{< figure src="/ox-hugo/fleming10_charge_ampl_piezo.png" caption="Figure 2: Electrical model of a piezoelectric force sensor is shown in gray. Developed charge \\(q\\) is proportional to the strain and hence the force experienced by the sensor. Op-amp charge amplifier produces an output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}}
+
+The output voltage \\(V\_s\\) is equal to
+\\[ V\_s = -\frac{q}{C\_s} = -\frac{n d\_{33}F\_s}{C\_s} \\]
+that is, the scaling between the force and voltage is \\(-\frac{n d\_{33}F\_s}{C\_s}\ [V/N]\\) .
+
+---
+
+We can also use a voltage amplifier.
+In that case, the generated charge is deposited on the transducer's internal capacitance.
+
+The open-circuit voltage of a piezoelectric force sensor is:
+\\[ V\_s = \frac{n d\_{33} F\_s}{C} \\]
+
+-   \\(C\\) is the transducer capacitance defined by \\(C = n \epsilon\_T A / h\\) in [F]
+    -   \\(A\\) is the area in \\([m^2]\\)
+    -   \\(h\\) is the layer thickness in [m]
+    -   \\(\epsilon\_T\\) is the dielectric permittivity under a constant stress in \\([F/m]\\)
+
+We obtain
+\\[ V\_s = g\_s F\_s, \quad g\_s = \frac{n d\_{33}}{C} \\]
+
+
+## Noise of a piezoelectric force sensor {#noise-of-a-piezoelectric-force-sensor}
+
+As piezoelectric sensors have a capacitive source impedance, the sensor noise density \\(N\_{V\_s}(\omega)\\) is primarily due to current noise \\(i\_n\\) reacting the capacitive source impedance:
+\\[ N\_{V\_s}(\omega) = i\_n \frac{1}{C \omega} \\]
+
+-   \\(N\_{V\_s}\\) is the measured noise in \\(V/\sqrt{\text{Hz}}\\)
+-   \\(i\_n\\) is the current noise in \\(A/\sqrt{\text{Hz}}\\)
+-   \\(C\\) is the capacitance of the piezoelectric in \\(F\\)
+
+The current noise density of a general purpose LM833 FET-input op-amp is \\(0.5\ pA/\sqrt{\text{Hz}}\\).
+The capacitance of a piezoelectric stack is typically between \\(1 \mu F\\) and \\(100 \mu F\\).
+
+# Bibliography
+<a class="bibtex-entry" id="fleming10_nanop_system_with_force_feedb">Fleming, A., *Nanopositioning system with force feedback for high-performance tracking and vibration control*, IEEE/ASME Transactions on Mechatronics, *15(3)*, 433–447 (2010).  http://dx.doi.org/10.1109/tmech.2009.2028422</a> [↩](#c823f68dd2a72b9667a61b3c046b4731)
+
+
+## Backlinks {#backlinks}
+
+-   [Actuators]({{< relref "actuators" >}})
+-   [Force Sensors]({{< relref "force_sensors" >}})