add notes on piezoelectrics

uni/mmme/2051_electromechanical_devices/piezoelectrics.md

@@ -0,0 +1,66 @@
+---
+author: Akbar Rahman
+date: \today
+title: MMME2051 // Piezoelectrics
+tags: [ piezoelectrics, op_amps ]
+uuid: ed7d0899-478d-4f0d-b0e9-634cdbb5b48a
+lecture_slides: [ ./lecture_slides/MMME2051EMD_Lecture7.pdf ]
+lecture_notes: []
+exercise_sheets: [ ./seminar_worksheets/MMME2051_Lec7_Top1_Quiz.pdf, ./seminar_worksheets/StrainGaugeHomework.pdf ]
+---
+
+Piezoelectricity is the charge that gets accumulated in some materials upon application of
+mechanical stress
+
+$$Q \propto F$$
+
+This relation allows the measurement of force using electric signals.
+
+![](./images/vimscrot-2023-03-16T11:15:41,326771312+00:00.png)
+
+\begin{align*}
+Q &\propto F \\
+Q &= k_1F \\
+  &= k_1Ma \\
+\frac{\mathrm d Q}{\mathrm dt} &= i = k_1M\frac{\mathrm da}{\mathrm dt}
+\end{align*}
+
+# Integrating Amplifier
+
+Measuring current is expensive and difficult.
+Integrating the current helps to measure a voltage instead, which is easier.
+This is done using the following amplifier:
+
+![](./images/vimscrot-2023-03-16T11:22:04,554599428+00:00.png)
+
+\begin{align*}
+V_\text{out} &= A_{OL}(V^+-V__) = -A_{OL}V__ \\
+V__ &= V_\text{out} - V_C
+\end{align*}
+
+As input resistance of op amp is infinite:
+
+$$i_f = -i_n = -k_1M\frac{\mathrm da}{\mathrm dt}$$
+
+From the capacitor equation:
+
+$$i_f = C_f \frac{\mathrm dV_C}{\mathrm dt} = -k_1M\frac{\mathrm da}{\mathrm dt}$$
+
+Integrating both sides gives
+
+$$V_C = -\frac{k_1M}{C_f}a$$
+
+And it can be found that
+
+\begin{align*}
+V_\text{out} &= -A_{OL}(V_\text{out} - V_C) \\
+V_C &= -V\text{out} \frac{1+A_\text{OL}}{A_{OL}}
+\end{align*}
+
+To get
+
+$$V_\text{out} = \frac{k_1M}{C_f}a$$
+
+This circuit can be stacked to get velocity and displacement:
+
+![](./images/vimscrot-2023-03-16T11:28:48,428685773+00:00.png)