67 lines
1.7 KiB
Markdown
Executable File
67 lines
1.7 KiB
Markdown
Executable File
---
|
|
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.
|
|
|
|
|
|
|
|
\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:
|
|
|
|
|
|
|
|
\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:
|
|
|
|
|