---
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)