SaM - MEMORIZE (Piezoelectricity)

List of things to memorize:

SaM - Piezoelectricity

• SaM - Piezoelectric Effect
• SaM - Piezoelectric Effect in Details • Direct and Inverse Piezoelectric Effect
• SaM - Complete Piezoelectric Device (Actuator + Sensor)
• SaM - Compact Piezoelectric Coefficient Matrix for PZT • Dependence of Piezoelectricity on Temperature • Curie Temperature
• SaM - Piezoelectric Accelerometer
• SaM - Ultrasonic Transducers (Piezoelectric Device)
  • SaM - Coarse Approximation of the Lumped Parameter Model of an Ultrasonic Transducer
• SaM - Quartz Oscillator
• SaM - AT-Cut Quartz
  • SaM - Crystallographic Axis
  • SaM - Types of Cut Quartzes
• SaM - Quartz in a 3 Point Oscillator
  • SaM - Behavior of the Quartz Ocillator at High Frequencies
• SaM - Quartz as a Sensor • QCM (Quartz Crystal Microbalance)

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SaM - Piezoelectric Effect

• Piezoelectric effect formula:$\{\begin{array}{lc}\vec{D}=\overline{\overline{{\epsilon }^T}}\vec{E}+\overline{\overline{d}}\overline{\overline{T}}& \text{(direct piezoelectric effect)}\\ \overline{\overline{S}}=\overline{\overline{s}}\overline{\overline{T}}+\overline{\overline{d^T}}\vec{E}& \text{(inverse piezoelectric effect)}\end{array}$
• Terminology:
• $\vec{D}$, $\overline{\overline{\epsilon }}$, $\vec{E}$ : displacement of the electric field, electric permittitivity matrix, electric field.
• $\overline{\overline{T}}$, $\overline{\overline{S}}$ : stres and strain
• $\overline{\overline{d}}$ : piezoelectric coefficient matrix.
• $\overline{\overline{s}}$ : inverse piezoelectric coefficient matrix, or yieldness coefficient matrix.

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SaM - Piezoelectric Effect in Details • Direct and Inverse Piezoelectric Effect

• Piezoelectric effect formula:$\{\begin{array}{lc}\vec{D}=\overline{\overline{{\epsilon }^T}}\vec{E}+\overline{\overline{d}}\overline{\overline{T}}& \text{(direct piezoelectric effect)}\\ \overline{\overline{S}}=\overline{\overline{s}}\overline{\overline{T}}+\overline{\overline{d^T}}\vec{E}& \text{(inverse piezoelectric effect)}\end{array}$Inverse piezoelectric effect : even if there is no stress applied to a piezoelectric material, but it is present an external electrical field ⇒ then we will see a deformation, or strain of the material itself.
• Different material with piezoelectric property:
  • Quartz $(Si{O}_2)$ : small losses, almost a perfect dielectic, ${\epsilon }_r^{\prime }\in [4÷5]$, ${\epsilon }_r^{\prime \prime }=0.001$.
  • PZT (Lead Zirconate Titanate, polycrystal) : large piezo-coefficients $(d_{33}=500\frac{\text{pC}}{\text{N}})$
  • PVDF (polymer) : low cost
• Acutally:NOT_SURE_ABOUT_THIS (this is my reasoning)
  • The quartz has: ${\epsilon }_r^{\prime }\approx 4.0,{\epsilon }_r^{\prime \prime }=0.001$ and $d_{12}\approx 2\frac{\text{pC}}{\text{N}}$.
  • The PZT has: ${\epsilon }_r^{\prime }\in [500÷2000],{\epsilon }_r^{\prime \prime }\in [0.01÷0.05]$ and $d_{12}\approx 500\frac{\text{pC}}{\text{N}}$.
  • Their complex electric permettivity ratio is almost the same $\frac{{\epsilon }_r^{\prime }}{{\epsilon }_r^{\prime \prime }}\sim 10^{-3}$ but the PZT has higher $d$ value, meaning if we take into the direct and inverse piezoelectric effect formual, it will result in a higer response given the same electric field $E$ and the same stress value $T$.
    The same is true for the capacitance value $C_E=\epsilon \frac{A}{h}$ and charge output $Q=d\cdot F$, the PZT will have higher value for both.
    ==So for “raw performance” the PZT is better==.
  • However we have studied how we can vary some characteristics of the quartz, if we cut it in a different way.
    Also we have seen how the AT-cut quartz has a linear relationship between its thickness $(t)$ and the wavelength $({\lambda }_{\omega })$: $t\approx \frac{{\lambda }_{\omega }}{2}$.
    So the quartz give us "more flexibility".

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SaM - Complete Piezoelectric Device (Actuator + Sensor)

• Actuator structure:
  

  • A slice of piezo-material in between of two slices of conductive material, like a capacitance.

• Direct piezoelectric effect formula with no external mechanical forces:$\overline{\overline{S}}=\overline{\overline{d}}\vec{E}$

• Sensor structure:
  

  • Acts again as a capacitance.

• Charge enclosed in the volume of the sensor:$Q=d\cdot F$

• Equivalent lumped parameter system of the sensor:
  

• Complete equivalent lumped parameter system:

• Different material with piezoelectric property:

  • Quartz $(Si{O}_2)$ : small losses, almost a perfect dielectic, ${\epsilon }_r^{\prime }\in [4÷5]$, ${\epsilon }_r^{\prime \prime }=0.001$.
  • PZT (Lead Zirconate Titanate, polycrystal) : large piezo-coefficients $(d_{33}=500\frac{\text{pC}}{\text{N}})$
  • PVDF (polymer) : low cost

• Terminology:

  • $\overline{\overline{d}}$, $d$ : piezoelectric coefficient.
  • $F=k\cdot x$ : force, elastic coefficient and displacement.

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SaM - Compact Piezoelectric Coefficient Matrix for PZT • Dependence of Piezoelectricity on Temperature • Curie Temperature

• Piezoelectric coefficient matrix in compact notation for PZT:${\overline{\overline{d}}}_{\text{PZT}}=\left[\begin{array}{lccccc}0& 0& 0& 0& d_{15}& 0\\ 0& 0& 0& d_{24}& 0& 0\\ d_{31}& d_{32}& d_{33}& 0& 0& 0\end{array}\right]$
• ==Piezoelectric materials have to be operated under the Curie temperature==.
• Curie Temperature: is the temperature at which, usually, there is a phase transition from an asymmetrical to a symmetrical crystal.
• For PZT this temperature is $T_C=250°\text{C}$.

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SaM - Piezoelectric Accelerometer

• Structure:
  
• Equivalent Circuit:
  
• Output formula:$V(s)=\frac{s{R}_{T}kd}{1+s{R}_T{C}_T}\cdot H(s)\cdot a(s)$
• Terminology:
  • ${\omega }_n$ : natural frequency $\left({\omega }_n\simeq {\omega }_0=\sqrt{\frac{k}{m_s}}\right)$
    • $k$ is the spring consntant of our accelerometer, and $m_s$ the sismic mass.
  • $C_T=C_E+C_L$
  • $R_T=R_L\parallel R_A\approx R_L$
  • ==$C_L$ and $R_L$ depend on the wire’s lenght (bad)==.
  • $a(s)$ : laplace transform of the acceleration $a(t)$.
  • $H(s)$ : accelerometer’s transient function $\left(H(s)=\frac{m_S}{k}\frac{1}{\frac{m_S}{k}{s}^2+\frac{\lambda }{k}s+1}\right)$
• Bode Plot:
  
  
• Frequency range:$f\in \left[\frac{10}{R_T{C}_T},\frac{{\omega }_n}{10}\right]$
• The overall system is AC-coupled. ⇒ ==You cannot sense static acelerations==.
• Lower cutoff frequency:NOT_SURE_ABOUT_THIS
  The professor wrote $f_L=\frac{1}{2\pi C_T{R}_L}$
  This is probably wrong, since we have already defined $f_L=\frac{10}{C_T{R}_L}$
• Constant sensitivity inside the frequency range:$\alpha =\frac{V(\omega )}{a(\omega )}=\frac{m_s\cdot d}{C_T}$
• Problem: the sensitivity depends the capacitance $C_T$, wich depends on the cable capacitance.
  Solution: make the amplifier (read-out electronics) a part of the sensor itself:
  
  • This particular kind of piezo-accelerometer are called IEPE or ICP.

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SaM - Ultrasonic Transducers (Piezoelectric Device)

• Structure:
  
• Structure:
  • A piezoelectric slice of material (like quartz).
  • An added dumper $\lambda$.
  • A source voltage for exiting the piezoelectric.
  • A read-out (an amplifier) to read the output from the piezolectric.
• Workings:
  1. We exite the piezoelectric, which is deformed inverse piezoelectric effect, and we make it vibrate.
  2. The vibration generates a wave.
  3. The wave bounces off the target, and returns to the piezoelectric, make it vibrate again.
  4. We measure the new vibration with the read-out.
• $V_g$ and output graph:
  
• Electrical circuit for actuator and sensor:
  (This is a coarse approximation)
  
  • The two diodes are a protection circuit, for read-out part, it protects the amplifier when using the device as a transducer.
• Used as an actuator (both diodes are ON):
  
  • Output graph of $\dot{x}(t)$
    
  • Relative output $\left(\left|\frac{\dot{x}(\omega )}{V}\right|\right)$ graph:
    
• Used as a sensor (both diodes are OFF):
  
  • Output charge formula:$Q_p=kdx$
  • Transfer function:$\frac{V}{F_{ext}}=\frac{R_{p}d}{1+j\omega R_p{C}_p}{H}_u(\omega )$Where:$H_u(s=j\omega )=\frac{s}{\frac{m}{K}{s}^2+\frac{\lambda }{K}s+1}$Please note the difference with the transfer function $H(s)$ of the acceleroemeter, do not confuse them.
  • Plot of $H_u$ and $\frac{R_{p}d}{1+j\omega R_p{C}_p}$, the complete output graph is their multiplication:
    (This device should be excited with a frequency $f\approx \frac{{\omega }_N}{2\pi }$)
    

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SaM - Coarse Approximation of the Lumped Parameter Model of an Ultrasonic Transducer

• “The lumped parameter approximation of this device is an extreamely coarse approximation, since the the wavelenght ${\lambda }_w$ is NOT larger than the dimension of the crystal ($h$)“.
• So ${\lambda }_w\ngtr h$ ⇒ coarse approximation.

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SaM - Quartz Oscillator

• Lumped parameter system:
  
• Wavelenght-thickness real world relationship:$t\approx \frac{{\lambda }_w}{2}$
• AT-cut quartz as a sensor, equivalent electrical circuit:
  
  • Formulas:$\begin{array}{l}{L}_{eq}=\frac{m}{k^2{d}^2}\\ R_{eq}=\frac{\lambda }{k^2{d}^2}\\ C_{eq}=k{d}^2\end{array}$
  • If we want to consider the added “mechanical impedance of the medium”:$\frac{Z_M}{k^2{d}^2}$Usually negligible.
  • The piezoelectric behavior of this structure depends on the surface orientation with respect to the crystallographic axis.
• Terminology:
  • $m,\lambda ,k$ : mass, dumping coefficient, elastic coefficient of the piezoelectric slice.
  • $d$ is the piezoelectric coefficient.
  • $Z_M$ : mechanical impedance of the medium, as an example when we talked about the ultrasonic transducer we said that it used an “added damper”, in that case $Z_M$ would represent this added dumper.

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SaM - AT-Cut Quartz

• Lumped parameter system:
  
• Real world values for $\omega =10\text{MHz}$:$\begin{array}{l}{L}_{eq}=100\text{mH}\\ C_{eq}=15\text{fF}\\ R_{eq}=20\text{ohm}\end{array}$Remember: $1f=10^{-15}$.
• Reactance $X_{eq}$, given: $R_{eq}=0$ and $Z_{eq}=j{X}_{eq}$:$X(\omega )=-\frac{1-{\omega }^2{L}_{eq}{C}_{eq}}{1-{\omega }^2{L}_{eq}\frac{C_E{C}_{eq}}{C_E+C_{eq}}}$
• If we consider that $C_E\ll C_{eq}$, and low frequencies:$X(\omega )=-\frac{1}{\omega C_E}$
• Frequencies to remember:$\begin{array}{l}{\omega }_s=\sqrt{\frac{1}{L_{eq}{C}_{eq}}}\\ {\omega }_p=\sqrt{\frac{C_E+C_{eq}}{C_E{L}_{eq}{C}_{eq}}}\end{array}$
• We have also seen the previous formula written as:$X_{eq}=-\frac{1}{\omega C_E}\frac{1-\frac{{\omega }^2}{{\omega }_s^2}}{1-\frac{{\omega }^2}{{\omega }_p^2}}$
• Plot of $\frac{1-\frac{{\omega }^2}{{\omega }_s^2}}{1-\frac{{\omega }^2}{{\omega }_p^2}}$:
  
• Plot of $X_{eq}$:
  
• ==There is a really small range in which the quartz acts as an inductance==.
  ==Moreover this range is set by the physical properties of the quartz, that define ${\omega }_s$ and ${\omega }_p$==.
• Wavelenght-thickness real world relationship:$t\approx \frac{{\lambda }_w}{2}$
• Plot if $R_{eq}=0$:
  
• Plot if $R_{eq}\ne 0$:
  
• Real world measures:
  • The percentage difference (calculated like a relative error) of ${\omega }_s$ and ${\omega }_p$ is $10^{-5}\%$, (really small difference between ${\omega }_s$ and ${\omega }_p$).
  • So let’s take for example ${\omega }_s=10\text{MHz}=10{}^{{}^{\prime }}000{}^{{}^{\prime }}000\text{Hz}$ then ${\omega }_p=10{}^{{}^{\prime }}000{}^{{}^{\prime }}100\text{Hz}$.

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SaM - Crystallographic Axis

• $\left[hkl\right]$ defines a vector.
• $\left(hkl\right)$ defines a plane $\perp \left[hkl\right]$.
• $⟨hkl⟩$ defines a direction $\parallel \left[hkl\right]$.
• Typical cell for Silicon:
  
• Real World Measure:
  • Typical surfaces: $\left(100\right)$, $\left(111\right)$, $\left(110\right)$, used in micromachining technology
  • For Silicon: $a=543\text{pm}$ (Remember : $p=10^{-12}$)
• Terminology
  • $a$ : length of the unit cell.

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SaM - Types of Cut Quartzes

• The way we cut the quartz, will relfect how it works:
  
• ==Cut-Quartzes work in in frequency up to $30$ MHz==.
  ==The “thickness shear mode” has the highest frequency range==.
• This frequency changes with temperature, and all cuts have a slope around $T_A$ (ambience temperature):
  

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SaM - Quartz in a 3 Point Oscillator

• Using an AT-cut quartz in a Colpits oscillator:
  (This circuit will oscillate only at frequency in which the quartz acts as an inductance, so for $f_0$ in the interval $[{\omega }_s,{\omega }_p]$)
  
• $Q$-factor value:
  (Adding a capacitance in series or in parallel to our quartz, will reduce this value, not goood)$Q=\frac{1}{{\omega }_s{R}_{eq}{C}_{eq}}$
• Adding a capacitance in series or in parallel to the quartz:
  (Adding a capacitance in series or in parallel can reduce the frequencies $f_s$ and $f_p$, giving us more control, good)
  • No added capacitance, base graph:
    
  • Added capacitance:
    
    • In series: we increase $f_s$.
    • In paralle: we reduce $f_p$.
    • In both cases we reduce the range $[f_s,f_p]$

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SaM - Behavior of the Quartz Ocillator at High Frequencies

• AT-cut quartz’s impedance:$Z_{eq}=C_E\parallel (L_{eq}+R_{eq}+C_{eq})$
• Real world values of an AT-cut quartz for $\omega =10\text{MHz}$:$\begin{array}{l}{L}_{eq}=100\text{mH}\\ C_{eq}=15\text{fF}\\ R_{eq}=20\text{ohm}\end{array}$Remember: $1f=10^{-15}$.
• AT-cut quartz special frequencies:$\begin{array}{l}{\omega }_s=\sqrt{\frac{1}{L_{eq}{C}_{eq}}}\\ {\omega }_p=\sqrt{\frac{C_E+C_{eq}}{C_E{L}_{eq}{C}_{eq}}}\end{array}$
• AT-cut quartz reactance formula:$X_{eq}=-\frac{1}{\omega C_E}\frac{1-\frac{{\omega }^2}{{\omega }_s^2}}{1-\frac{{\omega }^2}{{\omega }_p^2}}$
• Plot of $\frac{1-\frac{{\omega }^2}{{\omega }_s^2}}{1-\frac{{\omega }^2}{{\omega }_p^2}}$:
  
• More precise equivalent system for an AT-cut quartz at high frequencies:
  
  • Instead of:
    
• Additional components formulas:$\begin{array}{l}{L}_{m_i}=\frac{1}{8}\frac{m}{k^2{d}^2}\\ C_{m_i}=\frac{8}{(i\cdot \pi {)}^2}k{d}^2\\ R_{m_i}=\frac{(i\cdot \pi {)}^2}{8}\frac{\lambda }{k^2{d}^2}\end{array}$And:${\omega }_{s_i}\approx i\cdot w_{s_1}$Where: $i=1,2,\dots$
• Real world measure:
  • “Low frequency”, in this case means: $<10\text{MHz}$.
  • “High frequency”, in this case means: $[20\text{MHz}÷30\text{MHz}]$, or more.

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SaM - Quartz as a Sensor • QCM (Quartz Crystal Microbalance)

• Structure and equivalent system:
  
• The name of this sensor is: QCM (Quartz Crystal Microbalance).
  It is based on the concentration of the target species (that gets absorbed by the functionalization layer).
• Functionalization layer impedance formula:$Z_M=\Delta m\Rightarrow \frac{Z_M}{k^{2}d}=\frac{\Delta m}{k^{2}d}$
• Changes to the equivalent components of this sensor, with respect to the normal AT-cut quartz:$L_{eq}^{\prime }=L_{eq}+\frac{\Delta m}{K^{2}d}\Rightarrow \{\begin{array}{l}{\omega }_s^{\prime }=\frac{1}{\sqrt{L_{eq}^{\prime }\cdot C_{eq}}}\\ {\omega }_p^{\prime }=\sqrt{\frac{C_E+C_{eq}}{C_E{L}_{eq}^{\prime }{C}_{eq}}}\end{array}$
• Terminology:
  • $\Delta m$: mass of the “functionalization layer”
  • $K$ : cosntant defined by the geometry of the piezoelectric crystal
  • $d$ : piezoelectric coefficients.
  • $L_{eq}=\frac{m}{k^2{d}^2}$ defined previously.

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