SaM - AT-Cut Quartz

Remeber:

We can model an AT-Cut Quartz, which is a special cut along a crystallographic axis of a quartz, like so:

• $C_E$ is the capacity brought by placeing two metal layers on top of the quartz.
• We have given some common values for the electromechanical equivalent, considering $\omega =10\text{MHz}$:$\begin{array}{l}{L}_{eq}=100\text{mH}\\ C_{eq}=15\text{fF}\\ R_{eq}=20\text{ohm}\end{array}$Remember: $1f=10^{-15}$.
• The cut has a precise orientation with respect to the lattice.
• Therefore, we know also which one of the piezoelectric coefficient has to be taken into account in order to describe the phenomena.

We can plot the reactance, given that: $Z_{eq}=C_E\parallel (L_{eq}+R_{eq}+C_{eq})$ and for sempliciity we will consider:

1. $R_{eq}=0$
2. Instead of plotting $Z_{eq}$ (the impedance) we will plot $X_{eq}$ (the reactance) where: $Z_{eq}=j{X}_{eq}$.

Here is the unsimplified formula of $X_{eq}$:$X(j\omega )=-\frac{1-{\omega }^2{L}_{eq}{C}_{eq}}{1-{\omega }^2{L}_{eq}\frac{C_E{C}_{eq}}{C_E+C_{eq}}}$And if we consider that $C_E\ll C_{eq}$, and for low frequencies, we can simplifiy to:$X_{eq}=\frac{1}{\omega C_E}$ And here’s the simple plot:

• So the behavior of the quartz is the normal behavior of the capacitance

Now let’s define two imporant frequencies:${\omega }_s=\sqrt{\frac{1}{L_{eq}{C}_{eq}}}$And:${\omega }_p=\sqrt{\frac{C_E+C_{eq}}{C_E{L}_{eq}{C}_{eq}}}$This is what happenes at high frequency when $w=w_s$ and $w=w_p$:

• ==In a small frequency interval, we have a huge shift in amplitude, and the behavior of the quartz becomes normal behavior of an inductance, than returns to that of a capacitance==.
• $w_p$ and $w_s$ are very close since $C_E\gg C_{eq}$.
• We define for convinience:$A(\omega )=\frac{1-\frac{{\omega }^2}{{\omega }_s^2}}{1-\frac{{\omega }^2}{{\omega }_p^2}}$And outside this really small range $[{\omega }_s,{\omega }_p]$ the value of $A(\omega )\simeq 1$.

We have seen some numeric examples:IMPORTANTE

• 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 ${\omega }_p=10{}^{{}^{\prime }}000{}^{{}^{\prime }}100\text{Hz}$.

If we perform some calculations, we will find that: $t=\frac{{\lambda }_w}{2\pi }$. But actually due to some errors the real beahaviour is a bit different:$t=\frac{{\lambda }_w}{2}$==So if a slice of AT-Cut Quartz, the thickness of the slice will decide the wavelength==.IMPORTANTE

We have considered $R_{eq}=0$, and to recap here is the $|Z(\omega )|$ plot that we have seen:
But what if $R_{eq}\ne 0$? ⇒ Then $Z_{eq}$ is not a pure reactance anymore, it has real and imaginary part:

• The red line is an inducatance with a greater $R_{eq}$ with respect to the blue line.
  So: $R_{e{q}_{\text{RED}}}>R_{e{q}_{\text{BLUE}}}$
• The width of the arc that forms (looking at the amplitude plot) is related to the $Q$-factor.
  The $Q$-factor decreases if $R_{eq}$ increments.

Even if we take an $R_{eq}$ really small, so a small window in which the device behaves like an inductance, still the amplitude $|Z|$ changes a lot:

• So we have a special component that acts like a conductance, then like an inductance (for a small range of frequency) and then again like a conductance.
• ==More important the small range in which it acts as an inductance is set by the physical properties of the quartz, that define ${\omega }_s$ and ${\omega }_p$==.
• NOTE: the small range is NOT $[{\omega }_s,{\omega }_p]$, it is even smaller, it is contained in this range and the actual range depends on $R_{eq}$.
  If $R_{eq}$ is small enough we consider to have an inductance-behaviour exactly in this range $[{\omega }_s,{\omega }_p]$.

---

So I take as an example the so-called AT-Cut quartz. ![[Pasted image 20230720170030.png|]]

• The cut has a precise orientation with respect to the lattice.
• Therefore, we know also which one of the piezoelectric coefficient has to be taken into account in order to describe the phenomena.
• $C_E$ is the capacity brought by placeing two metal layers on top of the quartz.

We evaluate $Z_{eq}=C_E\parallel (L_{eq}+R_{eq}+C_{eq})$, but we consider $R_{eq}=0$ (it is typically very small). So it is just the imaginary part:

So we can plot $X(j\omega )$:

• So the behavior of the quartz is the normal behavior of the capacitance.

But even if the values are extreamely low, we can define:${\omega }_s=\sqrt{\frac{1}{L_{eq}{C}_{eq}}}$And more importantly:${\omega }_p=\sqrt{\frac{C_E+C_{e}q}{C_E{L}_{eq}{C}_{eq}}}$ This is what happenes at high frequency when $w=w_s$ and $w=w_p$.

• $w_p$ and $w_s$ are very close since $C_E\gg C_{eq}$.
• We define for convinience:$A(\omega )=\frac{1-\frac{{\omega }^2}{{\omega }_s^2}}{1-\frac{{\omega }^2}{{\omega }_p^2}}$
• And outside this really small range $[{\omega }_s,{\omega }_p]$ the value of $A(\omega )\simeq 1$.

Let’s see some numeric examples:

• $\sqrt{\frac{K}{m}}$ is defined as the mechanical natural frequency, it is often used and found in mass-spring-dumper system.

But we can write $K$ the elastic coefficient as:$K=\frac{T_{11}A}{{\epsilon }_{11}t}$Since:

• $T_{11}=\frac{F}{A}$ ⇒ $T_{11}A=F$.
• ${\epsilon }_{11}=\frac{\Delta x}{x}$ and in this case the variation quantity $x$ is $t$ ⇒ ${\epsilon }_{11}t=\Delta t$
• ==NOTE: This coefficinents are given by the cut “type”==.

So we obtain the (inverse) most common Hooke’s Law: $K=\frac{F}{\Delta x}$

Also:$m=\rho V=\rho At$Where:

• $\rho$ : density.
• $V$ : volume.

So we can write ${\omega }_s$ as $\frac{1}{t}\sqrt{\frac{c_{11}}{\rho }}$, as you can see here:

• We find that: $t=\frac{{\lambda }_w}{2\pi }$
• But actually due to some errors the real beahaviour is similar but it is:$t=\frac{{\lambda }_w}{2}$

==So if a slice of AT-Cut Quartz, the thickness of the slice will decide the wavelength==. This is an extreamly convinient property.

If $R_{eq}=0$: (as we have seen):

What if $R_{eq}\ne 0$? Then $Z_{eq}$ is not a pure reactance anymore, it has real and imaginary part:

• In red a greater $R_{eq}$ than in blue.
• The width of the arc that forms (looking at the amplitude plot) is related to the $Q$-factor.
• The $Q$-factor decreases if $R_{eq}$ increments.
• As I said before we wold require (usally) $R_{eq}<20\text{ohm}$, otherwise strange phenomena can happen.
• If the phase does not change, then we will not have a change from capacitive-behaviour to inductive-behaviour, more on that later.

If the $Q$-factor is large enough, so if $R_{eq}$ is small enough, we will have a small frequency, in which the system behaves like an inductance, so the phase is kept at $90°$ for a bit, we can see it in the figure below:

• Even if this range is small, the amplitude $|Z|$ changes a lot (as we can see in the previous figure), so we have some liberty over the values $Z_{eq}$ can assume.

So we have a special component that acts like a conductance, then like an inductance (for a small range of frequency) and then again like a conductance:

• More important the small range in which it acts as an inductance is set by the physical properties of the quartz, that define ${\omega }_s$ and ${\omega }_p$.
• NOTE: the small range is NOT $[{\omega }_s,{\omega }_p]$, it is even smaller, it is contained in this range and the actual range depends on $R_{eq}$.
  If $R_{eq}$ is small enough we consider to have an inductance-behaviour exactly in this range $[{\omega }_s,{\omega }_p]$.

---