SaM - Matching Layer for Acustic Impedance

Remeber:

Suppose we take an ultrasonic transducer made of PZT and we attach it to another undefined material called, their acustic impedances have this values:

• $Z_{\text{PZT}}=32.5\frac{\text{kg}}{{\text{m}}^2\text{s}}$
• $Z_{\text{?}}=430\frac{\text{kg}}{{\text{m}}^2\text{s}}$
• The unit of measure $\frac{\text{kg}}{{\text{m}}^2\text{s}}$ is called a $\text{Rayl}$.

Since we know that the wavelength has this formula:${\lambda }_w=\frac{v_l}{f_0}$And that given an intial value for $v_i$ (the longitudinal velocity of the wave), we have the ratio with the resulting transmitted wave is:$\frac{v_T}{v_i}=\frac{2Z_{\text{PZT}}}{Z_{\text{PZT}}+Z_{\text{?}}}$The same ratio is true for the wavelength. ==So for this particular example the resulting wavelength and velocity are only about $14\%$ of the initial values==.

If we wish to increase this ratio, we could consider inserting a matching layer, with an acustic impedance of $Z_A$, between the PZT and the other material ”?”, like so:

• So the formula, now becomes (ignoring the reflections, only the transmitted parts):$\frac{v_T}{v_i}=\frac{2Z_{\text{PZT}}}{Z_{\text{PZT}}+Z_{\text{A}}}\cdot \frac{2Z_{\text{A}}}{Z_{\text{A}}+Z_{\text{?}}}$Where:
  • $Z_A$: acustic impedance of the matching layer
  • $Z_{\text{PZT}}$: acustic impedance of the PZT layer $\left(Z_{\text{PZT}}=32.5\text{Rayl}\right)$.
  • $Z_{?}$ : acustic impedance of the other material, or outside material $\left(Z_{\text{?}}=430\text{Rayl}\right)$.
• ==If we suppose $Z_A=200\text{Rayl}$ we obtain that the ratio becomes $126.33\%$ of the intial value, so we have increased it==.
  Without the matching layer: $\frac{v_T}{v_i}=0.1405$, with the matching layer $\frac{v_T}{v_i}=0.1775$.
• As a rule of thumb we can consider a single acustic impedance $Z_M$, that combines the impedances of the material+matching layer:$Z_M=\sqrt{Z_{\text{A}}{Z}_{\text{PZT}}}$
• Given the thickness of the PZT layer: suppose $t_{\text{PZT}}=\frac{{\lambda }_{\omega }}{2}$, we usually take the thickness of the matching layer as $t_A=\frac{{\lambda }_{\omega }}{4}$, so half of the PZT thickness (in the figure $t_A$ = $t_M$).NOT_SURE_ABOUT_THIS

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So keeping in mind that the wavelength is an important parameter, since it decides how the wave reacts to the medium. W go back for a moment to our transducer, to the structure that we have described at the beginning:

• So we have said that the transducer is based on a piezoelectric slice, which is usually PZT.
• A backing which is used to enhance the damping called damper.
• then we have a matching layer, and now we will explain how, and why, this matching layer is needed it.
• $Z_{PZT}$ acustic impedance of PZT.
• $Z_M$ acustic impedance of the target material (for example air).
• $Z_A$ acustic impedance of the matching layer.
• If we select a matching layer with a certain acustic impedance $Z_A$ we will have that the propagation speed, so also the wavelenght can be changed (reduced) at will.
• So transmit a wave to the matching layer then to the target material (example air), so the final wavelength, propagating inside the target material will be:${\lambda }_M=\cdot \frac{Z_{PZT}-Z_A}{Z_{PZT}+Z_A}\cdot \frac{Z_A-Z_M}{Z_A+Z_M}\cdot {\lambda }_i$NOT_SURE_ABOUT_THIS
• NOTE: This opearation redcues the wavelength, it cannot increase it.NOT_SURE_ABOUT_THIS
• There is a name for acustic impeadance unit of measure: “Rayl”.
• Professor’s note:
  “We have very different acoustical impedance between the PZT and air, a large part if the wave is reflected back instead of being transmitted to the air.
  The a matching layer reduces the phenomena of reflection, and the matching layer is a material with an impedance which is selected according to this rule here:”