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MisterX

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## Homework Statement

Consider a quartz crystal and a PZT ceramic filter both designed with resonant frequency equal to 1 MHz. What is the bandwith of each? Given Young's modulus (Y) and density (ρ) for each, and that the filter is a disk with electrodes and is oscillating radially, what is the diameter of the disk for each material? Consider only the fundamental mode.

Quartz: Y = 80 GPa ρ = 2.65 g/cm^3

PZT: Y = 70 GPa ρ = 7.7 g/m^3 (I think this was a typo, author probably meant g/cm^3)

## Homework Equations

[itex]resonant frequency = \frac{1}{2\pi\sqrt{LC}}[/itex]

"where L represents the mass of the transducer and C the stiffness"

[itex]anti-resonant frequency = \frac{1}{2\pi\sqrt{LC'}}[/itex]

where C' is the equivalent capacitance of C from the resonant equation in series with the parallel plate capacitance of the transducer.

I remember bandwidth being said to be the difference between resonant and anti-resonant frequencies. This seems quite different than definitions of bandwidth used in other situations.

speed of mechanical vibrations in a medium:

[itex]v = \sqrt{\frac{Y}{ρ}}[/itex]

standing wave condition:

[itex]n(\lambda/2) = length[/itex]

relation between wavelength, wave speed, and frequency:

[itex]\lambda = v/f[/itex]

## The Attempt at a Solution

I'm not sure how to calculate the bandwidth without the parallel plate capacitance, or some other information from which it might be deduced, and other information such as L or C. The model I was given is an equivalent circuit of the parallel plate capacitance in parallel with a series LCR. I asked how to get the component values for the equivalent circuit, but I remember only being taught to find C or C (parallel plate) when both resonant and anti-resonant frequencies were given, and one of those two capacitances was given as well.

To get the diameter of the transducers, I would get the wavelength in each material, and then using the standing wave equation I would acquire the radius (which would be length in the above standing wave equation).

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