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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 fundemental 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]antiresonant 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 antiresonant 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 antiresonant 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 aquire the radius (which would be length in the above standing wave equation).
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