# What Is the Bandwidth and Diameter of Quartz and PZT Piezoelectric Filters?

• MisterX
In summary: IIn summary, the question asks for the bandwidth and diameter of a quartz crystal and PZT ceramic filter with a resonant frequency of 1 MHz. The bandwidth can be calculated by finding the difference between the resonant and anti-resonant frequencies, but additional information such as the parallel plate capacitance is needed. The diameter of the transducers can be calculated by finding the wavelength and using the standing wave equation, and then using the wave speed equation to determine the wave speed in each material.
MisterX

## 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

$resonant frequency = \frac{1}{2\pi\sqrt{LC}}$
"where L represents the mass of the transducer and C the stiffness"

$anti-resonant frequency = \frac{1}{2\pi\sqrt{LC'}}$
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:
$v = \sqrt{\frac{Y}{ρ}}$

standing wave condition:
$n(\lambda/2) = length$

relation between wavelength, wave speed, and frequency:
$\lambda = v/f$

## 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).

#### Attachments

• PiezoelectricEquivCircuit.png
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Then to get the diameter I would just multiply the radius by 2. I'd do this for both materials, and then use the wave speed equation to get the wave speed in each material.

## 1. What is a piezoelectric filter?

A piezoelectric filter is an electronic device that uses the piezoelectric effect to filter out certain frequencies from an electronic signal.

## 2. How does a piezoelectric filter work?

A piezoelectric filter works by using a piezoelectric crystal to convert mechanical vibrations into electrical signals. These vibrations are caused by the input signal, and the crystal filters out certain frequencies based on its size and shape.

## 3. What are the advantages of using a piezoelectric filter?

Piezoelectric filters have several advantages, including high selectivity, low insertion loss, and small size. They are also less susceptible to temperature changes and have a wide frequency range.

## 4. What are some common applications of piezoelectric filters?

Piezoelectric filters are commonly used in electronic devices such as radios, televisions, and mobile phones. They are also used in medical equipment, sensors, and acoustic transducers.

## 5. Are there any disadvantages to using a piezoelectric filter?

One potential disadvantage of piezoelectric filters is their susceptibility to mechanical stress and shock, which can affect their performance. They can also be more expensive than other types of filters.

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