# Prelab 11

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## 1) Overview

Our lab this week uses a piezoelectric resonator to measure small masses. A piezoelectric resonator is a crystalline mechanical spring-mass-damper resonator that can be excited to vibrate mechanically by applying an electrical signal at its electrical terminals. If external mass is attached to the resonator, then its spring-mass resonance frequency will decrease. By building an electrical oscillator tuned to this mechanical resonance frequency, we can observe the shift in the *electrical* resonance frequency of the overall system due to the added *mechanical* mass. To simplify things, we will observe the shift in oscillator frequency using an oscilloscope.

For its convenience and availability, in this lab we will use the piezoelectric resonator that is inside the piezoelectric transmitter and receiver used in the Doppler ultrasound velocimetry labs, part numbers 40T12 and 42R12 from Jameco Electronics, respectively. These transducers probably contain an economical lead-zirconate-titanate (PZT) piezoelectric crystal. Alternatively, we could use other piezoelectric crystals such as quartz. In fact, quartz crystals are the more common piezoelectric material used for high-accuracy mass microbalances, and high-precision oscillators in general.

In this pre-lab, we examine the model for a piezoelectric resonator. In the process we will determine how the electrical characteristics of the resonator vary with added mass.

## 2) Resonator Equivalent Circuit Model

A piezoelectric resonator is built by placing a piezoelectric crystal between two metal electrodes. This explains why its electrical schematic symbol looks like a capacitor with a block of material in between the capacitor plates, as shown below. When the capacitor is excited, an electric field is applied across the crystal, which, by virtue of its piezoelectric character, causes the crystal to expand or contract. Thus, an electrical voltage causes a mechanical displacement, similar to the transducers we have seen in lecture (speaker, energy harvester).

Simultaneously, again by virtue of its piezoelectric character, mechanical motion of the crystal induces charge accumulation on the capacitor plates. In other words, the transducer also works from the mechanical to electrical domain (i.e., it is reciprocal). We can develop a dependent source model of the resonator, and by transforming the mechanical elements through the sources, we arrive at a purely electrical model of the system, as shown below.

As discussed in lecture, the equivalent circuit model shown above is based on the “mobility” analogy in which velocity is represented by voltage and force by current. In the model, C_e is the original electrode-to-electrode capacitor. The mechanical elements are L_m, which represents the mechanical spring, C_m, which represents the mechanical mass, and R_m, which represents loss, including thermoelastic loss in the piezoelectric crystal, and “loss” due to acoustic energy transmission.

Finally, like the speaker transducer discussed in lecture, if the mechanical motion of the crystal is coupled acoustically to the surrounding air, then an acoustic transmitter is created (like the transmitter on your ultrasound PCB). Similarly, if pressure variations in the surrounding air cause the crystal to expand and contract, then an acoustic receiver is created (like the receiver on your ultrasound PCB). Both the transmitter and receiver used in the ultrasound velocimetry labs have a bowl-shaped horn attached to the crystal in order to enhance the acoustic coupling.

## 3) Impedance

For $C_m$ > $C_e$, the resonator impedance $Z$ varies with frequency as shown below.

For the question below, enter your answer in terms of Lm for L_m, Cm for C_m, Ce for C_e, j, and w for \omega.

For the special case of R_m=\infty, determine the impedance Z.

Z=

You should find that the impedance becomes zero at the lower resonance frequency, and becomes infinite at the upper resonance frequency, as shown in the above figure, though of course, in the figure above, R_m is not infinity, and so the zero and infinite impedance magnitudes are not reached in practice at the two resonance frequencies.

The datasheet for our piezoelectric transmitter and receiver provides a plot of the impedance magnitude and phase versus frequency as pictured below. Observe the similarity of the black impedance-magnitude curve to that shown above. Also take note of the 90^\circ bump in the red impedance-phase curve that occurs around the resonant frequencies. We will take advantage of this feature when designing the oscillator and analyzing its transfer function during lab.

Using numerical analysis methods, the 6.002 staff was able to extract approximate model parameters for our piezoelectric resonators from the plot above:

• R_m = 6.9 M\Omega
• L_m = 170 mH
• C_m = 88 pF
• C_e = 5 pF

Given these parameters, calculate the expected lower and upper resonance frequencies of the piezoelectric resonator, for the special case of R_m = \infty. Note that the frequency pairs are nearly the same for the ultrasonic transmitter and receiver so you need not worry about which transducer to use in lab.

What is the lower resonant frequency (in kHz) of our piezo transceiver (calculated to 4 significant digits)?

What is the upper resonance frequency (in kHz) of our piezo transceiver (calculated to 4 significant digits)?

How will the resonant frequencies change if extra mass is added to the piezoelectric resonator?