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Lab 11

The questions below are due on Friday November 15, 2019; 05:15:00 PM.
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Goals:In this lab we will use a piezoelectric resonator to make a mass balance capable of measuring tiny changes in mass!

1) Preliminaries

Make sure you have gone thru the prelab.

2) Overview

In this lab we will use the piezoelectric resonator that is inside the piezoelectric transmitter and receiver used in the Doppler ultrasound velocimetry labs. You may use either the transmitter or the receiver transducer as they are nearly identical. An equivalent circuit model for the resonator is shown below.

Today we will build an oscillator using the piezoelectric resonator. We will then apply small volumes of water to vary the effective mass of the resonator and observe changes in the oscillator frequency using an oscilloscope. A similar technique is commonly employed to measure extremely small masses deposited on a resonator surface during micro/nanofabrication or as a result of biochemical reactions. Let’s see how sensitive we can make our balance, and what is the smallest mass we can detect!

3) Oscillator

The circuit shown below, known as a Pierce oscillator, is commonly used with a quartz crystal to generate the clock frequency for digital systems. In such applications the inverting amplifier is normally replaced by a digital inverter with a parallel bootstrap resistor. But, for our purposes, we will use the more familiar op-amp-based implementation.

The goal of the oscillator is to sustain oscillation at a frequency determined by the piezoelectric resonator. In this way, we will be able to directly measure the frequency and observe its change as we add/remove mass to/from the resonator. Whether or not the oscillator will oscillate is determined by the sinusoidal steady-state transfer function around the loop identified above. If there is a frequency at which the gain around the loop is greater than unity, while the phase shift around the loop is a multiple of 360 degrees, then the oscillator will oscillate at that frequency. This is because a signal at the oscillation frequency injected into the loop at any point will travel around the loop and constructively reinforce itself causing unbounded growth. This process is started by random thermal fluctuations, and is eventually limited by amplitude saturation of the op amp. Since the transfer function around the loop is in part determined by the piezoelectric resonator, the mass of the resonator will affect the oscillator frequency.

To verify that the oscillator should oscillate, we use the left-hand circuit shown below. In this open-loop circuit, the loop is broken as shown in green at the input to the inverting op-amp amplifier. This allows us to inject a signal into the amplifier at v_{IN}, and measure the output of the circuit around the loop at v_{OUT}. Again, we expect to see that v_{OUT} is bigger than v_{IN} at the frequency for which the relative phase shift between the two signals appears to be zero; in actuality the phase shift will be 360 degrees. Once we have confirmed that the circuit provides a gain greater than unity together with a phase shift of “zero” at some frequency in the open-loop state, we can close the loop by connecting v_{OUT} to v_{IN} as shown below in pink in the right-hand circuit. We then measure the oscillating voltage, v_{OSC}, also shown below in pink.

To begin, let's build the open-loop circuit on a breadboard using the schematic shown below, and then measure its frequency response in the following section. This is the same circuit as shown above with component values labelled, and the loop unraveled for clarity. The op-amp pin numbers are given on the schematic, and the op-amp pinout description is also given below. We use large amplifier gain resistors R1 and R2 to reduce the loading by R1 on the upstream network of RS, C1, the piezoelectric resonator and C2. Large loading (small R1) reduces the loop gain, and the phase shift provided by the network, both of which are detrimental to oscillation. Also, recall that the oscilloscope probes have a parasitic resistance of ~10 M\Omega and capacitance of ~10 pF to ground. This could be important when measuring v_{OUT}. Fortunately, C2 is large, which reduces the impact of the scope parasitics on circuit operation.

You will not need the Teensy or OLED for this lab, but you should use the same breadboard and free up at least 20 rows of open workspace. You should get one TLV2371 op-amp and one disassembled piezoelectric transmitter or receiver from the staff desk. We have removed the “grill” and acoustic “horn” from the transducers to leave the top electrode exposed. It does not matter whether you use a receiver or transmitter, and its orientation in the circuit is also not important for proper circuit operation.

Construct the open-loop circuit on your breadboard making sure that you have everything wired correctly.

But, DO NOT POWER ON your circuit until it has been checked by a 6.002 LA/TA. In particular, note that you must supply the op-amp with +/- 5 V, requiring two channels of the power supply. Be ready to explain how you are generating the dual-supply during your checkoff.

Checkoff 1:
Discuss understanding of how mass affects the resonant frequency. What else might affect the frequency? Where do gain and phase shift come from? Verify circuit looks correctly wired before powering on.

4) Open-loop Transfer Function

Now that we have the open-loop circuit built, we can test it to find the frequency at which the loop phase shift is zero (namely the frequency at which the input and output are in-phase) and verify that the gain is greater than unity at that frequency.

After having your circuit checked by a 6.002 LA/TA, power it on with the dual +/- 5V power supply. Set up the signal generator to produce a 500-mV peak-to-peak-amplitude sine wave around the previously calculated resonance frequencies of the piezoelectric oscillator.

Apply the signal generator voltage at v_{IN}. Using two oscilloscope channels, probe both the input v_{IN} and the output v_{OUT} of the open-loop circuit with respect to ground. Sweep the signal generator frequency in 100-Hz steps in either direction over a range of about +/- 10 kHz around the calculated resonance frequencies to find the frequency at which the input and output signal peaks line up, as shown in the example below.

Ch1 (yellow) is the input from the signal generator and
Ch2 (blue) is the output from the piezo

This is the desired zero-phase-shift (or 360^\circ-phase-shift) frequency, which should be near the resonance frequency of the piezoelectric resonator.

Record the zero-phase-shift frequency (as displayed on the signal generator) and measure the gain of your system by taking the ratio of the peak-to-peak output amplitude to the peak-to-peak input amplitude.

If the gain is not greater than unity, you must increase it by changing the op-amp gain until the net open loop gain is greater than unity. Recall that to avoid loading the piezo, you should keep R_1 as high as possible, so make sure to only change R_2. Not having a large enough gain will cause any initial oscillations to decay when the circuit is used in its closed-loop configuration, and you won’t see any output!

Do you see why there can be a frequency at which there is zero phase shift? Recall that the inverting amplifier provides a 180^\circ phase shift. Then, the R_S-C_1-Resonator-C_2 network provides another 180^\circ phase shift. Since phase shifts add when cascaded, the net shift is 360^\circ, which is the same as 0^\circ.

What is the resonant frequency of your piezo (in kHz)?

What is the open-loop gain of your oscillator circuit?

Let’s take a look at what happens when we add mass to the resonator. Obtain a pipette filled with water and a paper towel from the staff desk. Carefully drop a little water on the exposed electrode of the piezo transducer as shown in the image below. Repeat the procedure outlined above to find the new zero-phase-shift frequency. How did it change? Is this what you expected? Verify that the gain is still greater than one at the new frequency.

5) Closed-loop Response

Now that we have verified in open-loop that the circuit should oscillate, let’s close the loop. First, turn off and disconnect the signal generator. Then remove the oscilloscope probes and turn off the dual power supply before you modify the circuit. You may find it best to wick up the water with a paper towel to avoid making a mess!

Convert your circuit to become an oscillator as shown in the schematic below with changes highlighted in blue. Pay attention to the new point at which to probe the output, and note that there is no input because the circuit should now oscillate on its own.

If you needed to use a different resistor value for R_2 to make the loop gain greater than unity, use that new value.

Once you have modified your oscillator, use the oscilloscope to probe the node labelled v_{OSC}. You can use the oscilloscope MEASURE feature to obtain a real-time measure of the oscillation frequency without using the cursors or counting divisions. Click on the MEASURE button near the top and then use the buttons along the right side of the screen to select the desired measurement. Using those buttons, toggle the channel setting (top button) until it is set to the channel on which you are measuring v_{OSC}. Toggle the measurement setting until it shows FREQ, then press back. Turn on the power for the op-amp supply. While the MEASURE menu is open, you should now see the frequency measurement updating as the waveform changes. You may need to adjust the horizontal divisions to show several clear cycles of the output for an accurate measurement. The waveform should look similar to that pictured below.

Frequency measurement of the oscillator output (on Ch3, pink)

Look at the frequency measurement on the oscilloscope for the oscillator you have built. How does it compare to the resonance frequencies determined earlier in the open-loop system? Add a drop of water similar in size to that which you added earlier and look at the frequency measurement again. How does it compare to the open-loop measurements? What happens if you add two water drops instead of one?

Checkoff 2:
Explain what you see. Watch the frequency change as a staff member wicks up the water!

6) Cleanup

Before you leave, it's time to clean up again! Steps for cleanup:

  • Return the piezo mass sensor to the staff table.
  • Carefully pick up your system and place into its plastic case.
  • Throw away loose wires on your desk.
  • Throw away paper, food, etc. on your desk.

Checkoff 3:
Show your cleaned-up lab space to a staff member.