Signal Processing
Division of Physics & Applied Physics PH2198/PAP218 Physics Laboratory IIa
Signal Processing Risk of Electrical Shock Ensure all wiring are secure before turning on power supply
Signal Processing Signal processing is a topic which both disciplines of Engineering and Physics broach on. There is however, a difference on the emphasis placed on the study and subsequent application of knowledge in this field. While engineers tend to be more concerned with the choice of components, construction, and characterization of device built, physicists are more aware of the functions, operational limits, signal-to-noise ratios and the maximum possible performance of the entire measurement system, which is composed of different devices working in tandem.
Aim • • • • •
To read up on Signal Processing from the compulsory reading materials To obtain the frequency dependence of the Preamplifier To obtain the frequency dependence of the Low-Pass Filter To obtain the frequency dependence of the Lock-In Detector To measure Rwire per unit length as a dependent variable of the supplied input voltage
1
Signal Processing
1. The lock-In Amplifier Amongst the different modules available at our disposal, the lock-in amplifier is perhaps the most worthy of a mention. When Robert Dicke from MIT first employed phase-sensitive detection to measure both the amplitude and phase of a small AC signal, he introduced a very powerful method for experimental physicists to enhance the signal-to-noise ratio of signals. During those days, all lock-in amplifiers were homemade. In contrast, highly sophisticated lock-in amplifiers are now commercially available, and they have ultra-low noise, large dynamic range, extreme stability and are completely automatic. However, the price to pay for such ease and simplicity results in novice users being clueless about the working principles. Using the equipment as a mere ‘blackbox’ results in incomplete mastery of signal processing, which is definitely not a desired outcome of education.
PLEASE SPEND THE NEXT 30 MINUTES GOING THROUGH THE “READING MATERIALS” BEFORE PROCEEDING. IT WILL HELP YOU PERFORM THE EXPERIMENT CORRECTLY.
Figure 1: A typical commercially-available lock-in amplifier.
2. The Experiment In this experiment, we shall employ several modules offered by the TeachSpin SPLIA1-A unit to measure the resistance-per-unit-length of a short piece of copper wire. The other equipment (which ought to be) available to you include a digital multimeter, a function generator, a dualchannel Cathode-Ray-Oscilloscope (CRO), coaxial cables, T-connectors, a pair of cable cutters and a set of precision screwdrivers.
2
Signal Processing
Figure 2: The apparatus.
2.1 The Device Under Test (DUT) A simple circuit board and a length of copper wire have been provided for you (Figure 3). As the contacts underneath the circuit are exposed, it would make much common sense to take care against resting it on electrically-conductive surfaces. Decide on your desired length of copper wire and cut to length. Switch the multimeter to the “continuity” mode, whereby audible beeps are emitted if the ends of the test probes are in contact with each other electrically. This mode is usually denoted by the symbol: . Unless your test probes are in contact with the freshly cut surfaces, the multimeter should not beep, as there is an enamel insulating coating on the copper wire. Scrap and expose the ends with the wire cutter (Figure 4), and confirm electrical contact with use of the multimeter.
3
Signal Processing
Figure 3: The copper wire, and the circuit boards.
Figure 4: The copper wire, with ends exposed.
2.2 Initial Investigations of Resistance Switch the multimeter to the ohmmeter mode and note the reading when the test probes are in direct contact with each other. Compare this value to that when the test probes are on the exposed ends of the copper wire. COMMENT ON THE DIFFERENCE. Insert and secure both ends of the wire into the contact-holder of the circuit board (Figure 5). Check for continuity thereafter by probing the securing screws.
4
Signal Processing
Figure 5: Connecting the copper wire to the circuit board. When in place, the circuit should follow Figure 6. Ensure that the connections are correct and that there are no short-circuits by using the multimeter.
Figure 6: The Circuitry for Device Under Test (DUT).
5
Signal Processing The resistance of the copper wi re can then be computed as: +
+
4
In this case, the approximation is good to a factor of at least 10 . The small v lue of resistance cannot be read by the multime ter to a sufficient accuracy. An alternative me hod would be to pass large voltages and/or cur rent through the wire, but that method will c ause undesirable resistive heating. Yet another ethod would be to measure for an extremely lo g length of wire, but this would cause difficultie with logistics. With the method described l ter, extremely short lengths of conducting materials can be measured with very little curr nt, making the method viable even on nanos tructures such as carbon nanotubes and nanoscal transistors.
3. On Phase and
ock-In Detection
Lock-in detection requires a r eference signal in additional to the experim ntal signal. The
method is able to extract small AC signals embedded in a noisy background by exploiting the orthogonality of sine and cosin functions. Consider the following:
The reference signal needs to b e a sine wave. The Lock-In Detector performs reference and signal, while the Low-Pass Filter averages the output from the duration of time as specified by the time constant. The larger the time tendency for the DC output to luctuate, at the drawback of being insensitive t the signal.
the mixing of the Lock-In across a onstant, the less rapid changes in
6
Signal Processing
4. The Investigations Your report should focus on the tasks given below. The points in bold prior, are to draw your attention to issues which might be useful in your discussions with respect to the following.
FOLLOW THE INSTRUCTIONS CAREFULLY. POTENTIALLY HIGH-VOLTAGES CAN DAMAGE THE SENSITIVE EQUIPMENT . Perform the following investigations: 1. Aim: Obtain the frequency dependence of the Preamplifier (“Preamp”). Procedures: Connect the “OUTPUT” of the external function generator to the AC-coupled input (“+ • INPUT”) of the Preamp. Ground the other input of the Preamp (“ − INPUT”). Connect the output of the Preamp to CH1 of the oscilloscope. Press the F1 button on the function generator and set the frequency of the function • generator to 1 kHz. Then press the F3 button. Notice you can now toggle between “50 Ohm” and “Hi-Z” for the output impedance by pressing F5. Select “50 Ohm” for the output impedance. Set an output signal amplitude of 0.010 Vp-p (peak-to-peak voltage). Press ENTER. Now, press F5 again to switch the output impedance to “Hi-Z”. You should see that the output signal amplitude is now automatically set to 0.020 V p-p. Switch on the output of the function generator by pressing the ON button above the OUTPUT. •
•
•
Set the Preamp Gain to 1. Confirm that the oscilloscope shows a 0.02 V p-p waveform. If not, your oscilloscope “Volts/Div” may not be calibrated. Turn the “VAR” knob completely clockwise to the position “CAL”. 3 Now set the Preamp Gain to 10 . Vary the output frequency of the function generator from 1 Hz to 1 MHz. For each function generator output frequency setting that you choose, record the corresponding peak-to-peak output of the preamplifier measured by the oscilloscope. You will need to adjust the controls on the oscilloscope to obtain “nice” waveforms. Do choose appropriate function generator frequency settings so that you have sufficient data points – you will need to plot a graph of Vp-p vs. frequency with a non-linear scale for the x-axis as shown in Fig. 1 of the Reading Material. Plot this graph and comment on it.
Questions: a) In real physics experiments the input signal to the preamplifier is usually of very small magnitudes (mV or smaller). What precautions can be taken here to minimize distortions to the weak signal before it reaches pre-amplification? b) Why do we need to employ a pre-amplifier in view of the fact that amplification is available in subsequent devices? 3 c) Though you have been asked to set the Preamp Gain to 10 and keep it at this setting
7
Signal Processing while you vary the frequency, is the actual Preamp Gain constant throughout your choice of frequencies in reality? d) If you do not ground the “ − INPUT” of the Preamp, what happens to the output voltage of the oscilloscope? Suggest reasons for the difference. 2. Aim: Obtain the frequency dependence of the Amplitude Detector (set at Amplitude Detector Gain = 20), AND Low-Pass Filter (set at a Low-Pass Gain = 5). Again, use the external Function Generator for this Task. Procedures: Remove all connecting cables from Task 1 to avoid confusion. We will not use the pre• amplifier for Task 2. We will use the same settings for the function generator as we did for Task 1. In the LOCK-IN/AMPLITUDE DETECTOR module panel, we can toggle between the • LOCK-IN DETECTOR and AMPLITUDE DETECTOR modes via a flip switch. Choose the AMPLITUDE DETECTOR mode. In this mode, the unit will perform full-wave rectification to the incoming AC signal. Check that the Amplitude Detector Gain has been set to 20, and Low-Pass Gain factor • has been set to 5. For this task, we will allow the function generator to provide the external trigger to the • oscilloscope. Do this by connecting the function generator SYNC-OUT to the EXT connector of the oscilloscope, as shown in Fig. 7 below. Ensure that the SOURCE of the oscilloscope is set to that of EXT. Set up the rest of the circuit as shown in Fig. 7. Make sure that CH1 and CH2 of the oscilloscope are AC-coupled.
INPUT SIGNAL
CH1 CH2
Figure 7: The block diagram for the frequency dependence of the Low-Pass Filter. Note that CH1 of the oscilloscope reads the direct output of the function generator; CH2 reads the same signal after passing through the Amplitude Detector; and the digital voltmeter reads the same signal after passing through BOTH the amplitude detector AND low-pass filter. The purpose of the low-pass filter is to extract the dc component of its input, that is, the full-wave rectified signal coming out of the Amplitude Detector. •
•
•
As in Task 1, vary the frequency of the function generator and examine the outputs from BOTH the oscilloscope (CH1 & CH2) and voltmeter. Verify that your oscilloscope outputs agree with Figures 3(a) and 3(b) of the Reading Material, i.e. the Amplitude Detector acts as a full-wave rectifier! Plot the frequency dependence of the Low-Pass output, as given by the digital voltmeter readings. Comment on your graph.
8
Signal Processing Questions: a) What is the relationship between the signal visualized on the oscilloscope, and that of the DC value from the low-pass filter? b) Though you have been asked to set the Low-Pass Gain to 5 and keep it at this setting while you vary the frequency, is the actual Low-Pass Gain constant throughout your choice of frequencies in reality? Illustrate your answer for a few chosen frequencies. 3. Aim: Obtain the frequency dependence of the Lock-In Detector (of Lock-In Gain = 5, and Low-Pass Gain = 10). Continue to use the external function generator to generate the input signal. Procedures: You are strongly recommended to first go through the Reading Material on the • principles of Lock-In Detection, else you will just go through the motions without any understanding and appreciation of this very powerful technique. Use the flip switch to select the LOCK-IN DETECTOR mode. • Use a frequency range of 5 Hz to 5 kHz for this Task. • Set the input signal from the Function Generator to be 0.6 Vp-p. • Ensure that Lock-In is set to AC-coupling; and that it is not operating as an amplitude • detector (Figure 8). Set the oscilloscope trigger to be in sync with that from the (normal) output from the Reference Oscillator, i.e. set the oscilloscope SOURCE to be “CH1”. Set up the circuit as shown in Fig. 8 below. •
CH1 Function Generator
Input Signal Phase Shifter Input
CH2
Figure 8: The block diagram for the frequency dependence of the Lock-In Detector. •
•
•
For each frequency, adjust the relative phase between the reference and signal, via the Phase Shifter module, by using both the “Phase” and “Fine Phase” knobs, until the voltmeter reading is a maximum. The CH2 oscilloscope trace should then look like Fig. 3(b) of the Reading Material. To check, switch the phase by 90°. You should get a zero reading on the voltmeter. Make fine adjustments of the phase to obtain the best zero, then switch the phase back by 90° your voltmeter reading now should really be maximum. Write down the approximate value of this relative phase in your notebook. Plot the frequency dependence of the Low-Pass Lock-In output, as given by the digital voltmeter readings. Comment on your graph. Choose a frequency, say 1000 Hz. Plot a graph of DC voltmeter reading versus phase. Hints: Phase = Course Phase + Fine Phase
9
Signal Processing
Questions: a) By referring to the Reading Material, why does this “correct phase’ give a maximum dc output on the voltmeter? b) Is this “correct phase” the same for different frequencies? c) Does your graph of DC voltmeter versus phase agree with the Reading Material? d) Choose three frequencies 100 Hz, 1 kHz, and 5 kHz. Though you have been asked to set the Lock-in Gain to 5 and keep it at this setting while you vary the frequency, is the actual Lock-In Gain constant for these frequencies in reality?
4. Aim: Perform a formal measurement of Rwire per unit length, as a dependent variable of the supplied input voltage (Figure 9). Procedures: 3 Set the Preamp Gain = 10 , Lock-In Gain = 10, and Low-Pass Gain = 5. • The low-pass filter, when coupled together with the Preamp and Lock-In in this Task, • gives a total ideal Gain of 50000. Use a frequency of 1000 Hz for the reference frequency. At this frequency, the actual combined gain of these three devices can also be taken as 50000. Use a large input voltage (say 2 V p-p) to help in your fine-tuning of the correct phase • shift. Adjust the phase until you obtain the “rectified” output signal (CH2) in “lock-in detector” mode, like Fig. 3(b) of the Reading Material. Adjust the “HOLDOFF” knob on the oscilloscope to stabilize the waveform. Hints: (1) Use the equation on Page 6 to obtain R wire , (2) Note that V out from that • equation has been amplified by the Preamp Gain, Low-Pass Gain and Lock-In Gain when you read it from the voltmeter, (3) Refer to the Reading Material, and results from your previous Tasks, for a good choice of reference frequency. (Use short cable here) Input Signal Function Generator
CH1 CH2
Phase Shifter Input
Figure 9: The block diagram for the measurement of Rwire. Questions: a) From the results of your previous tasks, is 1 kHz a good choice of reference frequency? Explain. Hint: also refer to the Reading Material. b) How do you judge if the signal was discernable from background noise? If you have not read the compulsory reading, please do so. Important and essential information critical to this experiment are found there.
10
