User:Boleszek/Notebook/Physics 307l, Junior Lab, Boleszek/2008/09/08

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Oscilloscope Lab

 * The purpose of this lab is to acquire working knowledge of the operation of a digital oscilloscope. In particular I am to explain some things and make a few measurements.

Initial Setup
I turn the function generator and the oscilloscope on. Using standard BNC (Bayonet Neill Concelman) connectors I connect the SINE OUT of the function generator to Channel A on the oscilloscope. I do not use the SYNC OUT of the function generator because this output is preceded by a circuit (using what is called transistor-transistor logic) which converts any generated signal, regardless of its output voltage, to a 5V signal at the same frequency as the actual output signal. This SYNC signal is used to synchronize (go figure) any other instruments used in the measurement process (like a lock-in amplifier, for example) with the output of the generator. I make sure the output signal of the generator is not to large in magnitude, then I make sure my trigger marker is in the region of the signal's amplitude. I do not need to use a protective T terminator at the end of the BNC because modern day instruments, like the digital scope, have high enough input impedances to prevent too high a current from frying the circuit. Now I'm ready to make measurements.

Explanations
AC and DC coupling refer generally to the mode (AC or DC) in which a signal is acquired and displayed. AC coupling effectively filters out the DC component of a signal by using a high-pass filter (DC can be thought of as an extremely low-frequency signal), whereas DC coupling can be used to allow the whole signal to pass through (AC+DC) or just the DC (the offset). The real difference between these two, though, is the circuit that underlies their effects. An AC coupled circuit contains a capacitor in series with the signal and a resistor (usually the input impedance of the next stage of the circuit) in parallel. Ideally The series capacitor should have a high reactance so that high frequency content is easily transmitted through. A DC coupled circuit, on the other hand, is just the opposite with a capacitor in parallel and a resistor is series. The parallel capacitor should have a low reactance so that low frequency content easily passes through the slowly charging capacitor. The oscilloscope has 5 options for triggering sources: CH1, CH2, Ext, Ext/5, AC Line. Triggering literally triggers the displaying of the waveform over time when a certain criterion of electrical nature is met. These "criteria" are determined by the specific type of trigger we choose to use: edge(rising, falling), video, and pulse. For the best display we usually choose to use the input signal as the trigger so the the frequencies of both signal and trigger are synchronized.
 * The difference between AC and DC coupling
 * Triggering

Measurements and Observations
Using a BK Precision 4017A 10MHz Seep/Function Generator I create a 200Hz sine wave with DC offset=0. The output level (voltage magnitude) knob on the generator does not have indicators so I only know V_pp (peak to peak voltage) by observing the readout in the MEASURE menu on the scope screen which turns out to be 4.08V. Then I press the cursor button and place one at the lowest dip and the other at the highest peak and, to my surprise, the cursors measure 4.09V (not bad at all for manual measurements). Then I switch to the time cursors at measure the period by enclosing one full period of the sine wave. The period is .0977s which makes sense because I set my output frequency at about 100Hz. For this measurement I choose to use a rising edge trigger set at 1V for displaying the waveform and observe that the display ceases to be stable when I lower the output voltage below 1V. When channel 1 is set to AC coupling the waveform tends to align its axis with the y-axis of the scope display. As I increase the DC offset of the function generator I notice that at high enough voltage the signal's top rounds off and at low enough voltage the bottom of the signal rounds off. These effects still confuse me. Since they are not due to changes in frequency, but voltage, it isn't the coupling capacitor that causes this attenuation, or is it? How does the capacitor react to high and low voltages? I'm not sure. In order to measure the fall time of a signal I choose to use a square wave, since a voltage drop is easily observable as a slanted square (well, actually we're told to do this, but it makes sense to me anyways). I notice that the scope will be unsure in its measurement of fall time above a frequency of about 100Hz (it displays something like "19.567μs?"), and I assume this is because there is not enough time for the scope to calculate the rate at which each square wave decays when they go by so fast. While using the MEASURE function I observe that the lower the input square wave frequency, the longer the measured fall time, which indicates to me that there's something wrong with the measurement because the fall time is a characteristic of the capacitor not the square wave and should be independent of the wave (the wave just gives us a way to measure it). At about 8Hz the fall time is measured to be 54.5ms and at about 18Hz it looks to be 21.9ms. Instead of using the measuring capabilities of the scope I can measure the fall time my self keeping in mind that it is defined to be the time it takes for the function to reach 10% of its initial value. So, using cursors to measure the height of a 3Hz wave that turns out to be 5.04V high, I take the top cursor down to the voltage 500mV (about 10% of 5.04). Now I align a time cursor with the point of intersection and the other one with the sharp rise of the square wave to find the time taken for the function to reach 10% of it's max. The change in time (in other words, the fall time) turns out to be 53.00ms. From the website sited below I found a nice formula for the time constant: τ≈2.2T where τ is the fall time and T is the time constant. So T=53ms/2.2=24.0909ms. I realize, now as I edit this page from the comfort of my own home, that So I'm afraid I have no error bars. OOps....
 * Measuring signal from Function Generator
 * AC coupling
 * Measuring fall time
 * Time Constant
 * 1) I failed to make sure what the step size of the cursors was
 * 2) I did not make numerous measurements of fall time and average them to find a more accurate answer

FFT
The Fast Fourier Transform is an algorithm used to compute discrete and finite approximations of the infinite Fourier transform of a periodic signal. There are more than one kind of FFT algorithm, but I do not know which one is used by the scope. The Fourier transform is a consequence of two things: These properties of sines and exponentials can be used to show that a sine (or combination of sines) multiplied by an exponential and integrated will result in a sum of products where each product has n=m. I can't really explain this fully, but I know that this is expressed by Parseval's identity, which basically is a version of the Pythagorean theorem for infinite sums that says "(a+b)^2≠a^2+2ab+b^2 but instead (a+b)^2=a^2+b^2". For engineering and experimental purposes we simply need to know that the FFT takes an AC signal whose voltage changes in time and represents it as the sum of its fundamental (unchanging) frequency and higher harmonics. Though I didn't have much time to fool around with the FFT I got the chance to perform the FFT on a sine wave, a square wave, and a triangle wave. This was easily done by pressing the "measure" button on the scope and choosing the "FFT" function and then changing the generator waveform. To be honest I did not write any numerical values down at the time, but I do recall the qualitative aspects of what I saw. The sine wave FFT had the densest frequency content of the three, which makes sense because its Fourier expansion contains all possible harmonics. The square wave frequency content tapered down quicker than the sine, but slower than the triangle. In both the triangle and square waves it was visible that the frequency content was composed of alternating short bumps and tall spikes. The first mark was a tall spike, indicating the fundamental frequency, followed by a short bump, which ideally shouldn't even be present (filtering problems), followed by a tall spike (3rd harmonic), followed by a short bump(4th harmonic) etc. The triangle wave frequencies tapered down very quickly.
 * What is it?
 * 1) Euler's Formula, which represents sines and cosines as complex exponential quantities
 * 2) The orthogonality of sin(n) and cos(m) as well as e^n and e^m when n≠m.
 * What I measured

Acknowledgments

 * 1) -I read Wikipedia's articles and capacitive coupling, high/low pass filtering, and FFT.
 * 2) -Professor Koch explained to me what triggering was.
 * 3) -I used [http://www.hobbyprojects.com/electronic_formula/time_constants.html] to find a formula for time constant of an RC circuit.


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