1 Lock-in Amplifier Introduction

Transcription

1 1 Lock-in Amplifier Introduction The purpose of this laboratory is to introduce the student to the lock-in amplifier. A lock-in amplifier is a nearly ubiquitous piece of laboratory equipment, and can serve several functions. It is also the most sensitive piece of equipment that is commonly found in modern labs. Certainly there are more sensitive devices, but they are specialized. Lock-in amplifers are common. References: Melissinos and Napolitano, p. 144 (section 3.8). A basic lock-in amplifier experiment for the undergraduate laboratory, Libbrecht, Black, and Hirata, Am. J. Phys. 71, 1208, (2003). (Attached.) The lock-in amplifier we will use is produced by Stanford Reserch Systems, model SRS-810. It is a digital signal processor lock-in, meaning that much of the circuitry and processing inside is digital, whereas the classical lock-in is a purely analog device. Do not be confused, though. The input of this device is capable of nano-volt measurements! WARNING: This is probably the most sensitive device you will use this semester. Treat it carefully! First, you will demonstate the ability of a lock-in amplifier to find a (relatively large) signal buried in (a very large amount of) noise. This will be quick and straightforward. Next, and this is the real part of the lab, you will measure the resistance and inductance per unit length of a short piece of 20 gauge copper wire that I will supply. The resistance that you will measure is some few milli-ohms. This is much too small to see with an ordinary hand-held meter. Nonetheless, you will find that you can make the measurement with an uncertainty of only a percent or so. I will not tell you the inductance of the wire, but you will find that it is just as easy to measure. 2 Demonstration In the demonstration, you will see the ability of a lock-in amplifier to find a signal in the presence of a lot of noise. The lock-in can do this because the signal will be modulated at a specific frequency, whereas the noise is broad-band. By looking at exactly the frequency that the signal is modulated, the lock-in can reject most of the noise. In this demonstration, the signal will be the output of an LED that is modulated by an optical chopper. 1

2 On the lab bench is an LED and a photo-diode. Both are battery operated, so turn them both on. The LED puts out about 25 mw, so it s not very bright. Put the LED about 1 m from the photodiode. Turn off the room lights, and observe the light from the LED. Make sure it is pointed at the photodiode. Connect the output of the photodiode to the channel-1 input of the oscilloscope. Set the scope to auto-trigger, so that there is a trace I recommend a sweep time of 5 ms per division. Set the vertical zero to be at the bottom of the screen. Set the vertical gain so that the signal from the photodiode is around the middle of the screen. Using the switch on the box holding the LED, turn the LED on and off, and see that the signal on the scope goes to zero and back. Now, turn on the room lights and do all this again. You should find that there is no way (without moving the source or detector, or blocking the lights) that you can see the signal from the LED with the room lights swamping the signal. This is an important lesson! It is common that you want to see a signal in the presence of way too much background noise. So what do you do? Turn on the lock-in amplifier, and wait for it to finish its self-tests. Set the time constant to 1 second and the sensitivity to 1 Volt. Turn on the optical chopper, and connect the reference output of the chopper to the reference input of the lock-in. This enables the lock-in to look at signals modulated at a specfic frequency, and at a specific phase with respect to that frequency. Place the optical chopper so that the light from the LED goes through the blades of the chopper. WARNING: The chopper blades are spinning very rapidly. Take care not to get your fingers or other body parts caught in the blades! Connect the output of the photo-diode to the signal input of the lock-in amplifier. Now, slowly increase the sensitivity of the lock-in. Change it in steps, from 1 V, to 500 mv, 200 mv, 100 mv, 50 mv, etc. Take your steps slowly, so you don t saturate the input. You should only change the sensitivity about once per the amount of time you have programmed as the time constant of the lock-in (1 second in this case.) At some point, you should get a nice clear signal that is very constant. Now, turn off the LED with the switch on the box. Now, turn it back on. See the signal come and go, even with all the background noise from the room lights! This should be an impressive demonstration of the power of a lock-in amplifier. You may find that the phase of the signal is different from the phase of the reference coming from optical chopper. this is because as you change the position of the blades, the exact moment when they block and unblock the LED changes. The lock-in can find this phase shift. Just press the auto-phase button, and you may get a boost in signal. You can also try lowering the time constant to 300 ms or 100 ms. Doing so will result in less signal-to-noise, but you may have so much that you can spare it! The result will be a much more responsive device to changes in the output of the LED. 2

3 When you are done, turn off the battery operated LED and photo-detector, and turn off the optical chopper. Move the LED and chopper out of the way (being careful with the chopper.) 3 Measurements You have been given a short piece of 20 gauge copper wire. You are to measure the resistance and inductance per unit length of the wire. For your convenience, I stripped both ends, and soldered a 1.2 kω resistor to one end of the wire. Measure the length of the copper wire. Measure the resistance of the resistor that I soldered on. (Never trust the colored bands always check it with a meter.) Connect the lo output of the Wavetek model 182A function generator to the oscilloscope, and set the function generator to produce a 100 Hz sine wave with 1 V amplitude and zero DC offset. Now, use a BNC tee and connect the output of the function generator to both the reference input of the lock-in and across your wire and resistor. Use one of the cables that ends with easy-hooks to connect to the wire. Set the input sensitivity of the lock-in to 1 V and the time constant to 100 ms. Use another cable with easy-hooks to connect the the same voltage (wire and resistor) to the signal input of the lock-in. The lock-in will display the RMS voltage at its input, which should be about V if you correctly set the function generator amplitude to 1 V. Now, move one of the easy-hooks that is connected to the lock-in input so that it is looking only across your wire. The signal will be very low. Slowly increase the sensitivity, from 1 V in steps down, until you get a clean reading of the RMS voltage across your wire. The signal probably will not be very stable. Now, increase the timeconstant of the lock-in from 100 ms to 3 s. You will now always have to wait about seconds (3-5 time constants) after any change before making a measurement. But now, the voltage read by the lock-in should be stable, and roughly 5 micro-volts. You have just measured a 5 micro-volt signal with high precision! looks like this: The set-up Wavetek function generator Va 1.2 kω Clip here for full voltage Here for small voltage easy hookers wire V b SRS 810 lock in input reference Figure 1: Experimental arrangement for measuring voltage across a short piece of wire. 3

4 Complex voltage Now, change the phase of the lock-in by +90 degrees. You are now looking at the voltage that is 90 degress out of phase from the reference. This should be nearly zero the voltage across your wire is in phase with the driving voltage. If you think of the voltage across the wire as a complex function, V = Ae iωt, then the voltage measured at zero phase is Re[A] and the voltage measured at +90 is Im[A]. You should now repeat these measurements for different frequencies of the sine wave. Measure both real and imaginary parts of the voltage (both in-phase and +90 out of phase) across the resistor+wire (V a ), and across the wire only (V b ), at 10 Hz, 30 Hz, 100 Hz, 300 Hz, 1 khz, 3 khz, 10 khz, 30 khz, and 100 khz. (The maximum frequency of this particular lock-in is 100 khz.) Be careful when you change the hooks from the large voltage to the small it is good to avoid putting a large voltage on the lock-in when the sensitivity is set for a low voltage. You expect the data to be something like this: Freqency (Hz) V a (Volts) V b (µ-volts) in-phase +90 in-phase When I took the above data, I made an error. Your data should be better. Now, cut the wire so it is 1/2 its original length and repeat the measurement. 4 Analysis First, plot your data. I want you to plot the real and imaginary parts (in-phase and +90 ) of the small voltages across your wire as a function of frequency. For the data above, the plot is shown in Fig. 2. What does this mean? Well, it is clear that if we want the real resistance then we have to use the low frequency measurements. Something is making the voltage drop at high frequency. So, using your data, get a good average for the real part of the resistance of your wire at low frequencies. From the circuit in Fig. 1 we see that: V b = V a R 1 + R 2 R 2 4

5 Figure 2: Real (solid circles) and imaginary (open circles) voltage across wire as a function of frequency. where R 1 is the resistance of the 1.2 kω resistor and R 2 is the wire. approximation that R 2 R 1 we see Making the R 2 = R 1 V 2 V 1 This gives you the resistance of the wire. Your answer should be smaller than anything you could have measured with an ordinary instrument! Next, look at the imaginary part of the resistance as a function of frequency. For high frequency, you see that the voltage increases linearly with frequency, just as for an inductor. Using the relationship Im[V ] = Lω gives the self-inductance of the wire. How can you verify that you are measuring an inductance and not some other frequency-dependent effect, such as capacitance? Repeat this analysis for the 2/3 and 1/3 lengths of the wire. Show how the resistance and inductance depend on the length. Derive your best estimate for the resistance and inductance per unit length of this wire. Estimate your uncertainty in these results. Look up the conductivity of copper. The diameter of the wire is inches. What number does this give for the resistance per unit length, and how does that compare to your measured value? 5