9 Feedback and Control

Transcription

1 9 Feedback and Control Due date: Tuesday, October 20 (midnight) Reading: none An important application of analog electronics, particularly in physics research, is the servomechanical control system. Here the concept of feedback is generalized and used to control almost any physical variable. We shall spend several labs constructing and studying a servo for a simple system consisting of an LED and a photodiode. The concepts, however, are universal and apply to any servo you may need. 9.1 LED Driver This exercise will work toward the construction of the circuit in Fig. 6. This is considerably more complex than anything we ve done up to now, so we will build it up gradually. However, plan from the start for what you are doing: think about where you will place the op-amps on your circuit board, and how you will get power to them all. Use short wires where you can, to reduce circuit clutter. Lay out each sub-circuit in a clear and logical way. Also, we will be working with this circuit for the next few labs, so keep good notes in your lab book about how the components are laid out and what each sub-circuit does. The starting point is the sub-circuit of Fig. 1, which applies a current to a light-emitting diode (LED) proportional to an input voltage. An LED is an ordinary diode that is optimized and packaged to produce light in response to a forward current. We want to apply a little more current than the op-amp can provide, so we use a transistor follower. However, you should avoid letting V in exceed 5 V, and for continuous operation about 3 V is better for both the diode and the resistor. Wire up the circuit, using one of the large red LEDs. Note that one of the leads is shorter than the other: this distinguishes the polarity of the diode. You can check which lead is which using your DMM diode tester. Drive the circuit with your variable power supply and verify that you can adjust the brightness of the light by varying V in. Figure 1: LED driver circuit. 1

2 anode out cathode anode cathode Figure 2: Photodiode detection circuit. See also Fig. 3. Figure 3: Identifying the polarity of the PNZ335 photodiode. 9.2 Photodiode Now that we have a light source, we will build a servo-mechanism to stabilize it. The first step in that process to measure the light level, which we will do with the circuit of Fig. 2. Set this up with the tip of the LED pointed right into the sensing surface of the photodiode. (In real life, we would probably use a lens to collect light from the LED and focus it onto the photodiode, but simply putting them right next to each other will work well enough here.) Notice that the 20 pf capacitor on the op-amp feedback forms a low pass filter with a cutoff frequency of about 10 khz. This simplifies the high-frequency response of the system for now, but we shall eventually want to omit it. Also note that the orientation of the photodiode is of some significance. The pin layout is described in Fig. 3, along with the directionality of the photocurrent. For now, orient the photodiode so that the circuit output is positive, but we may need to change the polarity later on. Flipping around the photodiode itself would require you to move the LED, so instead make sure that the wires running from the photodiode remain accessible so you can switch them instead. With the LED off, you should be able to see a signal from the room lights, including both a dc and a 60 Hz component. With the LED on at 3 V, you should see a photodiode response of a few volts. 9.3 Summing Amplifier In order to design an effective servo system, we need to know the frequency response of the driver-detector combination. The ELVIS Bode tool can measure this, but in order to turn the LED on, we need to supply a dc voltage along with the ac signal in whose response we are interested. The ELVIS tool has no facility for that, so we will add a dc offset to the circuit using the summing amplifier of Fig. 4. (We will need a summing circuit in any case when we apply feedback from the photodiode to the LED, so it s no extra work to build it now.) Construct the sub-circuit and verify that the output provides an ac signal added to a 3 V dc level, as desired. Hook the output up to the input of the LED driver. Drive the 2

3 FGen 1N746A (3.3 V) 2k to LED driver Figure 4: Summing amplifier. circuit with your function generator at a frequency of 1 khz and amplitude of 1 Vpp. The photodiode output should show a response with comparable (within a factor of two or so) amplitude. Of course, the summing amplifier inverts the signal; that is why the zener diode is arranged to give a negative voltage reference. This means the photodiode signal should now be 180 degrees out of phase with the input. To compensate, you can either use the inverted op-amp polarity setting on the Bode analyzer tool, or you can reverse the photodiode leads. Now hook the circuit input up to the ELVIS Scope 0 connector and the photodiode output up to ELVIS Scope 1, while leaving the function generator also attached to the circuit input. For the measurements to follow, you can eliminate noise from the room lights by covering your circuit with a piece of paper. Run the Bode tool with an input amplitude of 1 Vpp, a frequency range from 10 Hz to 200 khz, and 10 points per division. Copy the data (for both phase and gain) into your Excel spreadsheet, since you will need it later. For now, plot the data in Excel, and verify that the features make sense. It is possible that the phase measurement will fall below -180, in which case the analyzer wraps the phase around to near Your plots will be easier to interpret if you unwrap the phase in Excel by subtracting 360 from the appropriate points. In fact, the 200 khz frequency limit on the Bode tool is a little lower than we would like. Further, the accuracy of the tool is not always good at the highest frequencies. To rectify this, take a few higher frequency points by hand, using the function generator and your scope. Obtain the gain and phase at 100 khz, 200 khz, 500 khz and 1 MHz. Add the values to the Bode plot you already had and make sure they agree reasonably well with what the analyzer tool gave you. Note that if did not switch the leads on the photodiode to make the output and input have the same polarity, you can invert the signal on the scope to compensate. This is all you need for now, but later on we will remove the capacitor from the photodiode circuit to get a faster response time. For reference, take the capacitor out and repeat the frequency response measurements just as above, using the analyzer tool and taking highfrequency points by hand. Save and plot the data in Excel. You should be able to see the complication we are avoiding for now by using the capacitor. When you are done, put the 3

4 G LED Photodiode - V err V set (a) H From photodiode V err V set 100k +15 V To summing amplifier 1k (b) Figure 5: Feedback loop for LED servo. (a) Block diagram. (b) Sub-circuit. capacitor back in. 9.4 Feedback When the circuit is not covered, it should be easy to see how the output signal is affected by the room lights. We suppose that this is a problem... perhaps we have an experiment at the location of the photodiode that requires constant illumination. The idea of the servomechanism is to feed back the photodiode signal to the LED driver in such a way as to compensate for the noise. If the room lights get brighter, the LED would get dimmer, and vice versa. Achieving this requires three steps. First, we need to establish a desired signal level, or set point. Second, we subtract the photodiode signal from the set point to obtain an error signal. Third, we amplify the error signal to an appropriate level and apply it to the driver input. A block diagram of this process is shown in Fig. 5(a), and the corresponding circuit diagram in 5(b). Construct this sub-circuit, and attach it to the main circuit as in Fig. 6. Before turning it on, however, consider the polarity of the feedback. We require the feedback to be negative: if the room light level increases, the circuit should reduce the LED current to compensate. The actual sign of the feedback depends on whether it passes through an even or odd number of inverting amplifiers in the loop, and also on the polarity of the photodiode itself. Examine Fig. 6 and try to determine which photodiode polarity is required. (Note that the polarity shown in the diagram may or may not be correct.) Set the photodiode polarity as you think is necessary, and adjust the 100k feedback pot initially to zero Ω so that no feedback is actually present. Turn on the circuit and monitor 4

5 the error signal on your oscilloscope. Adjust the set-point potentiometer so that the error signal is close to zero, and then gradually turn up the feedback resistance. If everything is working correctly, the error signal should move toward zero and the noise in it should be reduced. The noise should continue decreasing to a minimum level, but when you increase the gain further, the error signal starts to oscillate at a high frequency. If the error signal oscillates immediately, or as soon as you increase the gain resistor slightly, then you probably have the photodiode polarity wrong. This is a common problem, and servomechanisms often have an inverting switch to reverse the sign of the feedback when needed. Here you can achieve that end by swapping the photodiode wires. 9.5 Performance Analysis It may take some debugging, but you should be able to get the servo working. We would like to characterize how well it works, by measuring how well it attenuates noise at a given frequency. We can measure this most easily by feeding another noise signal into the LED driver, as shown in Fig. 6. Use the Bode analyzer to measure the response of the error signal to the noise input, from 10 Hz to 200 khz. This servo response is slow enough now that nothing interesting happens at higher frequencies, so you don t need to take any points by hand. First, remove the feedback by taking out the wire from the feedback amplifier to the summing amplifier, and measure the open loop response. You may want to cover the circuit again for this and the following measurement. Now reattach the wire, and set the feedback gain to a point a bit below where the oscillations set in, and take the Bode plot again. Load both sets of data into Excel and plot them on the same graph. The difference between the curves shows the amount of noise reduction that the servo provides. Calculate and plot the noise attenuation factor by dividing the closed-loop response by the open-loop response, bearing in mind that to divide signals, you can simply subtract the gains in db and subtract the phases. Don t be surprised if, at some frequencies, the noise is actually higher for the closed-loop curve. This is called noise peaking, and occurs when the servo is at the edge of its stability. If you increase the feedback gain until the system just starts to oscillate, you should find the oscillation frequency is close to the peak in the closed-loop noise. We can compare the servo performance to what we expect. If the system transfer function is G and the feedback transfer function is H, theory predicts that the servo should reduce the noise by a factor of 1 + GH. Here G is the response function of the LED/photodiode system that we measured in Section 9.3, while H is the transfer function for the feedback amplifier of Fig. 5(b). Measure H now, by replacing the input from the photodiode with the function generation and monitoring the output with the Bode analyzer. Set the input amplitude to be 0.1 V so the op amp doesn t saturate. Measure the response from 10 Hz to 200 khz and plot in your report. Using this and your earlier data, calculate the loop transfer function GH, and plot its magnitude and phase. Then calculate 1/(1 + GH), and compare it to the noise attenuation factor that your servo achieves. You should see that the two curves agree reasonably well. You should also find that the loop phase (the phase of GH) reaches -180 when the loop gain ( GH ) is a bit below zero db, and at a frequency slightly higher than where you 5

6 observed noise peaking and oscillation. This is because instability occurs when the phase is -180 at unity gain, and by turning the feedback gain up to nearly the point of oscillation, you put the circuit near the point of instability. The phase margin of a servo is defined as the difference between the loop phase and -180 at the frequency where the loop gain reaches 0 db. What is the phase margin for your circuit? What circuit element(s) cause this -180 phase shift? In the next few labs, we will attempt to improve this circuit in two main ways. First, we can try to speed up the feedback loop so that noise attenuation at higher frequencies is possible. Second, we can provide more loop gain at lower frequencies to improve the noise reduction there. Leave your circuit set up for the next lab. Also keep your report handy, because we will be using some of the data in it again. Noise input 1N746A +15 V 2k 2N pf 1M Set point V err 100k +15 V 1k Figure 6: Complete servo circuit. All op amps are LF411s, and unlabeled resistors are 10 kω. 6