6.302 Feedback Systems


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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science Feedback Systems Fall Term 2005 Issued : November 18, 2005 Lab 2 Series Compensation in Practice Due : Friday, December 9, 2005 PRELAB Prelab should be completed prior to going into the lab. You should make sure your design is correct before you start building your circuit. Please consult the TAs if you have any questions. Since this is a very long prelab, we implore you to start early. Note: this is an INDIVIDUAL project, however you are encouraged to discuss your compensation methods with other students. Introduction The purpose of this lab is to give you an opportunity to design a variety of series compensators for a system with wellknown dynamics. This prelab examines the system to be compensated, as well as the characteristics of three electrical networks which will be used as compensators. You will also analytically determine appropriate compensation strategies and select the proper component values to implement these compensators. In the following lab, you will verify that your compensators work as predicted. Familiarization and Standardization The heart of our system with well known dynamics will be a pseudo opamp, which is represented by the symbol in Figure 5. V + V Figure 5: Symbol for pseudo opamp The transfer function of the pseudo opamp is, meaning that = (V + V ). Like a standard opamp, the pseudo opamp is assumed to draw negligible current. In the laboratory, you will build a circuit which yields the following transfer function: = a 0 (τs + 1)(10 3 s + 1)(10 4 s + 1) 1. Using Matlab, construct Bode magnitude and phase plots of for each of the following cases: (a) a 0 = 10 5, τ = 1sec
2 (b) a 0 = 10 5, τ = 0.1sec (c) a 0 = 10 4, τ = 1sec (d) a 0 = 10 4, τ = 0.1sec Note that the phase crossover frequency ω φ (where = 180 ) is virtually independent of the values of both a 0 and τ. Additionally, note that the magnitude of at ω = ω φ is dependent on both a 0 and τ. 2. Consider the circuit in Figure 6. R f Figure 6: OpAmp configuration 1 (a) Show that this circuit can be represented by the block diagram in Figure 7. R f f 0 where f 0 = +R f. Figure 7: Block diagram configuration 1 (b) You are told that the loop transmission L(s) = f 0 has a phase margin of 0 when = 62kΩ and R f = 220kΩ. Find the value of τ in, given a 0 = (In the lab, you will set τ in this circuit by choosing a capacitor to produce a system on the verge of instability.) (c) Using your value for τ, find the phase margin when = 22kΩ and R f = 220kΩ. What are M p and ω n? (These resistor values in the above circuit will make up the uncompensated system in the actual lab exercise.)
3 Designing Different Compensators 3. For each of the following circuits, draw the block diagram, normalized so that the closedloop expression R f appears outside of the loop, as in Figure 8. R f f 0 H(s) Figure 8: General Block Diagram Form Note that the loop transmission is L(s) = G(s)H(s), and that H(s) may be unity. Find expressions for the appropriate constants (f 0, α, and τ L ) in terms of the component values and write the loop transmissions in the proper form for Figures 9, 11, and 10. (a) Reduced DC gain: L(s) = f 0 220kΩ 22kΩ R Figure 9: Reduced DC gain
4 (b) Lag compensation: L(s) = f 0 τ L s + 1 ατ L s kΩ 22kΩ R lag C lag Figure 10: Lag compensation (c) Lead compensation: L(s) = f 0 ατ L s + 1 τ L s + 1 C lead 220kΩ 22kΩ R Figure 11: Lead compensation
5 220kΩ 22kΩ Figure 12: Uncompensated OpAmp 4. The uncompensated system is shown in Figure 12. Note that the loop gain function for the uncompensated system is L(s) = f 0 = a 0 f 0 (τs + 1)(10 3 s + 1)(10 4 s + 1) ; a 0 = 10 5,f 0 = 1 11,τ 2 sec (1) (a) Compensate the loop gain function to have a phase margin φ M = 45 by reducing the DC openloop gain a 0 f 0. Find the value of f 0 which gives the desired phase margin. You should not needlessly sacrifice DC gain or bandwidth. (b) Use a lag compensator to achieve a phase margin φ M = 39 : L(s) = a 0 f 0 ( ) τl s + 1 (τs + 1)(10 3 s + 1)(10 4 s + 1) ατ L s + 1 Determine appropriate values for α and τ L. Again, you should not reduce the bandwidth more than necessary. Note that you may not change f 0. Using Matlab, draw root locus plots of the uncompensated and compensated systems, and explain the effect of your lag compensator. (c) Compensate the system to have φ M = 45 using lead compensation: L(s) = a 0 f 0 (τs + 1)(10 3 s + 1)(10 4 s + 1) ( ) ατl s + 1 τ L s + 1 You may not be able to meet the phase margin specification by using lead compensation alone; you may reduce the DC gain margin slightly (by reducing f 0 ) to meet the design goal. Note that α and f 0 are not independent. You should not reduce the DC gain any more than necessary. Find the appropriate values for f 0, α, and τ L. One strategy might be to maximize your phase margin using α = 11, then reduce f 0 (increase α) until you meet your phase margin specification. Again, draw root locus plots to demonstrate the effect of your lead compensator. (2) (3)
6 5. Predicted responses (a) For the uncompensated system and each of the compensated systems, determine the: i. Crossover frequency (ω c ) and bandwidth (ω b ) ii. Dominant closedloop pole locations (using Matlab) iii. Damping ratio (ζ), and natural frequency (ω n ) for 5(a)ii iv. Predicted percent overshoot (P.O.) and peak time (t p ) v. Predicted peak magnitude (M p ) and frequency (ω p ) Construct a table showing these values for each system, and make sure that this table comprises the front page of this prelab. Points will be deducted if this is not done. (b) For the uncompensated system and each of the compensated systems, use Matlab to plot: i. The openloop Bode plot ii. The closedloop Bode magnitude response iii. The closedloop unit step response 6. Component values By using the transfer functions found for each of the compensation circuits, determine the necessary values for each of the circuit components in order to successfully implement the compensators designed above: (a) R for the reduced gain compensator (b) R lag and C lag for the lag compensator (c) R and C lead for the lead compensator Make sure that you enter these values in the table at the front of the prelab as well.
7 LAB Introduction The purpose of this laboratory is to give you some experience compensating a system with wellknown dynamics. In the prelab exercises, you analytically determined three appropriate series compensation strategies: reduced gain compensation, lag compensation, and lead compensation. In the lab you will verify that they work as predicted. The process is similar to that seen in lecture, and the material is also covered in Roberge s Operational Amplifiers pages We have changed the numbers and are using an inverting rather than a noninverting topology, but the ideas are the same. This is an INDIVIDUAL project  you may not work with a partner when building your circuit. Equipment At the equipment desk, be sure to pick up the following: Lab 2 component kit, EZ hook leads, a decade capacitor, and some hookup wire. You will also need to pick up the parts that you calculated in the prelab for the various compensation schemes. The Pseudo OpAmp Your experimental system will be the following pseudo opamp, which gives wellknown and quite repeatable opamp like characteristics. The circuit illustrating the pseudo opamp is shown in Figure LM 301A 8 6 2kΩ 0.075µF 4.3kΩ 0.15µF Figure 13: The Pseudo OpAmp Your first step is to build the pseudo opamp circuit, being sure to use an LM301A (not a 741) for the opamp. The 301A has the same pinout as the 741 (see the drawings posted on the walls in the lab); do not forget to power the opamp with +15V to pin 7 and 15V to pin 4. For the RC network on the output of the 301A, use the precision components (1% resistors and 5% capacitors) included in the component kit. Use the decade capacitor connected by short leads for the capacitor denoted by C; you will adjust the value during the standardization portion of the lab. Use disc capacitors (0.01µF) to decouple the ±15V power supplies place them on the board between (1) +15V and ground, and (2) 15V and ground. We will symbolize the pseudo opamp as in Figure 14. Throughout the course of this lab, you will be asked to use the pseudo opamp in a number of circuits. It is imperative that you use one pseudo opamp circuit for all experiments. That is, if asked to compensate the pseudo opamp in two ways, do not build two completely separate pseudo opamp circuits. Also be careful not to burn out the real opamp in the
8 V + V Figure 14: Pseudo OpAmp Symbol circuit. The 301A reacts very poorly (usually by dying) if pins 1, 5, or 8 are shorted to almost any potential, including ground. If you destroy an amplifier after you have standardized it, you will have to start the lab over again. Standardization In all parts of the lab, the pseudo opamp will be used to form an inverting amplifier as shown in Figure 15. R f Figure 15: Basic Inverting Amplifier using Pseudo OpAmp If we assume that the 301A has a transfer function which is dominated by a single low frequency pole at s = 1 τ, where τ >> 10 3 sec, the open loop transfer will be of the following form: f(s) = a 0 f 0 (τs + 1)(10 3 s + 1)( ) where f 0 = + R f (4) The remaining two poles at 10 3 and 10 4 rps are associated with the RC network, including the output impedance of the 301A. Not surprisingly, we see that the openloop transfer function depends upon two of the real opamp characteristics its DC gain a 0 and the location of the low frequency pole ( 1 ). Hence, the openloop transfer function will in τ general vary somewhat depending on the characteristics of the particular circuit which you will build. In order to obtain experimental results which agree with your analytic prelab calculations, it will be necessary to standardize the opamp circuit which you have built. Since we will be concerned with the issues of compensation and the degree of stability in the lab, standardization will consist of making the behavior of the pseudo opamp in the vicinity of the anticipated crossover frequency a known function. Of the two parameters a 0 and τ, we have no control over a 0, but we have complete control over the value of τ by changing the compensation capacitor C (the capacitor between pins 1 and 8 of the 301A) using the decade capacitor. The following procedure can be used to standardize your pseudo opamp circuit:
9 1. Build the inverting amplifier circuit of Figure 15, setting = 62kΩ and R f = 220kΩ using 1% resistors. 2. The driving source impedance in this lab must be low. A resistive divider attenuating the signal generator output and located close to the pseudo opamp is suggested. A suggested attenuator is shown in Figure 16. Note that the output of the attenuator (To circuit) provides the input to your amplifier setup ( ). 1kΩ To Circuit Signal Generator 10Ω Figure 16: Attenuation Circuit 3. Apply a low frequency, small amplitude square wave to the attenuator. The assumptions required for linear analysis are severely compromised if the peak to peak magnitude of (the output of the attenuator) exceeds 50mV. (In most cases, inaccurate lab results are traceable to this sort of error.) 4. Adjust the value of the compensating capacitor C (the decade capacitor) so that the circuit is on the verge of instability. The longer the step response rings before dying out, the closer the poles are to the jω axis. The record for ring time is about 10 sec, but 0.2 to 0.5 sec is sufficient. A ballpark value for C is on the order of 5000 pf. Be sure that the circuit is ringing and not oscillating. Note that this procedure is the experimental equivalent to process of determining τ analytically in the prelab to achieve a system with zero phase margin. Do Not Change The Compensation Capacitor C For The Remainder Of The Lab
10 Initial Measurements Now connect your standardized pseudo opamp circuit as an inverting gain of ten amplifier by replacing the 62kΩ resistor with a 22kΩ resistor. Again, you should use a 1% precision resistor. This circuit will comprise your basic uncompensated system, with properties calculated in the prelab. This is illustrated in Figure kΩ 22kΩ Figure 17: Uncompensated System (Gain of 10 Inverting Amplifier) Measure the time and frequency domain responses of your amplifier (damped natural frequency ω d, peak time t p, and percent overshoot P.O.; peak magnitude M p, and peak frequency ω p.) Deduce the equivalent damping ratio and natural frequency, and compare these values with your predictions from the prelab. Compensation Your are now ready to compensate the system to improve its phase margin by: 1. Reducing the DC openloop gain 2. Lag compensation 3. Lead compensation You may not change the value of the compensating capacitor C or elements in the RC network of the pseudo opamp, or load the network unreasonably to implement your compensation. The circuits for each compensated system are shown in the prelab. You should use the component values calculated in the prelab for R (reduced gain), R lag and C lag (lag compensator), and R and C lead (lead compensator). Take time domain and frequency domain measurements for each compensated system, and deduce the damping ratio, natural frequency, and phase margin. You should measure the bandwidth for each system as well. Convince yourself that the step and frequency responses that you measure are in reasonable agreement with your prelab designs.
11 Writeup You should include the following: 1. Your prelab, including block diagrams, Bode plots, root loci, closedloop pole/zero locations, and predicted responses for the uncompensated and compensated systems. 2. The component values you used to implement each compensator. 3. Measured responses and calculated closedloop parameters (ζ, ω n, and ω b ) for each compensator. 4. A discussion of the advantages and drawbacks of each compensator (such as DC gain, bandwidth, transient behavor, etc.) Checkoff Make sure that you do not disassemble your compensator until after checkoff, as the TAs will want to see the circuit which you built. Checkoff will be scheduled during the week of December 5.
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