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Review of Feedback Systems CHAPTER 1 Review of Feedback Systems Marc Thompson Dr. Marc Thompson leads us to an appreciation of how the world has learned about FEEDBACK (negative) over the years, so we can understand how to do better feedback in our systems. /rap Introduction and Some Early History of Feedback Control A feedback system is one that compares its output to a desired input and takes corrective action to force the output to follow the input. Arguably, the beginnings of automatic feedback control 1 can be traced back to the work of James Watt in the 1700s. Watt did lots of work on steam engines, and he adapted 2 a centrifugal governor to automatically control the speed of a steam engine. The governor comprised two rotating metal balls that would fl y out due to centrifugal force. The amount of “ fl y-out ” was then used to regulate the speed of the steam engine by adjusting a throttle. This was an example of proportional control. The steam engines of Watt ’ s day worked well with the governor, but as steam engines became larger and better engineered, it was found that there could be stability problems in the engine speed. One of the problems was hunting , or an engine speed that would surge and decrease, apparently hunting for a stable operating point. This phenomenon was not well understood until the latter part of the 19th century, when James Maxwell 3 (yes, the same Maxwell famous for all those equations) developed the mathematics of the stability of the Watt governor using differential equations. 1 Others may argue that the origins of feedback control trace back to the water clocks and fl oat regulators of the ancients. See, e.g., Otto Mayr’s The Origins of Feedback Control , The MIT Press, 1970. 2 The centrifugal governor was invented by Thomas Mead c. 1787, for which he received British Patent #1628. 3 James C. Maxwell, “On Governors,” Proceedings of the Royal Society , 1867, pp. 270–283. www.newnespress.com CCH001-H8627.inddH001-H8627.indd 1 44/17/2008/17/2008 22:40:05:40:05 PPMM 2 Chapter 1 Invention of the Negative Feedback Amplifi er We now jump forward to the 20th century. In the early days of the telephone, practical diffi culties were encountered in building a transcontinental telephone line. The fi rst transcontinental telephone system, built in 1914, used #8 copper wire weighing about 1000 pounds per mile. Loss due to the resistance of the wire was approximately 60dB. 4 Several vacuum tube amplifi ers were used to boost the amplitude. These amplifi ers have limited bandwidth and signifi cant nonlinear distortion. The effects of cascading amplifi ers ( Figure 1-1 ) resulted in intolerable amounts of signal distortion. Harold Black graduated from Worcester Polytechnic Institute in 1921 and joined Bell Laboratory. At this time, a major task facing AT & T was the improvement of the telephone system and the problem of distortion in cascaded amplifi ers. In 1927, Black 5 was considering the problem of distortion in amplifi ers and came up with the idea of the negative feedback amplifi er ( Figure 1-2 ) . “ Then came the morning of Tuesday, August 2, 1927, when the concept of the nega- tive feedback amplifier came to me in a flash while I was crossing the Hudson River on the Lackawanna Ferry, on the way to work. For more than 50 years I have pon- dered how and why the idea came, and I can ’ t say any more today than I could that morning. All I know is that after several years of hard work on the problem, I sud- denly realized that if I fed the amplifier output back to the input, in reverse phase, and kept the device from oscillating (singing, as we called it then), I would have exactly what I wanted: a means of canceling out the distortion in the output. I opened my morning newspaper and on a page of The New York Times I sketched a simple diagram of a negative feedback amplifier plus the equations for the amplification with feedback. I signed the sketch, and 20 minutes later, when I reached the labora- tory at 463 West Street, it was witnessed, understood, and signed by the late Earl C. Bleassing. I envisioned this circuit as leading to extremely linear amplifiers (40 to 50 dB of negative feedback), but an important question is: How did I know I could avoid self-oscillations over very wide frequency bands when many people doubted such circuits would be stable? My confidence stemmed from work that I had done two years earlier on certain novel oscillator circuits and three years earlier in designing the terminal circuits, including the filters, and developing the mathematics for a carrier telephone system for short toll circuits.” 4 William McC. Siebert, Circuits, Signals and Systems , The MIT Press, 1986. 5 Harold Black, “Inventing the Negative Feedback Amplifi er,” IEEE Spectrum , December 1977, pp. 55–60. See also Harold Black’s U.S. Patent #2,102, 671, “Wave Translation System,” fi led April 22, 1932, and issued December 21, 1937, and Black’s early paper “Stabilized Feed-Back Amplifi ers,” Bell System Technical Journal , 1934. www.newnespress.com CCH001-H8627.inddH001-H8627.indd 2 44/17/2008/17/2008 22:40:05:40:05 PPMM Review of Feedback Systems 3 A typical closed-loop negative feedback system as is commonly implemented is shown in Figure 1-3 . The “ plant ” in this diagram might represent, for instance, the power stage in an audio amplifi er. A properly designed control system can maintain the output at a desired level in the face of external disturbances and uncertainties in the model of the plant. The goal of the feedback system is to force the output to track the input, perhaps with some gain and frequency-response shaping. vi a1 a2 an vo Figure 1-1 : Amplifi er cascade. v v ϩ e a v i Ϫ o v f f (a) GAIN FREQUENCY CHARACTERISTIC AS FUNCTION OF DEGENERATIVE 100 FEEDBACK 561 559 551 550 556 560 562 90 NO 555 552 568 R FEEDBACK R 80 Ro 2 565 564 557 KR Ro2 567 554 553 KR 70 OPERATING RANGE 558 o KR2 KR 60 o2 566 563 FEEDBACK 1 50 571 572 573 GAIN-DB f 40 GRID OF FIRST TUBE PLATE OF LAST TUBE 30 FEEDBACK 2 C G o L 20 V V1 V2 Ro E mV 10 102 2468 103 2 468 104 2468 105 FREQUENCY, CYCLES (b) Figure 1-2 : Classical single-input, single-output control loop, as envisioned by Black. (a) Block diagram form. (b) Excerpt from Black ’ s U.S. patent #2,102,671, issued in 1937. www.newnespress.com CCH001-H8627.inddH001-H8627.indd 3 44/17/2008/17/2008 22:40:05:40:05 PPMM 4 Chapter 1 Disturbance Input Output Amplifier/ Comparator Plant Compensator Feedback Element Figure 1-3 : Typical feedback system showing functionality of individual blocks. In this confi guration, the output signal is fed back to the input, where it is compared with the desired input. The difference between the two signals is amplifi ed and applied to the plant input. In order to design a successful feedback system, several issues must be resolved: • First, how do you generate the model of the plant, given that many systems do not have well-defi ned transfer functions? • Once you have the model of the plant, how do you close the loop, resulting in a stable system with a desired gain and bandwidth? Control System Basics A classical feedback loop, as envisioned by Black, is shown in Figure 1-4 . Note that there is an external disturbance in this system, the voltage vd . In this system, a is the forward path gain and f is the feedback gain. The forward gain a and feedback factor f may have frequency dependence (and, hence, the plant should be denoted as a ( s )), but for notational simplicity we ’ ll drop the Laplace variable s . vd ve ϩ v ϩ a ϩ v i Ϫ o v f f Figure 1-4 : Classical single-input, single-output control loop, with input voltage v i , output voltage v o , and external disturbance vd . www.newnespress.com CCH001-H8627.inddH001-H8627.indd 4 44/17/2008/17/2008 22:40:06:40:06 PPMM Review of Feedback Systems 5 Initially, let’ s set the disturbance vd to zero. The “ error ” term v e is the difference between the input and the fed-back portion of the output. We can solve for the transfer function with the result: ϭ vavoe [1-1] vvvei f vfv fo The closed-loop gain is (A ϵ closed-loop gain): v a A ϭϭo [1-2] v ϩ af i 1 or: FORWARD GAIN A ϭ [1-3] Ϫ LOOP TRANSMISSION 1 Note what happens in the limit of af ϾϾ 1: 1 A [1-4] f This is the key to designing a successful feedback system; if you can guarantee that af ϾϾ 1 for the frequencies that you are interested in, then your closed-loop gain will not be dependent on the details of the plant gain a ( s ). This is very useful, since in some cases the feedback function f can be implemented with a simple resistive divider . which can be cheap and accurate. Loop Transmission and Disturbance Rejection The term in the denominator of the gain equation is 1 ϩ af , where the term − af is called the loop transmission , or L.T . This term is the gain going around the whole feedback loop; you can fi nd the L.T.
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