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RAT Feedback Stability FAQ Project
(Notes: I've included some comments from the other Engineer type folks in the newsgroup (whom I have collectively dubbed "the Peanut Gallery"), whose insights the reader may find educational..if one wishes to see an analysis of a typical audio power amp, please scroll to the bottom of the page, where Mr Bench dissects a Dynaco Modified Wiliiamson amplifier, whose schematic is contained on this site)
Prior to creating a feedback amplifier and stabilizing it, a conventional amplifier is presented. This example creates a 2 stage 6SN7GTA/B amplifier, analyzes it for gain, finds the poles and zeros of the amplifier, and presents the frequency response from the analyzed amplifier.
BOUNDARY CONDITIONS:
2 Stage 6SN7GTA or GTB amplifier.
Filament: 6.3V AC or DC, 600 mA.
High Tension: 400 volts DC. Regulated and filtered.
Input: 1 volt RMS behind a source resistance of 10k.
Output Load: 31.6k and 100pF load capacitance.
Standard 1 or 5% valued components utilized.
The Circuit:
0.15 uF to first stage grid, 210k resistor grid to ground.
First stage cathode: Unbypassed cathode resistor of 887 ohms.
First stage plate (anode): 22k resistor to high voltage supply.
First stage plate to second stage grid: 0.22 uf.
Second stage grid 71.5k resistor to ground.
Second stage cathode: 715 ohms paralleled with 68 uF cap to gnd.
Second stage plate: 12k resistor to high voltage supply, 1 uF to the output.
DC Conditions:
The first stage is biased at 7 volts on the cathode and runs at 8 mA. The Plate voltage is 225 volts. Under these conditions, the 6SN7 has a plate resistance of 8000 ohms (neglecting the cathode degeneration temporarily), and a transconductance of 2.5 mS. The second stage is biased at 8 volts on the cathode and about 11.3 mA. The plate voltage is 265 volts. Under these conditions, the 6SN7 has a plate resistance of 7200 ohms and a transconductance of 2.8 mS. (Thats 2800 umhos to some of us.) Notice the second stage tube is dissipating slightly less than 3 watts, no problem.
Midband Gain: There is about 0.5 dB loss from the voltage division of the 10k source resistance and the 210k grid resistor. The first stage gain is about 4.2, or 12.5 dB. (The way to get this is to take the TOTAL plate circuit resistance divided by the total cathode circuit resistance... 8k paralleled with 22k paralleled with 71.5k (5.42k) divided by 887+400 (1/gm)). The second stage gain is about 11, or 21 dB. (12k paralleled with 7.2k paralleled with 31.6k (3.94k) divided by 357 (1/gm)). Thus, the overall midband gain is about 44, or 33 dB.
Low Frequency Poles and Zeros: The input coupling capacitor forms a pole at 5 Hz. (1/2*pi*r*c). There is a zero at DC. The coupling from the first to the second stage forms a pole at 10 Hz. There is another zero at DC. The coupling to the output forms another pole at 5 Hz. There is a zero at DC. The cathode bypass on the second stage forms a pole- zero pair. The zero is at 1.66 Hz, and the pole is at 5 Hz. This is a little trickier to figure, and can best be visualized as follows: As you DECREASE frequency, the capacitor begins to look like an open circuit, causing gain to decrease as you LOWER frequency. This is the pole position. At some yet lower frequency, the capacitor is invisible and some lower gain is produced. This is the zero. The pole is at f = 1/(2*pi*c*(Rk paralleled with (1/gm))). The zero is at f = 1/(2*pi*c*Rk). Summary: 3 zeros at DC, a zero at 1.66 Hz, 3 poles at 5 Hz, 1 pole at 10 Hz. High Frequency Poles: Due to the finite source resistance, there is a high frequency pole at the circuit input. This is 10k and the input capacitance of the first stage and any wiring capacitance. With a gain of 4.2, the 4 pF Cgp is multiplied to 17 pF, added to 2 pF Cgk and about 5 pF wiring capacitance or a total of 24 pF. With a 10k source, this produces a pole at 600 kHz. Due to the cathode degeneration, the effective plate resistance will be multiplied by the gain reduction due to degeneration, or about 25k. The high frequency resistance is therefore 22k paralleled with 25k paralleled with 71.5k, or about 10.2k. The capacitance is the second stage input capacitance or 11*4 + 2 pF plus wiring capacitance, plus first stage plate capacitance (1 pF), for a total of about 52pF. This produces a high frequency pole at 300 kHz. The output resistance is 7.2k paralleled with 12k paralleled with 31.7k or 3.94k. The capacitance is about 110 pf, dominated by the assumed load capacitance as specified above. This produces a third high frequency pole at 380 kHz.
Summary: 1 pole at 300kHz, 1 pole at 380kHz, 1 pole at 600kHz. Amplifier Gain and Frequency Response: The midband gain is 33 dB. This is down by 1 dB at 27 Hz, down 3 dB at 16.5 Hz, and the gain drops to unity at 2.8 Hz. At 2.8 Hz, there is about 275 degrees phase shift between input and output. The gain is also down by 1 dB at 160 kHz, down 3 dB at 260 kHz, and drops to unity at 1.2 MHz. At 1.2 MHz, there is about 230 degrees of phase shift between input and output. These numbers were obtained very simply by plotting the straight pole/zeros on semi-log paper, and applying the correction factors as previously described (3 dB error at the pole, 1 dB an octave away, 0.3 dB 2 octaves away etc) I believe SF is going to post a nice pole/zero article as the next lead into this subject. Steve Bench
EVERYTHING YOU NEVER WANTED TO KNOW ABOUT POLES & ZEROS
Pole/zero analysis is at the very center of electric circuit theory. Unfortunately, most textbooks on the subject are too rigorous to provide a clear picture of the physical concepts until the theory has been worked through successfully (!). For those readers not inclined to work through the theory, I here present a brief overview of the subject. There is no simple physical analogy for poles and zeros.
Poles and zeros are mathematical roots, or solutions, of the network function. The simplist way to think of poles and zeros is as follows: a pole creates a roll-off at a slope of 6db per octave; while a zero creates a roll-up at 6db per octave. This is true, however, only in the steady-state (a hypothetical state, as will be shown). Before looking at the network response for transients, however, it is useful to see how poles and zeros relate to the steady-state response. For a capacitor the impedance function = 1/sC. This function has a pole at s = 0 and a zero at s = infinity. We observe that a capacitor behaves as an open circuit at 0Hz (the pole) and as a short circuit at infinity (the zero). For an inductor the impedance function = sL. This function has a pole at s = infinity and a zero at s = 0. An inductor thus behaves in a manner opposite to that of a capacitor, i.e., it acts as an open circuit at infinity (the pole) and as a short circuit at 0Hz (the zero). Thus, in either case, we can see that a zero acts like a short circuit, while a pole acts like an open circuit in the impedance function (note that the poles and zeros are interchanged for capacitors and inductors due to the opposite reactions they have on frequency). At intermediate frequencies the network impedance has a finite, non-zero value.
So far I have been describing the steady-state behavior of the network function. In music, however, we get transients; therefore it is necessary to examine how poles and zeros operate in the time domain. This is where things get complicated; but if you read through to the end, you will find that the unique power of pole/zero analysis will become quite evident. In any linear amplifier, the gain as a function of frequency, G(s), is fully described by the transfer function, G(s) = Vout(s) / Vin(s). The term s indicates a complex frequency, sigma + jw, where sigma is the resistive part and jw the reactive part. The transfer function becomes more meaningful when we rearrange the equation so as to define the network response, Vout(s), such that Vout(s) = G(s)Vin(s). If there were no reactances in the circuit the frequency response would be infinitely wideband--in which case the jw part would vanish. In this case the network response would be Vin * G = Vout. The s term thus represents the effect on signal current of the network impedance RLC as a function of frequency.
Poles and zeros, therefore, are the critical frequencies of s that define the network function. What then, is the effect of poles and zeros given music signals? In any real circuit the network function, Vout(s), will contain both a steady-state and a transient component. This is so because even steady-state signals have starting and stopping points (especially in music). By the same token, transients can excite a free oscillation in the network response at the natural frequency of oscillation (resonant frequency). This occurs because a capacitance (the electric storage element) will interchange energy with an inductance (the magnetic storage element) at the frequency of resonance. A transient, being wideband, has the effect of exciting this critical frequency. The familiar case is a feedback amplifier wherein a square-wave input excites a unique frequency of oscillation at the output.
A pole/zero plot tells at sight whether the resonant response to a transient will be (a) underdamped; (b) overdamped; or (c) critically damped. Because both phase and magnitude are involved, poles and zeros are plotted on the complex-frequency plane (sigma vs jw, or s-plane). When the steady-state response is combined with the transient response, the poles determine the waveform of the time variation of the response, while the zeros determine the magnitude of each part of the response (i.e., the peak-to-peak envelope). A network function is completely specified by its poles and zeros (and the scale factor). The bottom line here is that the Vout(s) response of any linear network to an arbitrary input can be graphed once the poles and zeros have been plotted (!).
To summarize, then, locating the poles and zeros of a network function on the s-plane allows one to
(1) plot the output waveform and its envelope;
(2) plot the phase response;
(3) find the resonant frequency;
\(4) find the damping ratio, or zeta, of the impedance function;
(5) find the circuit Q (Q = 1 / 2 * zeta);
\(6) confirm critical damping at the output (zeta = 1); and
(7) confirm the stability of a feedback amplifier.
Of course, this assumes that one wishes to design a complete amplifier on paper. There are empirical methods that work just as well, but they are most often learned by cut-and-try methods. It goes without saying that a study of the theory leads invariably to a streamlining of the cut-and-try process--not to mention an understanding of the why's and the wherefor's of what makes what do what.
MATHEMATICA MISCELLANIA
It goes without saying that network analysis requires mathematical tools--in particular: algebra (to expand and reduce basic operations, to find roots, and to solve differential equations); the theory of functions and graphs (to plot waveforms as functions of time); trigonometry (to mark time and to find phase angles); differential and integral calculus (to freeze and sum quantities which are functions of time); complex number theory and vector analysis (to evaluate the phase shift between voltage and current waveforms and to plot and interpret poles and zeros); and, ideally, operational calculus such as the Laplace transform (to simplify operations and to transform frequency domain functions into time domain functions). The drudgery of learning math without concrete problems to solve is largely overcome by the study of network theory, which not only employs all of the tools shown above, but does so in a manner that links them together into a unified, coherent, and powerful system; wherein each branch serves to reflect and illuminate the others.
The foundations for pole/zero theory were developed largely by Euler, Wessel, Gauss, Argand, and Cauchy (working independently) circa 1800. The theory was subsequently developed and applied to the stability problems of oscillating mechanical systems by such theorists as Maxwell, Routh, Hurwitz, Minorsky, and Hazen. In 1932, Nyquist employed the theory to determine his stability criterion for feedback amplifiers. Today the theory finds wide application in the design of feedback control systems (everything from active suspensions to robot control arms).
SELECTED REFERENCES
Following is a list of annotated references that will take the interested reader about as far into pole/zero theory as he might wish to go. The references here are all established engineering textbooks. More recent references can be found at your local university bookstore. In addition to this list, I plan (soon, I hope) to post an annotated list of references aimed specifically at the problem of stabilization in feedback amplifiers.
1942 Murray F. Gardner and John L. Barnes, Transients in Linear Systems (John Wiley & Sons, Inc., NY). Gardner was a professor of electrical engineering at MIT; Barnes was a professor of engineering at UCLA. This is among the most influential of all books on electric circuit theory. The book stems from the graduate lectures of Vannevar Bush at MIT. Bush became famous for his invention of the differential analyzer, which is widely acknowledged to have ushered in the modern computer age circa 1930. In addition to its didactic function, this book is also intended to serve as a reference for practicing engineers. The authors show how to use a table of transform pairs to solve many common engineering problems. This book was among the first to replace the older operational calculus of Heaviside with the Laplace-transform notation (see also Valley and Wallman [Section 1.4]). To buttress this radical notion, a thorough review of the mathematical literature relating to engineering problems is given in Appendix C. Finally, the bibliography comprises fifteen pages of tightly compacted references.
1955 M.E. Van Valkenburg, Network Analysis (Prentice-Hall, Inc., Englewood Cliffs, NJ). Van Valkenburg was Professor of Electrical Engineering at the University of Illinois. The author emphasizes the pole and zero approach to network analysis. In so doing, he clearly explains the complex-frequency plane and its use in determining the damping ratio of overshoots; indispensable for predicting the transient response of feedback amplifiers. In addition, this book provides a thorough explanation of the Fourier transform and clearly shows its relation to the Laplace transform. Chapter 14 covers tuned amplifier networks using a pentode tube as a current source. The final chapter is an extensive treatment of stability in feedback amplifiers. This book, while not considered seminal, was for many years one of the standard texts on the subject. Its reputation is well deserved as it teaches the basics in a way that none of the other books on this list quite approach. The 8th printing appeared in 1962.
1957 John D. Ryder, Engineering Electronics (McGraw-Hill Book Co., Inc., NY). As usual, Ryder has a way of cutting to the essential elements of a subject and making them immediately accessible. See chapter 7 for a concise summary of pole-zero theory as it relates to stability considerations in feedback amplifiers.
1957 Hugh Hildreth Skilling, Electrical Engineering Circuits (John Wiley & Sons, Inc., NY). Skilling was a professor of electrical engineering at Stanford. This book was developed as a new approach to classroom teaching at Stanford. Thus, it does not present the rudiments of its subject in the usual way, as by allotting equal space to each element of the discussion; rather, the most fundamental concepts are allotted the most space, and are the most thoroughly developed, especially with regard to graphical illustration. Indeed, this book excels in the area of graphical presentation. Many key concepts, with which other authors often assume familiarity, are illustrated by means of a series of cunningly contrived graphical representations. This becomes evident in the chapter on complex algebra, and in the chapters on network analysis and resonance that follow. There is even a chapter dedicated to explaining the three basic types of diagrams (chapter 8), which also tells how to map simple functions on the complex-frequency plane. Another notable chapter is dedicated to nonlinear elements (chapter 13), a subject that is conspicuously absent from most other elementary texts on network analysis. The waveforms of certain nonlinear passives are depicted (i.e., a resistor, an inductor, a detector, and a transistor), and the method known as the power series is used to evaluate each waveform. An example is then given of the application of Taylor's method for approximating a function out to three terms. The terms are plotted graphically to show the approximation to the true function. This is a rare presentation of a valuable method of analysis that can be applied to, e.g., the transfer curve of a triode or pentode. There are further chapters on Fourier analysis, on transient response (Laplace transform), and on the complex frequency plane. This book is almost uniquely valuable in its ability to present many key aspects of basic circuit theory with a "beginner's mind". Rivals Van Valkenburg's book in this regard, while providing new insights of its own.
1959 Egon Brenner and Mansour Javid, Analysis of Electric Circuits (McGraw-Hill Book Co., Inc., NY). Both authors were Assistant Professors of Electrical Engineering at the City College of New York. This book represents a breakthrough to the modern approach to network analysis. The authors argue their points persuasively in the preface. The idea is to tie-in transient concepts early on in the student's career and to defer the use of transforms for later study. This places the emphasis on simple, fundamental problems. Such problems can be readily solved by classical means--i.e., by means of graphical analysis or by differential equations. The method culminates in the study of pole-zero configurations. The foundation is thus laid for a later study of transformation methods and of servo-mechanisms.
1959 David K. Cheng, Analysis of Linear Systems (Addison-Wesley Publishing Co., Inc., Reading, MA). Cheng worked in the Department of Electrical Engineering at Syracuse University. His book emphasizes the block diagram approach to network analysis, in which complex systems are reduced to block diagrams by means of transfer functions. Depicts the common transfer functions that appear, for example, in most of the classic TIM articles. Many illustrations and examples are provided. In preparation for this study, there are chapters on solving differential equations and another on analogous systems. For those familiar with mechanical physics these two chapters will make the electrical analogies immediately clear. The 2nd printing appeared in 1961.
1960 Samuel J. Mason and Henry J. Zimmermann, Electronic Circuits, Signals, and Systems (John Wiley & Sons, Inc., NY). This book is a companion volume to their Electronic Circuit Theory. Section 3.7 analyzes the triode circuit. Section 3.8 expands this analysis to derive the voltage gain of a triode feedback amplifier. Section 6.22 shows how to make successive differentiations in order to simplify the Fourier transformation. This allows one to quickly jump from the time domain to the complex-frequency domain, where network problems are more easily solved. The spectra of many common waveforms are then analyzed by means of the Fourier transform. Profusely illustrated and annotated.
1966 Henry Ruston and Joseph Bordogna, Electric Networks: Functions, Filters, Analysis (McGraw-Hill Book Co., Inc., NY). Ruston was Associate Professor of Electrical Engineering at Brooklyn's Polytechnic Institute; Bordogna was Winterstein Assistant Professor of Engineering at The Moore School of Electrical Engineering, UOP. This relatively large book (550+ pages) covers the subject of network theory about as thoroughly as one could reasonably expect--and it does so without bogging down in abstruse detail. Abbreviations and symbols are given immediately at the front of the book, where they belong. As the book moves from basic network concepts to complex frequencies to network synthesis to network analysis, the authors take pains to explain the concepts in minute detail. In spite of this, or perhaps because of it, the presentation of fundamentals is not always as accessible as in Van Valkenburg's book (above). After the introduction of the concept of complex-frequency, the tie-in is made immediately to the representation of frequency-dependent impedances (i.e., reactive elements) as mathematical poles and zeroes on the complex plane. The mechanics of finding poles and zeroes is treated more thoroughly than in any book known to me. For example, tables are given that show the polynomial, the trigonometric form, the phase plot of each pole or zero configuration. Each configuration is shown on the complex plane and is then correlated to a number of common filter functions. This method of parallel analysis provides a physical insight into the nature of filters not otherwise obtainable. An enormously successful book that is almost uniquely clear on this difficult subject.
1967 Jose B. Cruz, Jr. and M.E. Van Valkenburg, Introductory Signals and Circuits (Ginn and Company, Waltham, MA). Cruz was a professor at the University of Illinois. By this time, Van Valkenburg had moved on to Princeton. This book presents an introductory approach to circuit analysis by means of signals and waveforms in simple networks. In other words, it explains in a straightforward manner what happens to a particular waveform when it passes through a particular circuit. The authors stress differential equations as the preferred tools for evaluating the concepts. Transforms are deferred for the later study of transients in linear systems. Contains an extensive bibliography, conveniently organized into the various branches of circuit theory.
1974 M.E. Van Valkenburg, ed., Circuit Theory: Foundations and Classical Contributions (Dowden, Hutchinson & Ross, Inc., Stroudsburg, PA, 1974). This book is a collection of twenty-five seminal articles on electric circuit theory. It is Volume 8 in a series entitled Benchmark Papers in Electrical Engineering and Computer Science (by the same publisher). The intent of this compendium is to trace the development of circuit theory since its inception. Two of the articles are themselves histories of circuit theory. The editor introduces each chapter with a short essay in which he describes the significance of each paper. A recurring theme in these essays is that familiarity with the original contributors "frequently offer[s] insight and clarity not found in textbook treatments." Perhaps that is because, in so many cases, the originator of an idea best understands what it takes to grasp the theory behind it; being the only one who approached the idea from a blank slate.
Scott Frankland
(Uncle Ned Notes: Some comments from the Peanut Gallery follow)
While I do not take issue with what you have said, I suspect that a very rigorous treatment of this subject will lose all but those trained in the dicipline of engineering or at least higher mathematics. Firstly, I would like to state that I have done some, but not done a great deal of servo, feedback work in the past, but have done none in recent years, so am admittedly a bit rusty, esp at differential equations, but did master the subject fairly well about 35 years ago and still remember much of it. My impressions of the three primary analysis techniquies are as follows:
1. Differential Equation Solution: the most rigorous and effective mewthod of dealing with servomechanism analysis, or feedback systems. Is not very intuitive, especially for the average hobbist, unless perhaps trained in engineering or mathematics. Without an understanding of differential equations, the average hobbist will likely very soon become totally lost.
2. Nyquist Stability Criterion: tells you whether the network will be stable, but is not very useful for optimizing the compensation network, a kind of a go, no-go approach.
3. Bode plot: probably the most intuitive and easily understood method for analyzing the stability of feedback networks. Its graphical, intuitive approach will lead you to an adequate solution for all but very high performance systems. In the case of very high performance servo or feedback systems, the differential equation method is more capable of optimizing for best performance in terms of speed of response, minimum overshoot, minimum settling time, etc. However, it requires characterizing all system components in the loop in differential euqation terms. The Bode method, being based on phase-amplitude relationships graphically presented is much easier to visualize and is an adequate method to a solve relatively simple networks such as feedback audio amplifiers. Here, with relatively simple bench test equipment you can measure the transfer function (phase-amplitude characteristics) of the amplifier and go directly to the Bode Plot and derive a suitable lead/lag network to compensate for the amplifier transfer function requiring only the understanding of a few relatively simple concepts without getting buried in complex math that will likely obscure the subject for most. A drawback is how to deal with graphs in a forum such as this. For those who would really like to understand this subject, but already are getting that overwhelmed feeling, I would suggest that a good text that explains the Bode Plot solution method would be your best means of understanding this subject. Once you get the idea of how RC networks, and the like, shift the phase and and affect the amplitude of a signal, it is fairly straingtforward to intuit the Bode method. I suspect that among the texts mentioned, that one, or more, explain the Bode Plot method. I recognized several of the texts that Scott mentioned as ones I used when studying engineering many years ago, but would have to break them out and review to recall which text would be most appropriate. Anyway, I just had the feeling that many interested parties possibly already had that sinking feeling and I just wanted to point out that there is a relatively straightforward method for dealing with this subject that most can comprehend if willing to put in a little effort. Dan Marshall
Just so people don't feel cheated by the title of the thread, here are the performance data for a 12AT7/6201 operated in SRPP (what I have always called "totem pole" operation):
B+ =3D 310 V (this is non-critical)
Bottom tube cathode resistors =3D 1 K ohm
Bottom tube cathode bypass capacitor =3D 1000 uF/6.3 V (figures in parentheses are with unbypassed cathode)
Top tube cathode resistor =3D 1.5 K ohm
Output network: 5 uF coupling capacitor with 100 K ohm load resistor to ground
Feedback for closed-loop figures: the bottom-tube grid is used as a summing junction. 100 K ohms from the signal source to the bottom grid, 470 K ohms from the load side of the output coupling capacitor to the bottom grid. (This configuration makes a nice line amplifier, bearing in mind that it inverts absolute polarity so the speaker leads must be reversed.)
(Uncle Ned Notes: Some of the symbols did not translate on following)
Open loop gain =3D 36 =3D 31 dB (=3D 25 =3D 28 dB)
Open loop output impedance =3D 8 K ohms, resistive
Closed-loop gain =3D 4.7 =3D 13.5 dB
Closed-loop output impedance =3D 300 ohms, resistive (1.1 K ohms)
Closed-loop bandwidth: -3 dB at 110 Kc (105 Kc)
Distortion: driving 10 K ohms to +10 dBv
IMD =3D < 0.1 % driving 10 K ohms to +30 dBv,
IMD =3D < 1 %
Maximum output =3D greater than 100 V p-p (> +33 dBv)
Capacitive loading: using 2 Kc squarewave signal, drives 1000 pF without visible change; 2500 pF produces just visible rounding; 10,000 pF (.01 uF) produces definite rounding but still good shape.
Note that the closed-loop bandwidth reflects the rather high effective source impedance of the summing operation. Fed from a lower source impedance (even the plate of a 12AX7 is substantially lower) the bandwidth is correspondingly higher. Anonymous
Those who want to look at some primary sources might consider:
(1) H.W. Bode, "Relations Between Attenuation and Phase in Feedback Amplifiers," July, 1940 Bell Technical Journal
(2) H.W. Bode, Network Analysis and Feedback Amplifier Design (1945, Van Nostrand) These two by Bode are the loci classicii of feedback theory . (Ed note: "loci classicii" sounds suspiciously like the Spanish for "all-time raving loonies"...perhaps a good description of guys who can write entire books on Network Analysis?)
(3) Radiotron Handbook, 4th ed., pp. 356-388 Probably not a good introduction to the subject.
(4) James K. Roberge, Operational Amplifiers: Theory and Practice (1975, Wiley) (Ed note: Op Amps? WTHDTGIH?) Chapters 5 and 13 are a good overview with a balance of intuitive and mathematical treatment -- recommended.
(5) Horowitz and Hill, The Art of Electronics (1980, Cambridge) pp. 136-140 give an intuitive idea what you are trying to do.
REALLY BROAD OVERVIEW:
Basically, you need to start with an idea of what you are trying to do. (Ed note: Looks like we have a candidate for management ranks here..OTOH, some folks do need to be reminded..) If a feedback amplifier is unstable, it is because the "negative" feedback signal is not in phase with the input signal at all frequencies due to phase shift in the forward path of the amplifier and/or in the feedback path itself, at frequencies where the amplifier still has gain. (If a non-feedback amplifier is unstable, it is because there is feedback somewhere.) In tube amplifiers the forward path has nonzero phase shift at both ends of the band. My discussion below focuses on the high end, but the same principles are applicable mutatis mutandis to the low end.
BASIC CONCEPTS -- LOOP GAIN & PHASE:
Compensation tailors the open-loop gain and phase response and the feedback-loop attenuation and phase response so that the "round trip" phase shift is less than 180 degrees (typically, less than 135 degrees or so) at all frequencies where the "loop gain" is greater than unity. The loop gain (a frequency-dependent parameter) is the open-loop gain of the amplifier multiplied by the attenuation of the feedback loop. For example, if at some frequency an amplifier has an open-loop gain of 100 and the feedback loop is a voltage divider with a 9 K ohm resistor to the amplifier output and a 1 K ohm resistor to ground, then the loop gain is 100 x .1 = 10 at that frequency. If you changed the feedback loop by connecting the output straight to the inverting input you would create a voltage follower, and the loop gain at that frequency would be 100 x 1 = 100. If you raised the closed-loop gain by using a 99 K/1 K voltage divider, the loop gain would be 100 x .01 = 1 at that frequency.
BASIC CONCEPTS -- RESPONSE POLE, DOMINANT POLE:
Open-loop response tailoring is usually done with a "dominant pole" in the open-loop transfer characteristic -- a 6 db per octave rolloff in the open loop frequency response that starts at a frequency low enough that the loop gain falls to unity somewhere near the second-lowest pole in the forward transfer characteristic. Each response pole introduces 45 degrees of phase shift at its -3dB frequency, increasing to 90 degrees at higher frequencies. (If the loop gain as tailored by the dominant pole reaches unity exactly at the second pole frequency, the second pole will contribute an additional 45 degrees of phase shift to the dominant pole's 90 degrees, for a total phase shift in the forward path of 135 degrees at the unity-loop-gain frequency.) This dominant pole can be introduced by shunting the high frequencies to AC ground with a capacitor somewhere in the amp, or by using local frequency-selective NFB around one or more of the amplifier stages. The second method is generally preferable because if done right, it introduces a zero into the transfer function that exactly cancels the phase shift of the amplifier's lowest natural pole, giving the designer a larger phase/frequency budget to work with. To design compensation "on paper," you need to be able to rather completely characterize the open-loop phase & frequency response of the amplifier, stage by stage. This is not something the home constructor can do with limited test equipment. To design compensation by "cut and try," you need to know at least where in the amplifier the lowest open-loop response pole is generated (preferably also where the second-lowest pole is created, and its approximate frequency). Often this can be determined by educated guess and a bit of calculation, but sometimes there are unexpectedly low poles due to overlooked factors or to the transformer design.
There are a couple points of clarification I would like to add to the initial FAQ post:
BASIC CONCEPTS -- POLES AND ZEROS:
These are the frequency domain frequency manipulating concepts. A pole causes the frequency response to tip "clockwise" at 6 dB per octave (20 dB per decade). A zero causes the frequency response to tip "counterclockwise" at 6 dB per octave. At the "corner" frequency, the response is altered by 3 dB and by 45 degrees. (A clockwise bend is 45 degrees lag, a counterclockwise bend is 45 degrees lead). Poles can be "bought" for your intended circuit fairly cheaply, while zeros usually "cost". An example of a pole is an RC network: series resistor, capacitor to ground. This has the pole at the frequency f=1/(2*pi*r*c). It also has a zero at "infinite frequency". A coupling network (series c, r to ground) has a zero at DC (zero frequency) and a pole at the expected f=1/(2*pi*r*c).
Example:
RIAA phono characteristic: Pole at 50 Hz, Zero at 500 Hz, Pole at 2120 Hz. (Usually there is also at least one zero at DC and a matching pole at 10 to 20 Hz).
Frequency Response Rule of Thumb: Frequency response is usually graphed dB wise on the vertical axis, and logarithmic frequency wise on the horizontal axis. In this manner, a pole or zero is modeled as a straight line whose slope is 6 dB per octave.
Example:
Pole at 1000 Hz: Frequency is "flat" below 1 kHz, and the slope is 6 dB per octave above 1kHz (e.g. -6 dB at 2 kHz, -12 dB at 4 kHz, -20 dB at 10kHz). The "real" or "exact" frequency response departs from this idealized straight line approach in a rather "nice" way. The "error" is 3 dB at the corner and decreases to 1 dB at an octave away, and decreases to 0.3 dB 2 octaves away, 0.1 dB 3 octaves away, 0.03 dB 4 octaves away etc.
Thus, in our example (pole at 1kHz), response is "flat" at DC, down 0.1 dB at 125 Hz, down 0.3 dB at 250 Hz, down 1 dB at 500 Hz, down 3 dB at 1 kHz, down 7 dB at 2 kHz (6 dB + 1 dB error term), down 12.3 dB at 4 kHz and down 18.1 dB at 8 kHz.
BASIC CONCEPTS -- CIRCUIT ELEMENTS FOR ALTERING FREQUENCY:
1. Series R, C to ground (low pass): Single pole at 1/(2*pi*R*C)
2. Series C, R to ground (high pass): Zero at DC, pole at 1/(2*pi*R*C)
3. Series R, L to ground (high pass): Zero at DC, pole at R/(2*pi*L)
4. Series L, R to ground (low pass): single pole at R/(2*pi*L)
5. Series LRC.2 Complex poles. Presents low impedance (R) at resonance, which is at 1/(2*pi*sqrt(L*C)). Impedance increases away from resonance.
6. Parallel RLC. 2 Complex Poles. Presents high impedance (R) at resonance which is at 1/(2*pi*sqrt(L*C)). Impedance decreases away from resonance.
7. Transformers: Separate installment (later). Other Notes: Stability and feedback. The issue with an accumulated 180 degree phase shift is that 180 degrees phase shift transforms the negative feedback into positive feedback. With positive feedback and greater unity "loop gain" the circuit oscillates! The 135 degree accumulated phase shift mentioned in the FAQ provides 45 degree "margin" against this oscillation. The "goal" of negative feedback compensation is to get to less than unity LOOP gain before 180 degrees of phase shift has accumulated (preferably with some phase margin - 45 degree margin given in the FAQ).
Hi All, Here's a proposal for modeling the frequency effects of a transformer. Comments regarding appropriateness or accuracy are welcome. --------------------------------------------------------------------
Feedback and Compensation FAQ:
Transformer Frequency Response Model:
Transformers are more complex than RC networks from a frequency response view. The following formulas allow you to predict the frequency response of your circuit with transformer coupling.
Definitions: Lp = Primary inductance
LL = Leakage inductance
Cp = Primary winding capacitance
Cs = Secondary winding capacitance
n = Turns Ratio
f = Low Frequency pole
fr = High Frequency resonant point
rp = Plate resistance (also add primary winding resistance)
Rl = Load resistance (also add secondary winding resistance)
Q = Circuit Q of the high frequency pole pair
Low Frequencies:
Pole: Unterminated transformer case: f = rp/(2*pi*Lp)
Terminated transformer case: f = Rl*rp/(2*pi*Lp*(Rl + (n*n*rp)))
Zero: There is a zero at DC
High Frequencies:
Poles:
There is a set of "complex" hi frequency poles. This leads to peaking at high frequencies as has been discussed in this newsgroup. The formulas given allow you include the effect of these on your feedback circuit. Since the pole-pair is "complex" I have to introduce the concept of "Q". Higher Q leads to more peaking, lower Q to less peaking.
The following table provides an insight into how much high frequency peaking to expect:
Q Peaking (dB) -3dB point
W.R.T. "resonant" freq. 2 6 dB 1.5 1.41 3 dB 1.4 1 1 dB 1.25 .8 0.3 dB 1.1 .7 0 dB 1 .6 no peaking 0.8 .5 no peaking 0.65
Resonant Frequency: fr = 1/(2*pi*sqrt(LL*(Cp+(n*n*Cs)))
Circuit Q: Q = (2*pi*fr*LL)/(rp paralleled with (n*n*Rl))
Example: A 1:1 transformer couples a 6SN7 plate into the grid of a class AB1 triode. The grid resistance is assumed high. (rp = 7.7k)
Transformer characteristics
Lp = 25 Henry
LL = 0.05 Henry
Cp = 100 pF
Cs = 100 pF
Winding resistance assumed negligible. (ed note: The DC resistance of typical transformers used in this fashion are rarely "negligible", a Peerless output xfmr used with a 6CG7 has approx 1,400 ohms DC resistance. But for the sake of this argument, let's play along to keep it simple)
Use "unterminated" cases:
Low Frequency pole is at 7700/(2*pi*25) = 49 Hz.
Hi Frequency Resonance is at 1/(2*pi*sqrt(0.05*200pf) = 50.3 kHz Q is (2*pi*50300*0.05)/7700 = 2.05
From the table, the -3 dB point is 1.5*50300 = 75.5 kHz.
There will be a 6 dB peak in the response at about 50kHz.
Steve Bench
(Ed note: More Peanut Gallery comments follow, then we return to the regularly scheduled programme)
BASIC CONCEPTS -- IN PHASE OR OUT OF PHASE?
The reader needs to keep in mind that some references consider negative feedback (NFB) "in phase" and some consider it "out of phase" with respect to the input signal. This is just a matter of how you look at it, and once one has a good idea of what is going on it should not be confusing. Ultimately, the music input and the NFB signal are subtracted from each other, and the difference between them drives the amplifier. Usually, the amplifier has an "inverting input" (in tube amplifiers, often the cathode of the input tube) that does the subtracting. In this case, the NFB signal is the same polarity as the input signal. In some older designs, the NFB is added to the input signal by floating the "ground" side of the input transformer secondary on the feedback network. In this case, the subtraction is achieved by reversing the NFB polarity (adding a negative quantity is the same as subtracting a positive quantity of the same absolute magnitude). So, when you read that stability requires that the loop phase remain within 180 degrees at all frequencies where the loop gain is greater than unity, it means with respect to whatever polarity the design needs in order to subtract the NFB from the input signal. Anonymous
What you say relates to the feedback equation A / 1 + or - AB, and it seems clear enough. You might next want to tie in gain margin and phase margin. Knowing the limits for each seems a basic requirement for setting up compensation. This all leads to finding the gain crossover frequency (the frequency at which the feedback factor has fallen to unity), a rather fundamental concept where stability is concerned. Regards, ~SF~
Hi, The only point of clarification here the example. A rising input on the grid causes the plate voltage to fall (inversion as you have stated), however, a rising signal applied as a feedback signal to the cathode causes a rise in plate voltage. Thus, as you have stated, NEGATIVE feedback signal is the same polarity as the input. Steve Bench
Feedback Stability FAQ Project
ABSTRACT
This "installment" analyzes a conventional Williamsom Amplifier to provide insight into an already "compensated" feedback amplifier. Using the "Dynaco Modified Williamson" amplifier as an example, we'll go through a fairly simple analysis of this amplifier. This installment is NOT intended to teach you how to compensate a feedback amplifier, but provides some familiarity for later installments, as it walks through an already compensated and *proven* design.
We will discuss power supply considerations, then analyze the DC operating points of the amplifier, identify the feedback networks, determine stage gains, find the low and high frequency poles and zeros of the amplifier, discuss how the amplifier is compensated, determine the ammount of feedback used, and the expected stability of the amplifier. Although this can be done "engineering wise" using SPICE or various CAE tools, this approach is not taken; rather, the amplifier is very simply analyzed so that the audience, who are NOT expected to be practicing engineers, can use this as a "real" example. As such, there will be some minor errors, as would be expected from a simple analysis. However, the end result is reasonably close to reality.
BACKGROUND
As was hinted at in the abstract, the methodology I will use to go through the analysis is to first review the power supply, in order to determine if this can be considered a DC source and ignored for AC considerations. We will then find the operating points of the various gain stages, as that is needed to determine gain and impedance levels. We will identify "blocks" to analyze, which will determine what the feedback networks are. We will then determine the midband gain of the amplifier, both open and closed loop. (This also tells you how much feedback is being applied). We will determine the low and high frequency poles and zeros of the amplifier. Finally we will discuss how the feedback to the amplifier is accomplished, and discuss the open and closed loop performance of the amplifier.
POWER SUPPLY
The main power supply is a full wave tube rectifier, feeding a CLC filter section and a second LC filter section. The rectifier etc will have a output resistance of a few hundred ohms. Therefore, the first LC filter section "resonates" at 16 Hz with a Q of about 5, causing the impedance of this filter to be over 1000 ohms at 16 Hz. This will be decreased by the input impedance of the second LC filter, which is resonant at 11 Hz, with a Q of about 10. The net effect is the supply will wiggle around substantially when the circuit is attempting to reproduce 8 Hz. (Not that it would, but this can affect stability if the feedback network causes the system to go through unity gain at this frequency.) The second LC filter will cause a few thousand ohm impedance to be presented to the driver and input stages at 16 Hz. Since the load impedances are substantially higher than this, there is probably minimal effect on system stability, since this cannot change the stage gains by anything substantial. Note, however, that if the loads were much lower, we would have to take this into account.
It should be noted that the effects of power supply on the operation of an amplifier, particularly with feedback applied are sometimes forgotten, with sometimes not-so-good results. By the way, if the tube rectifier were replaced with silicon, the lower output impedance would raise the Q of the LC circuit, potentially causing additional problems. THIS point is often missed during "upgrades", and can lead to disapointing results.
DC OPERATING POINTS
The output stage is ultra-linear, running 450 volts on the anode and g2, biased at -35 volts. Contrary to Mullard's recomendations, there is no series g2 resistors. This stage will draw about 50 mA quiescent current, which is probably OK. AB1 operation is assumed (grid NOT driven positive).
The driver stage is a push-pull class A 6SN7 stage. Drawing the load line shows the bias is 4 volts, a current (per section) of 5.5 mA, and a plate voltage of 150 volts (Vp=150, Vg=0, Vk=4). Under these conditions, the 6SN7 has a gm of 2.5 mS, and an rp of 8000 ohms. Note that the 47k load resistors will be dissipating about 1.5 watts. The input stage draws about 4 mA, with 2 volts bias, for a plate voltage of 90 volts. (Vp=90, Vg=0, Vk=2). Under these conditions, the 6SN7 has a gm of 2.3 mS and an rp of 8.7k. Due to the unbypassed cathode, the effective output resistance is about 17.4k ohms. Note that the voltage at the junction of the 20 uF, 47k and 33k will have about 288 VDC on it.
The phase inverter operates with about 4 volts bias and 4 mA current. This puts 94 volts on the cathode, 90 on the grid, and 232 volts on the plate. There will be 326 volts at the junction of the 20 uF and the two 22k resistors. For a split load phase inverter the output resistances are unequal: the cathode side is (Rk*(rp+RL)/(rp+RL+Rk*(mu+1)) = 1.3k and the anode side is (RL*(rp+Rk*(mu+1)))/(rp+RL+Rk*(mu+1)) = 20k.
GAIN STAGES - AMPLIFIER MIDBAND GAIN
The output stage, since it already incorporates some level of feedback because of the "ultra linear" approach, is a little tricky, so we use a different technique. As this produces about 55 watts, at full power, there is about 30 volts developed at the 16 ohm tap. With 35 volts of bias, assume 24 volts RMS produces this. (1.4x25 = 35). At the output XFMR PRIMARY, this will be about 240 VRMS, so the EL34s are operating with a gain of 10, and a circuit gain from grid to "speaker" of 1.25 (2 dB).
The 6SN7 driver stage, because it is class A, has no AC voltage on the cathode, so the stage gain is simply gm*Rl where gm is 2.5 mS and Rl is 8k parallel with 47k, parallel with 100k, or 6.4k, for a stage gain of 16 (24 dB). The phase inverter produces a gain of roughly .98, which is close enough to 1. (0dB gain).
There are 2 "gain issues" with the input stage, as this is the connection point to the global feedback network. The gain from the grid to plate is (47k parallel 8.7k)/(470+(1/gm)) or 8 (18 dB). The gain from cathode to plate (which is the global feedback tap) is about 9 (more or less 1 + g-p gain) (19 dB). Notice that the main feedback components (100 pF, 10k and 82 pF) have been ignored. There are 2 feedback loops. The 10k 82pF from output to input wants to make the closed loop midband "loop gain" 10k/(470 parallel 1/gm) or about 45 (33 dB). This would be the closed loop gain, cathode-output, with "infinite" open loop gain. The open loop loop gain is 9*16*1.25=180 (45 dB). Thus, the circuit has about 12 dB of global midband feedback. Also notice that the "minor" loop caused by the 100 pF capacitor would want to make the closed loop gain 6.4k/(470 parallel 1/gm) or about 28 (29 dB). More on this later. The overall open loop gain in the midband input to output is 8*16*1.25=160 (44 dB). The closed loop gain will be about 33 dB, so the sensitivity for full power output is slightly lower than 0.7 volts.
LOW FREQUENCY POLES AND ZEROES
The first stage has a low frequency pole-zero pair due to the 20 uF in the plate circuit. This produces a small boost in low frequencies, as the first stage has slightly more gain at "DC" than it does when the 20 uF is effective. At DC the stage gain is (8.7k parallel 80k/(470+1/gm)) or 8.6 (19 dB). The 1 dB "lift" occurs at about 0.5 Hz. The phase inverter bottom side has a zero at DC and a pole at 1.3 Hz. The top side has this same set and one additional pole-zero pairs due to the 20 uF capacitor. At very low frequencies, there is actually 6 dB of gain on the plate side of the phase inverter, as the plate load is 44k instead of 22k. The pole is at .36 Hz, the zero is at .72 Hz. This is a distortion producing mechanism at very low frequencies, although arguably it is too low in frequency to be a concern. It may, affect stability and need be accounted for. (It may be worthwhile to place a 300 volt zener across the 20 uF cap to eliminate this additional pole-zero). The driver to output is a pole-zero combination as well. The pole is at 6.4 Hz, the zero is at 0.64 Hz. (Caused by the 1 meg in parallel with the 0.25 uF coupling cap.) Note that this is a nice trick that aids stability at low frequencies. As there is no AC voltage at the junction of the 100k resistors in the EL34 grid path, the 40 uF does not contribute to the amplifier response or stability. (This statement is not true during clipping of for effects of distortion, which I'm currently ignoring). The transformer contributes the final low frequency effect. The frequency is controlled by the impedance level and the primary inductance. For the A-430 transformer, I couldn't find the constants, but I'll assume an Lp of 50 Hy, an LL of 4.7 mH, and .5 nF primary capacitance. The loaded LF pole is therefore at 16 Hz, and becomes the dominant pole.
HIGH FREQUENCY POLES AND ZEROES
There is a pole due to the input resistor and tube capacitance. Using a gain of 8, the miller capacitance is 32 pF, added to 2 pF Cgk and 6 pF wiring capacitance puts this pole at 400 kHz. Due to the cathode degeneration, the effective input stage "plate side" resistance is 12.3k. This produces a pole at about 1.2 MHz. The cathode follower pole-zeros are again different for the plate side and the cathode side. At the plate, the effective resistance is 20k and combined with a 72 pF capacitance (4*16 + 2 + 6) produces a pole at 110 kHz. The cathode side has the same capacitance, but an effective resistance of about 1.3k ohms, putting the pole at almost 2 MHz which can probably be ignored. In some sense, this is bad, because the 2 amplifying paths do not have the same high frequency response, which is a distortion producing mechanism as well as complicating the feedback issues. The driver is interesting, since one of the "feedback" capacitors ties to the bottom driver plate. This causes another "asymmetric" pole, but this also happens to be the correct phase for negative feedback, and it also happens to be in the path that "lacked" a high frequency pole in the phase inverter stage, potentially "compensating" for it. There are 2 pole frequencies: the upper section is 2 MHz, the lower is 250 kHz. The EL34 grid circuit is "isolated" by 1000 ohms. The input capacitance is 11 pF, the miller capacitance is 1.1*10 pF and with wiring the whole thing is about 32 pF. Against 1k+6.4k, this produces a pole at 670 kHz. The feedback circuit produces a pole zero pair at 200 kHz and 18 MHz. The transformer contributes a pair of poles at high frequencies due to the distributed capacitance and leakage inductance. If I did the math right, this pair is located at 100 kHz with a Q of about .8, producing essentially no peaking.
FEEDBACK AND STABILITY ANALYSIS
There are 2 feedback loops to deal with on the high frequency side, due to the 100 pF between input cathode and driver plate, and from output to input cathode. There is only one loop to deal with at low frequencies. On the low frequency side, the good news is that the asymmetries mentioned in the LF P-Z section occur well after unity loop gain is reached, so do not significantly affect the low frequency stability. Also there is no real affect caused by the power supply inductors, as the 8 to 11 Hz region ends up not being too critical. The low frequency "bode" plot is described as: "flat" 45 dB gain to 16 Hz. One pole due to the transformer at 16 Hz. A second pole due to the output stage grid coupling cap at 6.4 Hz. This causes the "unity gain" (actually 33 dB gain) point to be reached at 5.6 Hz, providing about 40 degrees of phase margin and/or about 10 dB gain margin. (This means the feedback could be increased by lowering the 10k resistor if desired). Note that all lower frequency poles are unimportant. However, this gain and phase margin is somewhat affected by the power supply inductors, which are resonant in the 8 to 11 Hz region, so the stated margins are probably somewhat eroded. Hence it would probably be unwise to increase the feedback, without "fixing" that problem. On the low frequency side, the open loop 3 dB point is at 16.5 Hz, and the closed loop 3 dB point is about 6 Hz. On the high frequency side the "minor" loop is first analyzed. This is the loop from the driver plate to input cathode. This would "want" to make the closed loop gain (1+(6400/470 parallel 435) = 29 or about 29 dB at very high frequencies. Note this adds a zero to the loop response (loop gain increases with frequency) but a pole-zero to the closed loop (circuit gain decreases at high frequencies, until the "pole" is reached). The HF poles/zeroes to be considered for the minor loop are 1.2M for the input plate, 2M for the phase inverter cathode, 110k phase inverter plate, cross coupled to the driver cathode the zero at DC and pole at 250 kHz. The "open loop" gain is 9x.98*16 = 141 (43 dB), the "closed loop" is 30 dB. Plotting this out puts the "unity gain" point at 470 kHz with the next dominant pole at 1.2 MHz. There is p l e n t y of gain and phase margin. From an overall amplifier view, the effect is an amplifier with a pole at 63 kHz (which is the LF 3 dB point of the minor loop) and a zero at 470 kHz, and more effects at 1.2 MHz, 2 MHz etc. The global HF path is open loop 45 dB, closed loop 29 dB, pole at 63 kHz, 2 poles at 100 kHz (approximation-this is a pole pair not on the jw axis), zero at 200 kHz, zero at 470 kHz, pole at 670 kHz, pole at 1.2 MHz, pole at 2 MHz etc. Plotting this out shows the 3 dB point at 150 kHz with about 30 degrees of phase margin and about 9 dB amplitude margin. Fair stability. However, this is strongly controlled by the transformer, and the A-430 may perform substantially better than my "model". The stability looks like it could be improved by changing the minor loop 100 pF cap or the global feedback 82 pF to slightly higher values.
Conclusions
The Dynaco modified Williamson appears to be stable and compensated OK. (ed note: heh heh..I'm sure Mr Hafler is flattered that his dessign has held up so well.. ;-)) There is about 14 dB of feedback. We have found a couple of potential "improvements". We have seen the effects of transformer on the stability of an amplifier, and we have seen the importance of power supply operation in its effects on amplifier performance. Note that the analysis given is simplified so that anyone with some determination can follow this report as an example and analyze their own potential design. The results shown here are not *exactly* correct, as I've simplified the analysis as much as I could. (ed note: You did good, Steve) However, they are close enough to provide useful data.
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