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feedback & stability (long)
Here's a rather long blurb on feedback and stability. I'm not too
sure if it's too simple or too complicated! There's probably all kinds of mistakes, please check it out for screw-ups. Since there are ASCII schematics and diagrams, you'll need to view it with a fixed width font like "new courier". INTRODUCTION ............................. You have built an amplifier, or are planning one and you decide that maybe you should employ feedback. Feedback is a mechanism where you use excess gain in an amplifier to reduce "bad things" that are generated within the amplifier circuit. Those "bad things" are distortion, noise, poor frequency response, phase shift - anything that wasn't present in the input signal. Feedback has limitations. At the extreme ends of the amplifiers frequency response, those "bad things" can be made WORSE, often to the point where the amplifier decides to become a signal generator, at maximum possible output level. The art in using feedback is to maximize the amount of feedback in the audio spectrum (or range of frequencies that have your interest), and minimize the nasty effects where feedback makes things worse. There are a number of designs that avoid feedback in any measure, it is very difficult to avoid since many circuits and devices have "built-in" feedback. Triodes have built in feedback from plate to grid (pentodes eliminate that by using a screen grid to shield the main grid, thereby getting higher power gain). "Follower" topologies use feedback. Since these circuits wrap the feedback around just one gain stage, they are considered fairly benign. The evil feedback usually encompasses 2 or more gain stages. Poor feedback practices can cause any number of headaches, and when components, loads, or input signals are altered or undergo changes, the circuit can do some very unexpected behaviour. First I'll try to go through the feedback theory. Second I'll show how I set up feedback using a minimum of equipment and time. This method uses a lot of trial and error (adding components and tweaking them for best response). Remember, trial and error is a VERY powerful design technique, and should never be dismissed! Thirdly I'll go the whole 9 yards with Bode plots, Nyquist criteria, etc. This explains all the weird things that you would have used trial and error to solve, and it will allow you to do the complete design, providing that you know in detail all the circuit behaviour and parameters. Knowing that detail, and making many measurements make this 3rd method unacceptable (IMHO). .....You don't need Microsoft Excel for most of this, but it is handy for publishing the results, and is one of the easier ways to graph Nyquist diagrams. Otherwise graph paper, or just the list of measurements should do fine. THEORY ............................. I will make use of ASCII circuits, to see them you will need to change your font to a fixed width font like "Courier New". I will use "beta" or "B" as a positive value (not like Radio Designers Handbook ed.4 -RDH4), and will use a summing point with inversion as is more commonly used in today's textbooks. Here's the generic feedback configuration: summing point .--------------. + |Amplifier | input ---------------(X)------|gain= A |----------- output Vin - | Va | | | Vout | '--------------' | | .--------------. | | |feedback | | feedback '-------|network |---------' voltage |I/O ratio= B | Vfb=B*Vout '--------------' The summing point (X) adds the input and subtracts the feedback signal. In many circuits, the + point is the grid or transistor base, and the - point is the cathode or emitter. With an op-amp, they could be the inverting or non-inverting inputs. The summing point could also be a simple node (junction) if you are using current instead of voltage. I'll use the more common voltage feedback, derived directly from the output (it samples output voltage). With different topologies, you need to redo the math, especially if you are deriving the feedback from load current or/and using current summing at the input. There are "rules of thumb" when you use those topologies, but don't trust 'em...... do a bit of math to make sure. The amplifier is your amp..... with all its warts. Its gain is "A". Va is the input voltage to the amp (after the summing point). You can consider this as a single number at mid-band, but it must be expressed as a complex number or "phasor" as you get to the extreme ends. I prefer to use "phasors", they're easier. A phasor is simply a number PAIR, the amplitude and the phase difference between amp input and output. It's much like the wind measurement.... 40km/hr isn't much use by itself to a sailor, until they add "from NE", in which case it's a useful value. The complex pair equivalent would be a 14km/hr component from the north, and a 14 km/hr from the east (where north is the imaginary axis, and east the real axis). In electrical notation, say 4 volts, phase +45deg, or even better, 5 db gain and phase=+45 deg. Phasors expressed with db make the math nice to work with. Sometimes there is an additional amplifier stage between the input and the summing point. It will not be affected by the feedback. Make sure that this amp section is free of "warts". You'd hate to have set up the main amp to be very good, and then have crud injected by the additional stage. The feedback network could be a simple voltage divider where B=R1/(R1+R2), or something more complex with capacitors and other components. It produces a voltage Vfb = B * Vout The output(Vout) = A * Va (eq.1) but Va = Vin - Vfb (eq.2) Va = Vin - (B * Vout) (eq.3) substitute for Va using eq3 and recalculate eq1: Vout = A * (Vin - (B * Vout)) = A*Vin - B*A*Vout (eq.4) gather Vin's and Vout's Vout + B*A*Vout = A*Vin factor Vout Vout*(1 +B*A) = A*Vin Since the effective gain of the whole system is G, and is Vout/Vin moving things around you get: Vout = Vin * ( A/(1+B*A)) (eq.5) or: gain = G = Vout/Vin = A / (1 + B*A) The original gain (A or open loop gain) is effectively reduced to a lower "closed loop gain" (with feedback in place). It is reduced by the factor "1 + B*A". If that's all feedback did, it wouldn't be much use. Suppose your amplifier circuit without feedback produced an extraneous (unwanted) output, I'll call it "CRUD". Then: CRUD summing point .--------------. | + |Amplifier |Vao |+ input ---------------(X)------|gain= A |----(X)--.-- output Vin - | Va | | + | Vout | '--------------' | | .--------------. | | |feedback | | feedback '-------|network |---------' voltage |I/O ratio= B | Vfb '--------------' I have added the "CRUD" signal between the amp output and the real output. You can add the CRUD signal anywhere INSIDE the feedback loop. Vao = A*Va Vout = A*Va + CRUD (eq.1a) but Va = Vin - Vfb (eq.2a) Va = Vin - (B * Vout) (eq.3a) Vao = A * (Vin - (B * Vout)) Vao = A*Vin - B*A*Vout (eq.4a) Vout = A*Vin - B*A*Vout + CRUD gather terms, Vout( 1+B*A) = A*Vin + CRUD Vout = Vin * A/(1+B*A) + CRUD/(1+B*A) (eq.5a) Compare eq.5 and eq.5a, you'll see that the value of "CRUD" has been reduced by the factor (1 + B*A), just like the gain. So... if you designed your amp to have an open loop gain 10 times what you need, then reduce it with feedback to what you want, the noise, distortion and unwanted behaviour is also reduced. Similarly, the frequency response is extended, but with some reservations. If you go to the frequency extremes, the open loop phase response is such that the phase change can get to 180 degrees, and that means your feedback becomes positive feedback, which makes everything worse, or the system goes nuts and oscillates. LOW TECH METHOD of OPTIMIZING FEEDBACK ............................................ You need an oscilloscope, it should be able to respond to at least 10-50 times your intended high frequency limit. Similarly, it should be able to respond to 1/10 to 1/50 of your low frequency limit (DC response is preferred). You need a signal generator that can give a quality square wave at 1000 Hz. Put a resistor load on the amp, correct power rating and value. Your feedback circuit (whose transfer function or "gain" is "beta" or B) will look something like this: to summing point Vfb ------,---/\/\/\/\/--------- to output | R1 \ / \ R2 / \ | gnd In this case B = R2/(R1 + R2) voltage divider Since the effective gain is A/(1 + B*A) A= amp openloop gain, if A is large compared to 1, you can approximate the equation to: effective gain = 1/B = (R1 + R2)/R2; the error is 100%/A Usually I keep R2 at a fixed value, and change R1. R2 is often a cathode resistor for tube amps, and is awkward to change since biasing is affected. In op-amps R2 is often to be matched to values on the non-inverting side. So R1 is the one usually changed. For gains of 10- 50 (typical), R1 will be 10-50 times the value of R2. Here's the problem - as you increase the feedback, the term (1 + BA) increases. This is the factor by which the bad stuff goes away. BUT, as that factor increases, the amp is more likely to go nuts at some very high or very low frequency. My first defense against nutty behaviour is to place a capacitor in parallel with R1, and pick the cutoff frequency (where |Xc|=1/(2*pi*f*c)=R) to a reasonable value. This is a tough choice. You have a nice circuit, possibly with a very good output transformer, that claims to go way beyond your usual limit (20KHz for audio). Let's say your openloop amp (with transformer?) can have a cutoff of 150KHz. I claim you should set the feedback cutoff to approximately 25-40 KHz. One way or another, if your goal is to reduce distortion without crazy amp behaviour, you'll be dropping your cutoff back to this value. Here's how I go about it. Put a 1 KHZ square wave in the input, watch the output on the 'scope. Choose a reasonable signal level,that gives 2-5%of full output. Start off with no capacitor in parallel with R1, and R1 about 50-100 times that value of R2. R1 is easiest to be a trimmer resistor or potentiometer (pot). Decrease R1, watch out for overshoot.Add a capacitor in parallel with R1, calculated to give the desired cutoff. The over shoot should disappear. Decrease R1 further, until overshoot shows up. You will need to increase the capacitor to the proper cutoff. You will reach a point where R1 is as low as it can go without lowering the cutoff freq. to silly values. Drop R1 to the point where there is quite a bit of overshoot. Now it's necessary to modify the phase response of the amplifier. In order to do that, we must compromise its open loop response. I will place a capacitor (optional variable resistor in series with it - stepping network) from the first gain stage output (collector or triode plate) to ground. This is called a "lag network". In the case of transistor amps, you can put a cap from collector to base on an intermediate stage. Select capacitors that make the overshoot go away, without excessive rounding off the square wave edges. You can play with the optional series resistor as well. This is a "dominant pole", and reduction of gain at high frequencies. To really understand what's happening here, you need to have a Bode plot (phase, and gain). Pretty soon, you should have optimal feedback for the amp. Now you need to tweak the system for stability. Remove the amp's main load resistor, and use a .1uF - 4.7uF low ESR (Equivalent Series Resistance) capacitor as load. Keep the leads short. The overshoot should get real bad, and maybe the system will oscillate. Avoid that! Things can overheat real quick. Fiddle with the lag network (or stepping network) until it stops oscillating, or reduce the ringing. It's not necessary to eliminate the ringing, 'cause you don't normally have a very capacitive load (unless you have electrostatic speakers). Increase R1, play with the values of capacitor in parallel with R1, change the capacitor and optional series resistor that were in the first stage output. You don't need to eliminate the overshoot under these conditions, just keep the system well away from oscillating. Try to figure out what frequency the system likes to oscillate at. Sometimes you can bypass cathode resistors or emitter resistors that are "degenerative", with capacitors such that cutoff (Fc=1/(2*pi*R*C)) is at the oscillating frequency. This may shift the phase of the amp towards something more stable. It may help to put a snubber network across the output. Typically its a resistor and capacitor in series. R is about 1-3 times the usual load, and C is a value that gives a cutoff 2-3 times the max. useful frequency (60-70 KHz). I used 10 ohms in series with 0.22uF. It helped a bit. Watch out for the power rating of the resistor! If the amp oscillates, the resistor may be cooked! Usually you must back off the feedback to keep the system stable with strange loads. Once you think you have it set up (normal load), momentarily overload the amp (set your signal generator to a much higher output) and then quickly drop the signal to normal. Watch the amp's output for low frequency junk (monitoring the DC current of the output devices is a sensitive check). Things should immediately settle down, without low or high frequency jiggle. If they don't... you have a problem with low frequency phase shift. You should have one stage with poor low frequency response (often a transformer), the rest should have better low frequency responses. This establishes a "dominant zero", which with feedback should give better stability. At the end of this exercise, you have an amp whose frequency response is about 25-40 KHz, with a maximum amount of feedback, without going nuts with a capacitive load. Maximizing feedback, without losing too much stability is the goal. Determine the ratio of openloop (no feedback) gain to closed loop (max feedback) gain. This will be the factor 1 + BA, and all the bad things you amp can do will be reduced by this factor. You weren't able to extend your frequency response, but the compromise allowed you to further reduce distortion, noise, hum and other extraneous signals generated by the amp. DETAILED, THEORETICAL METHOD ................................... Now you need to make a whole lot of measurements on your system. Most importantly, you need to measure the open loop (no feedback) response of your amplifier. You need the amplitude of the output, and the phase of the output as compared to the input. These will establish gain and phase response of the system. The frequency range (sine wave) must extend beyond where the gain drops to 1.0 . Include the frequencies where the gain is 0.5 or lower, and you should be fine. Some of these frequencies are difficult to measure, it's probably easiest to use an oscilloscope (one that responds down to DC). A 'scope isn't all that accurate, however! I prefer using a 'scope and I'll measure the input and output voltages using peak-to-peak, simply because it's easier on a 'scope. Be careful if you switch between RMS and p-p to adjust the values. At very high frequencies, when the output is very low, you may have trouble measuring in the presence of hum from the amp output. Switch your 'scope triggering to "line", and measure the thickness of the trace. The hum should be stationary on the display. Phase measurements can be tricky. You need a 2 channel 'scope, it must have DC response (otherwise phase will be affected at low frequencies). Your triggering should be external, and the 'scope triggering input should come from the "sync" or equivalent (square wave output) from the signal generator. Your scope should also allow you to view each channel separately, in addition to both at the same time ("chop" or "alternate"). You need to see the reference channel by itself every once in a while, to establish which way to time the signals. Here's the way I do it: Ch. A: reference (input) Ch. B: output - Ch. A and B both set to DC, both zero'd so the baselines overlap. - amplitudes set on both channels to fill at least half the screen. - timebase set to show 1 period (approx) - start at mid band frequency (500 Hz). - waveforms should cross baseline at about the same time, and should be similiar in shape. If one is opposite (one goes up when the other goes down), switch the invert button on one of the 'scope vertical channels. - As the phase changes, similiar events (+ going signal crossing the baseline) for the two waveforms will shift. You need to measure and record the event's TIME shift (measured compared to reference). ref .'. / \__ / .' '.measured / / \ \ / / / \ \ / / --/---/-------\---\-------*---/-------------- / / \ \ / / / / \ \ / / \ '. .' | | \ /-- | | '.' | | t0 t1 In the above ASCII drawing (yuchh!) I have shown the reference and measured signals. The +going measured signal crosses the axis at t1, the reference at t0. Record the value (t1 - t0) which will be a positive value in time corresponding to a negative phase angle. It's negative because at t0 the reference is at zero degrees. At that same time (t0) the measured angle is at some negative phase angle. Reference=sin(wt), measured=sin(wt+theta). If we assume t=0, then theta must be negative. It's confusing, especially when you look at the orientation of the two waveforms. Be very careful measuring phase when it gets close to 180 deg or more. It's EASY to lose track of which one is reference. With a 'scope (externally triggered), you should be able to select one channel at a time to make sure you know which is reference. To calculate the phase angle (degrees), phase=-360*(t1-t0)*frequency. Above the midband you should see + time (- phase), and below midband you should see - time (+ phase). First we need the open loop response of the amplifier system being tested. Here is data taken from a tube amplifier that I'm playing with: Table 1.1 Open loop response Vin(Volts-pp)= 0.0148 delta-t V out gain gain phase Freq us v-pp db degrees 1.5 -225000 0.02 1.4 2.6 121.5 2 -200000 0.038 2.6 8.2 144.0 2.5 -165000 0.076 5.1 14.2 148.5 3.5 -130000 0.17 11.5 21.2 163.8 5 -100000 0.36 24.3 27.7 180.0 7.5 -54000 0.8 54.1 34.7 145.8 10 -35000 1.2 81.1 38.2 126.0 20 -12000 2.4 162.2 44.2 86.4 35 -4800 3.5 236.5 47.5 60.5 50 -2600 4 270.3 48.6 46.8 100 -700 5 337.8 50.6 25.2 200 -160 5.4 364.9 51.2 11.5 350 0 5.4 364.9 51.2 0.0 500 40 5.4 364.9 51.2 -7.2 1000 60 5 337.8 50.6 -21.6 3000 50 3.4 229.7 47.2 -54.0 4000 44 2.9 195.9 45.8 -63.4 5000 40 2.4 162.2 44.2 -72.0 7500 32 1.65 111.5 40.9 -86.4 10000 27 1.28 86.5 38.7 -97.2 15000 20 0.8 54.1 34.7 -108.0 20000 16 0.6 40.5 32.2 -115.2 30000 12.5 0.36 24.3 27.7 -135.0 40000 11 0.25 16.9 24.6 -158.4 50000 9.2 0.17 11.5 21.2 -165.6 75000 8 0.08 5.4 14.7 -216.0 100000 7 0.032 2.2 6.7 -252.0 150000 5 0.013 0.9 -1.1 -270.0 200000 3 0.0044 0.3 -10.5 -216.0 250000 2.6 0.0036 0.2 -12.3 -234.0 300000 2.2 0.0024 0.2 -15.8 -237.6 "delta-t" is the time between reference and measured events (t0-t1), in microseconds. The phase shift is quite excessive! It has been compensated (a dominant pole has been placed in the plate load of the first stage). Without the compensation, the phase shift is not as bad. However, without the compensation, I cannot apply very much feedback before the system gets unstable. This open loop circuit also has positive feedback inside it (between 1st and 2nd stage). this has slight effect at high frequencies, but has considerable effect at the very low frequencies due to capacitive coupling within the positive feedback loop. The following chart shows the slope of the above open loop gain vs. frequency, to show possible areas of trouble. You can see that the slope is bad at 75KHz, where the slope is 19 db/octave. In fact, with capacitive loads (which aggravate the stability), that's the frequency where the system wants to oscillate. It get worse at higher frequencies, but at that point, the gain is very low. The attenuation in db/octave is: db/octave= (db@f1-db@f2) * log-base-2-of(f2/f1) Table 1.2 Open loop gain-phase data gain phase Freq db degrees octaves db/octave 1.5 2.62 121.50 0.42 -13.43 2 8.19 144.00 0.32 -18.70 2.5 14.21 148.50 0.49 -14.41 3.5 21.20 163.80 0.51 -12.67 5 27.72 180.00 0.58 -11.86 7.5 34.66 145.80 0.42 -8.49 10 38.18 126.00 1.00 -6.02 20 44.20 86.40 0.81 -4.06 35 47.48 60.48 0.51 -2.25 50 48.64 46.80 1.00 -1.94 100 50.57 25.20 1.00 -0.67 200 51.24 11.52 0.81 0.00 350 51.24 0.00 0.51 0.00 500 51.24 -7.20 1.00 0.67 1000 50.57 -21.60 1.58 2.11 3000 47.22 -54.00 0.42 3.33 4000 45.84 -63.36 0.32 5.11 5000 44.20 -72.00 0.58 5.56 7500 40.94 -86.40 0.42 5.31 10000 38.74 -97.20 0.58 6.98 15000 34.66 -108.00 0.42 6.02 20000 32.16 -115.20 0.58 7.59 30000 27.72 -135.00 0.42 7.63 40000 24.55 -158.40 0.32 10.41 50000 21.20 -165.60 0.58 11.19 75000 14.66 -216.00 0.42 19.18 100000 6.70 -252.00 0.58 13.38 150000 -1.13 -270.00 0.42 22.67 200000 -10.54 -216.00 0.32 5.41 250000 -12.28 -234.00 0.26 13.39 300000 -15.80 -237.60 In much of the design literature, you are supposed break the above data into segments, where each segment is a multiple of 6db/octave, corresponding to a pole or zero. From the above real data, this is not going to be a simple task. The only clear sloped area is from 4KHz to 15KHz, and that's from my lag network. To get a measure of how prone the system is to oscillation, you can make a "Nyquist diagram". That is the measure of B*A in magnitude and phase, plotted on a polar scale (or on rectangular complex scale). You can also (more easily) make a "Bode diagram". This is a measure of B*A and phase, each one plotted vs. frequency, on a conventional X-Y graph. It is possible to compute "BA" from open and closed loop responses, but the results are very rough at low signal levels due to measurement problems. Trying to use that data in calculations generates a lot of error. It's not worth the effort to do it that way, unless you have some automated systems. You can get "B*A" by the following method: Disconnect the feedback network from the amplifier output. The system will be running open loop now. Insert your test signal into the feedback wire you just disconnected! Now your test signal goes through the feedback network (B or beta), and THEN through the open loop amplifier ("A"). The gain will be B * A ! Measure the response (gain and phase shift). When you do this, the output will be inverted because of the polarity at the summing point. You can flip the polarity with most scope input controls. Here are the results from doing this measurement with the amp above: Table 1.3 "BA, or loopgain, or T, or return ratio)" for Bode diagram for Nyquist diagram gain gain phase BA-real BA-imaginary Freq db 1.5 0.350 -9.1 162.0 -0.33 0.11 2 0.625 -4.1 154.8 -0.57 0.27 2.5 0.925 -0.7 144.0 -0.75 0.54 3.5 1.500 3.5 126.0 -0.88 1.21 5 2.400 7.6 108.0 -0.74 2.28 7.5 3.750 11.5 90.5 -0.03 3.75 10 4.500 13.1 72.0 1.39 4.28 20 7.500 17.5 54.7 4.33 6.12 35 10.000 20.0 36.5 8.03 5.95 50 11.000 20.8 28.8 9.64 5.30 100 12.500 21.9 12.6 12.20 2.73 200 12.500 21.9 2.9 12.48 0.63 350 12.500 21.9 -5.0 12.45 -1.10 500 12.500 21.9 -7.2 12.40 -1.57 1000 11.000 20.8 -25.2 9.95 -4.68 3000 6.750 16.6 -48.6 4.46 -5.06 4000 5.500 14.8 -54.7 3.18 -4.49 5000 4.750 13.5 -57.6 2.55 -4.01 7500 3.500 10.9 -64.8 1.49 -3.17 10000 3.000 9.5 -68.4 1.10 -2.79 15000 2.100 6.4 -75.6 0.52 -2.03 20000 1.650 4.3 -82.8 0.21 -1.64 30000 1.200 1.6 -90.7 -0.02 -1.20 40000 0.900 -0.9 -97.9 -0.12 -0.89 50000 0.700 -3.1 -102.6 -0.15 -0.68 75000 0.450 -6.9 -121.5 -0.24 -0.38 100000 0.250 -12.0 -133.2 -0.17 -0.18 150000 0.150 -16.5 -124.2 -0.08 -0.12 200000 0.083 -21.7 -129.6 -0.05 -0.06 250000 0.075 -22.5 -144.0 -0.06 -0.04 300000 0.070 -23.1 -151.2 -0.06 -0.03 By plotting B*A in db vs. frequency, and phase vs. frequency you get a "Bode diagram". The stability can be determined directly from this data. Determine how far away you are in degrees from +- 180 deg at gain=1 or 0db gain. In my case, unity gain (0db) has a margin of 180-94 (86) degrees at around 35KHz. At low frequencies, unity gain has a margin of 180-140 (40) degrees at around 2Hz. You can plot (B*A) and phase as a polar plot. This is a "Nyquist diagram". The B*A vector is pinned at one end to the origin, it swings around according to the phase, and the other end of the vector draws a line. Few people have that kind of paper, so you can convert it to complex and plot it on a rectangular scale using complex numbers. You can use simply the classic methods to convert phasors to complex numbers and back: phasor pair : magnitude, phase complex pair : real, imaginary real=magnitude*cos(phase) imaginary=magnitude*sin(phase) or magnitude=sqrt(real^2 + imaginary^2) phase=ATAN(imaginary/real) The last two columns in Table 1.3 are the rectangular complex values used to draw the Nyquist diagram on an XY scale. The following is a crude representation of the B*A data for my amp, graphed directly from the B*A phasor data converted to complex: Nyquist diagram for table 1.3 y axis (imaginary) | a=10Hz | b=100 Hz |.--''--.._ c=1 KHz / a \b d=10KHz { \ _______\ |_________x axis (real) / | { / polar plot of gain(B*A)vs phase \ / or complex plot of the BA vector |'-..__..-'c | d | This plot shows direction how the B*A magnitude is pointing (rotating about 0,0). In this case, B*A is pointing at 0 degrees around 300Hz, 90 deg just below 10Hz, and -90 deg. at 30KHz. The shape you get is like a heart with a rounded bottom, but rotated counterclockwise so the rounded bottom is now pointing to the right. The dimple at what used to be the top of the heart is now on the left, and the dimple is at the origin (0,0) of the plot. This origin corresponds to the zero gain at very low and very high frequencies. A DC amplifier wouldn't have the top half (DC gain constant at all low freq. Stability and tendency to oscillate depend on the behaviour of the lobes that are on the left, above and below the x axis, and have negative x values. If the Nyquist diagram touches or encloses where x=-1, y=0, the system may oscillate. How close it gets to that point determines the degree of instability and overshoot. Some Nyquist diagrams have multiple loops in this area - determining stability in those cases is a bit more complex. You really need to dig up a textbook in that case. The tangent of the curves (or angles of the lines) in this area also determine stability. The tangents that are close to x=-1,y=0 determine phase margins. The trick is to keep the lines from sloping to +180 or - 180 where gain=1. Often you'll see a circle drawn with a radius=1 (that's gain=1). Where your graph cuts that circle, the angle between the phase at that intersection and the -X axis is the system "phase margin". It's probably easier to use the Bode plots to establish this value. It turns out that you can do a better job using Nyquist diagrams and plotting all the poles and zeros, than analyzing using Bode plots. The increased complexity, math, and weird mapping puts that method out of reach for most hobbyists and "weekend designers". LAG COMPENSATION: Let's say your system with feedback is on the verge of oscillating. You can compensate the system by prematurely rolling off the frequency response, and altering the system phase response. This is done as described in the "LOW TECH METHOD" section above, by putting a capacitor to ground in the collector/plate circuit, or adding a local feedback loop with a capacitor between a stage input and output. .--------------. C2 | Gain stage | | | | current |-----.----| |-------. | output | | | | | '--------------' | | LAG compensation \ \ network. "stepping / R2 / R3 (optional) network" if R3 is \ \ zero ohms not zero / / normally | | | | gnd gnd R3 is replaced by a short, normally. If you use R3 it's called a "stepping network". Without R3, the -3db point (where the phase angle would be -45 deg) is fc=1/(2*pi*R2*C2). At an octave above fc, the phase angle would be about -65 deg. With R3 in the circuit, you need to calculate the phase/amplitude response for the different values of R2,R3,C2. In the example I used here, I built the amp with the simple lag network. I tried the "stepping network", but it wouldn't cut the gain enough to get the desired phase margin. Here's the before and after response for a lag compensation: Table 1.4 open loop response uncompensated compensated difference Freq Phase Gain Phase Gain gain phase 1.5 135.0 0.0 135.0 0.0 0.0 0.0 2.5 148.5 13.6 148.5 13.6 0.0 0.0 3.5 170.1 21.0 170.1 21.0 0.0 0.0 5 194.4 28.0 194.4 28.0 0.0 0.0 7.5 118.8 34.0 118.8 34.0 0.0 0.0 10 108.0 36.9 108.0 36.9 0.0 0.0 20 57.6 41.2 57.6 41.2 0.0 0.0 35 37.8 42.6 37.8 42.6 0.0 0.0 50 27.0 42.9 27.0 42.9 0.0 0.0 100 14.4 43.5 14.4 43.5 0.0 0.0 200 7.2 43.8 7.2 43.8 0.0 0.0 350 2.5 44.1 0.0 44.1 0.0 -2.5 500 0.0 44.1 -2.2 44.1 0.0 -2.2 1000 -3.6 44.1 -10.8 44.1 0.0 -7.2 3000 -13.0 44.1 -32.4 43.2 -0.9 -19.4 4000 -17.3 43.8 -41.8 42.3 -1.5 -24.5 5000 -21.6 43.5 -48.6 41.6 -1.9 -27.0 7500 -32.4 42.9 -64.8 39.1 -3.8 -32.4 10000 -43.2 42.6 -79.2 37.5 -5.1 -36.0 15000 -59.4 41.2 -94.5 34.6 -6.6 -35.1 20000 -72.0 39.6 -108.0 32.0 -7.5 -36.0 30000 -97.2 36.9 -129.6 27.6 -9.3 -32.4 40000 -115.2 34.2 -146.9 24.6 -9.5 -31.7 50000 -129.6 32.0 -162.0 21.2 -10.8 -32.4 75000 -172.8 26.0 -194.4 14.0 -12.0 -21.6 100000 -187.2 18.8 -216.0 6.8 -12.0 -28.8 150000 -194.4 13.1 -216.0 -2.5 -15.6 -21.6 200000 -201.6 5.1 -216.0 -12.0 -17.1 -14.4 250000 -216.0 1.9 -234.0 -15.1 -17.1 -18.0 300000 -237.6 0.0 -259.2 -18.4 -18.4 -21.6 Looking at the uncompensated system, the upper -3db point is around 15KHz. The compensated system's upper -3db point is near 5KHz. At first glance this seems quite unproductive! Let's assume you have about 20db of feedback. That means subtract 20 db from the openloop gains. What I have done is simulate a feedback network that is a 10:1 voltage divider, or B (beta) equals 0.1 (- 20db). When the result = 0db, check the phase angle..... If it's close to, or greater than +- 180 deg., the system is unstable. In the uncompensated amp, the result is close to 0db at 75-100 KHz, and the phase is near +-180 deg. This system will oscillate. Doing the same calculation for the compensated amp, the resultant gain (subtract 20 db) is 0 db at 40 KHz, but you are 30-40 degrees away from oscillating (phase margin about 35 deg). Not really great, but much better than having the amp go nuts! The primary reason for using lag compensation is to reduce the loopgain (B * A) at high frequencies, thereby giving a better phase margin (keeps you away from the -180 deg. at B*A=1 or 0db). The "stepping network" is a less extreme version of the lag network, and is used where the phase response is important beyond where the gain has been cut. It drops the phase lag back to zero at frequencies much higher. In many amps, once you get above the point where B*A is 1 (or 0db), it's well out of the frequencies of interest, and phase response isn't much of an issue (you've already set the phase margin for where B*A=1). From my limited experience, the stepping network works well where you have large margin of stability, but when you are squeezing a system hard, the plain lag network works better. AMOUNT of FEEDBACK VS. STABILITY (BODE Method) Here's a table of the same open loop response, but limited to the extreme frequencies, and showing the effect of -10db, -20db, -30db feedback. All I did was add the db's corresponding to the beta's of -10db, -20db, -30db. -10db corresponds to beta=.31, 20db beta=0.1, 30db beta=0.03. The 10db corresponds to a LOT of feedback. Table 1.4A (B*A), or loopgain beta (voltage divider in feedback) none (huge db) -20db -10db -30db compensated compensated compensated compensated Freq Phase Gain Phase Gain Phase Gain Phase Gain 500 -2.2 44.1 -2.2 24.1 -2.2 34.1 -2.2 14.1 .................................................. .................. 20000 -108.0 32.0 -108.0 12.0 -108.0 22.0 -108.0 2.0 30000 -129.6 27.6 -129.6 7.6 -129.6 17.6 -129.6 -2.4 40000 -146.9 24.6 -146.9 4.6 -146.9 14.6 -146.9 -5.4 50000 -162.0 21.2 -162.0 1.2 -162.0 11.2 -162.0 -8.8 75000 -194.4 14.0 -194.4 -2.0 -194.4 4.0 -194.4 -16.0 100000 -216.0 6.8 -216.0 -15.2 -216.0 -3.2 -216.0 -23.2 150000 -216.0 -2.5 -216.0 -22.5 -216.0 -12.5 -216.0 -32.5 200000 -216.0 -12.0 -216.0 -32.0 -216.0 -22.0 -216.0 -42.0 250000 -234.0 -15.1 -234.0 -35.1 -234.0 -25.1 -234.0 -45.1 300000 -259.2 -18.4 -259.2 -38.4 -259.2 -28.4 -259.2 -48.4 By inspecting the above data, you can see that the -10db feedback will most likely oscillate since the phase angle exceeds 180 deg. where B*A is 0db (or 1). The system with -30db feedback won't oscillate and has a phase margin of about 70 degrees. You can see that in this case, the -20db feedback is as close as we can get before the system is unstable. You can apply more aggressive compensation (in the lag network). This will allow you to put more feedback, but there's a point where you roll off B*A in the audio range. That means you aren't getting the benefit of feedback any more. It will be good for low frequencies, but have little effect at higher audio ranges. My guess is that you shouldn't set the lag network to cut in much below 10KHz for audio work. You do get the frequency response back again (and better) with the closed loop response. Here's the closed loop response for the same amp whose open loop response appeared in table 1.1: Table 1.5 closed loop gain gain phase Freq db degrees 1.5 10.0 226.8 2 17.9 172.8 2.5 19.5 135.0 3.5 20.9 100.8 5 22.0 82.8 7.5 23.4 67.5 10 25.1 68.0 20 26.5 30.2 35 27.3 20.2 50 27.5 12.6 100 27.7 7.2 200 27.7 2.9 350 27.7 0.0 500 27.7 0.0 1000 27.7 -3.6 3000 27.7 -8.6 4000 27.7 -11.5 5000 27.7 -14.4 7500 27.5 -21.6 10000 27.3 -28.8 15000 26.9 -43.2 20000 26.0 -57.6 30000 24.5 -86.4 40000 22.7 -103.7 50000 21.3 -122.4 75000 16.4 -183.6 100000 9.1 -201.6 150000 0.0 -216.0 200000 -8.0 -230.4 250000 -10.0 -234.0 300000 -13.5 -237.6 The -3db points are at 9 Hz and 30 KHz. The upper -3db point would have been a lot higher, but I dropped it down to 30KHz to give a better phase margin and improve stability. So far feedback sounds pretty wonderful except for it's ability to cause instability. You can fiddle with an amplifiers phase/gain response to maximize the amount of feedback, but there is a limit if you want to keep the closed loop response to reasonable values. I set the -3db point by applying a compensation in the feedback circuit. the feedback compensation does two useful things: it sets a reasonable overall system bandwidth, and it shifts the phase change the other way by using a "lead network". LEAD NETWORK in FEEDBACK Here's the feedback network with C1 added, which provides lead compensation, and also sets the system bandwidth: | | C1 .-----| |------. | | | | to summing point | | Vfb ------,---/\/\/\/\/--'------- to output | R1 \ / My amp: \ R2 R2=400 ohms (approx.) / R1=10 Kohms \ C1=390 pF | gnd The above values (R1, R2) set the closed loop gain to approximately R1/R2, or about 25. It's not exact, it depends also on the open loop gain (A). R1 and C1 set the -3db point to be approximately 40KHz (f=1/(2*pi*R1*C1). The important aspect of this network is that it gives a "lead" (positive) phase angle, opposite to the normal lagging phase angle you normally get at the upper frequency limit. Because this circuit defines "beta", it will affect B*A (the loopgain). It will shift the phase THE OTHER WAY, and improve the phase response, and margin of stability. In other words, it will make the system less prone to oscillate. Here's the above circuit's amplitude and phase response: TABLE 1.5A Response of feedback network ("beta") freq db phase 1.5 -26.96 0.00 2 -26.96 0.00 2.5 -26.96 0.00 3.5 -26.96 0.00 5 -26.96 0.01 7.5 -26.96 0.01 10 -26.96 0.01 20 -26.96 0.03 35 -26.96 0.05 50 -26.96 0.07 100 -26.96 0.13 200 -26.96 0.27 350 -26.96 0.47 500 -26.96 0.67 1000 -26.95 1.34 3000 -26.93 4.02 4000 -26.92 5.35 5000 -26.89 6.67 7500 -26.81 9.94 10000 -26.70 13.14 15000 -26.41 19.24 20000 -26.02 24.85 30000 -25.09 34.43 40000 -24.04 41.91 50000 -22.99 47.63 75000 -20.57 56.73 100000 -18.56 61.52 150000 -15.46 65.41 200000 -13.18 66.06 250000 -11.42 65.35 300000 -10.00 63.99 Multiplying the "beta" response by the open loop response "A" we get the loopgain (B*A). Multiplication of phasors is product of magnitudes, and sums of angles. Because I used db for magnitudes, I add them. That's the nice feature of db's! Here's the result (B*A) including the lead network in the feedback (beta) circuit: TABLE 1.5B Freq B*A B*A Hz db phase 350 24.3 0.5 500 24.3 -6.5 1000 23.6 -20.3 3000 20.3 -50.0 4000 18.9 -58.0 5000 17.3 -65.3 7500 14.1 -76.5 10000 12.0 -84.1 15000 8.2 -88.8 20000 6.1 -90.4 30000 2.6 -100.6 40000 0.5 -116.5 50000 -1.8 -118.0 75000 -5.9 -159.3 100000 -11.9 -190.5 150000 -16.6 -204.6 200000 -23.7 -149.9 250000 -23.7 -168.6 300000 -25.8 -173.6 Now after using the lead network in the "beta" circuit, when you check B*A for stability, ie., how close to +-180 degrees at BA=1 (or 0db), you can see we have 80 or 90 degrees phase margin. That's probably enough to tolerate a purely capacitive load. FEEDBACK ISSUES Feedback DOES have a related dark side. We expect feedback to reduce the overall gain. It does this in most of the bandwidth, but at the upper and lower (if your amp is not DC coupled) extremes, it INCREASES the overall gain. This happens when you look at the Nyquist diagram, and your response curves get inside the little unity gain circle. This is very common! Seeing as Nyquist diagrams are a pain to produce, check the phase/gain data for the amplifier described above at the frequency extremes, and compare the closed loop (feedback) and open loop (no feedback) responses. The feedback amp has about about 8db more gain at low frequencies, and 2 db more gain than the open loop system! Table 1.6 Amplifier Gain (not B*A) and phase - no feedback vs. feedback open loop closed loop gain phase gain phase Freq db degrees db degrees 1.5 2.62 121.50 10.0 226.8 2 8.19 144.00 17.9 172.8 2.5 14.21 148.50 19.5 135.0 3.5 21.20 163.80 20.9 100.8 5 27.72 180.00 22.0 82.8 ........................................ 150000 -1.1 -270.0 0.0 -216.0 200000 -10.5 -216.0 -8.0 -230.4 250000 -12.3 -234.0 -10.0 -234.0 300000 -15.8 -237.6 -13.5 -237.6 This means (for audio work) that the system is giving excessive amplification for "rumble" or radio frequencies. If you didn't pay attention to this behaviour and filter out those frequencies on the input (prior to the feedback loop), you could get some nasty behaviour. This is also a mechanism whereby RF (radio frequencies) can get into your system from the output (speaker) wiring. In some situations you might need to filter BOTH the inputs and outputs at high frequencies! Normally the system gain at these extremes is pretty low, and should be quite a bit lower than midband gain. You should be aware of this problem if you are exposed to unusual signal conditions such as turntable rumble, radio frequency interference, EMI, crosstalk, clock frequency noise (CD player, or digital circuits), etc. AND your feedback forces a high gain at the extremities. As I mentioned above, there IS a limit to how much feedback you can apply. My guess is that it is difficult to have B*A much in excess of 20-25db for a tube amplifier. Semiconductor amps can allow considerable more, especially with high bandwidth FET's/transistors. I have heard of designs with 250db of feedback! Those would be low frequency extremely high accuracy systems. Feedback systems can be very naughty when the system is nonlinear, and especially if the system has low phase margins. Poorly designed class B amplifiers with too much bias current or badly designed current "crossover" on output devices can change "A" on the fly, and cause problems. Near saturation, some circuits will cause the system to momentarily oscillate as phase/gain characteristics change. I have had some otherwise well designed amplifiers show this kind of trouble. The maximum gain under dynamic conditions would need to be calculated or measured, and the stability margin would be calculated or measured based on the worst-case value for the changing open-loop gain ("A"). Feedback systems want instant feedback..... delays really mess up stability. Normally you don't think of delays other than phase shifts, but some circuits will have "slew rate limiting". That's where the system can only change the voltage at a fixed rate (slewing), no matter how hard it's driven. During this time, the circuits on either side of the "slew rate limiter" are out of touch with each other, and must wait until things settle down. This behaviour can haunt badly designed semiconductor amplifiers. The delay time can be converted to phase at the frequency where stability margin is calculated, and considered as an additional lag. It will reduce the stability margin. The last 2 problems should prompt the designer to back way off on the amount of feedback! -Paul |
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feedback & stability (long)
Paul,
if you wrote this article yourself, I really admire your technical writing tallent! Perhaps the same transient response observation method can be used on the low frequencies (LF). Apply 1Hz rectangular signal to the input and observe the output. In a good amplifier it shall respond by neat exponential "teeth" (alternating polarity) on each edge of the input signal, with some overshoot. If you see prolonged undulation, like a swell on the sea, it is a problem. Feedback widens LF bandwidth to a few Hz. It is useless, causing nothing but rumble overloading tubes and magnetizing the transformer for no purpose. In this case I usually deepen feedback at LF just below mechanical resonance of a speaker by connecting a series RC (say 47uF + 120ohm) in parallel to the first stage cathode resistor where the feedback is applied. Now while the transformer is rolling off, this circuit is rising loop gain up, compensating phase advance of the transformer. In this configuration, output impedance is maintained low at LF, helping damping speaker resonance better. The dominant null now can be an interstage cap (set to about 30...40Hz cut-off), not the transformer. (There is no need to select an enormous interstage capacitor.) At 3...4Hz open loop gain will gently cross 0dB with ample phase margin. Regards, Alex "Paul G." wrote in message ... Here's a rather long blurb on feedback and stability. I'm not too sure if it's too simple or too complicated! There's probably all kinds of mistakes, please check it out for screw-ups. Since there are ASCII schematics and diagrams, you'll need to view it with a fixed width font like "new courier". INTRODUCTION ............................ You have built an amplifier, or are planning one and you decide that maybe you should employ feedback. Feedback is a mechanism where you use excess gain in an amplifier to reduce "bad things" that are generated within the amplifier circuit. Those "bad things" are distortion, noise, poor frequency response, phase shift - anything that wasn't present in the input signal. Feedback has limitations. At the extreme ends of the amplifiers frequency response, those "bad things" can be made WORSE, often to the point where the amplifier decides to become a signal generator, at maximum possible output level. The art in using feedback is to maximize the amount of feedback in the audio spectrum (or range of frequencies that have your interest), and minimize the nasty effects where feedback makes things worse. There are a number of designs that avoid feedback in any measure, it is very difficult to avoid since many circuits and devices have "built-in" feedback. Triodes have built in feedback from plate to grid (pentodes eliminate that by using a screen grid to shield the main grid, thereby getting higher power gain). "Follower" topologies use feedback. Since these circuits wrap the feedback around just one gain stage, they are considered fairly benign. The evil feedback usually encompasses 2 or more gain stages. Poor feedback practices can cause any number of headaches, and when components, loads, or input signals are altered or undergo changes, the circuit can do some very unexpected behaviour. First I'll try to go through the feedback theory. Second I'll show how I set up feedback using a minimum of equipment and time. This method uses a lot of trial and error (adding components and tweaking them for best response). Remember, trial and error is a VERY powerful design technique, and should never be dismissed! Thirdly I'll go the whole 9 yards with Bode plots, Nyquist criteria, etc. This explains all the weird things that you would have used trial and error to solve, and it will allow you to do the complete design, providing that you know in detail all the circuit behaviour and parameters. Knowing that detail, and making many measurements make this 3rd method unacceptable (IMHO). ....You don't need Microsoft Excel for most of this, but it is handy for publishing the results, and is one of the easier ways to graph Nyquist diagrams. Otherwise graph paper, or just the list of measurements should do fine. THEORY ............................ I will make use of ASCII circuits, to see them you will need to change your font to a fixed width font like "Courier New". I will use "beta" or "B" as a positive value (not like Radio Designers Handbook ed.4 -RDH4), and will use a summing point with inversion as is more commonly used in today's textbooks. Here's the generic feedback configuration: summing point .--------------. + |Amplifier | input ---------------(X)------|gain= A |----------- output Vin - | Va | | | Vout | '--------------' | | .--------------. | | |feedback | | feedback '-------|network |---------' voltage |I/O ratio= B | Vfb=B*Vout '--------------' The summing point (X) adds the input and subtracts the feedback signal. In many circuits, the + point is the grid or transistor base, and the - point is the cathode or emitter. With an op-amp, they could be the inverting or non-inverting inputs. The summing point could also be a simple node (junction) if you are using current instead of voltage. I'll use the more common voltage feedback, derived directly from the output (it samples output voltage). With different topologies, you need to redo the math, especially if you are deriving the feedback from load current or/and using current summing at the input. There are "rules of thumb" when you use those topologies, but don't trust 'em...... do a bit of math to make sure. The amplifier is your amp..... with all its warts. Its gain is "A". Va is the input voltage to the amp (after the summing point). You can consider this as a single number at mid-band, but it must be expressed as a complex number or "phasor" as you get to the extreme ends. I prefer to use "phasors", they're easier. A phasor is simply a number PAIR, the amplitude and the phase difference between amp input and output. It's much like the wind measurement.... 40km/hr isn't much use by itself to a sailor, until they add "from NE", in which case it's a useful value. The complex pair equivalent would be a 14km/hr component from the north, and a 14 km/hr from the east (where north is the imaginary axis, and east the real axis). In electrical notation, say 4 volts, phase +45deg, or even better, 5 db gain and phase=+45 deg. Phasors expressed with db make the math nice to work with. Sometimes there is an additional amplifier stage between the input and the summing point. It will not be affected by the feedback. Make sure that this amp section is free of "warts". You'd hate to have set up the main amp to be very good, and then have crud injected by the additional stage. The feedback network could be a simple voltage divider where B=R1/(R1+R2), or something more complex with capacitors and other components. It produces a voltage Vfb = B * Vout The output(Vout) = A * Va (eq.1) but Va = Vin - Vfb (eq.2) Va = Vin - (B * Vout) (eq.3) substitute for Va using eq3 and recalculate eq1: Vout = A * (Vin - (B * Vout)) = A*Vin - B*A*Vout (eq.4) gather Vin's and Vout's Vout + B*A*Vout = A*Vin factor Vout Vout*(1 +B*A) = A*Vin Since the effective gain of the whole system is G, and is Vout/Vin moving things around you get: Vout = Vin * ( A/(1+B*A)) (eq.5) or: gain = G = Vout/Vin = A / (1 + B*A) The original gain (A or open loop gain) is effectively reduced to a lower "closed loop gain" (with feedback in place). It is reduced by the factor "1 + B*A". If that's all feedback did, it wouldn't be much use. Suppose your amplifier circuit without feedback produced an extraneous (unwanted) output, I'll call it "CRUD". Then: CRUD summing point .--------------. | + |Amplifier |Vao |+ input ---------------(X)------|gain= A |----(X)--.-- output Vin - | Va | | + | Vout | '--------------' | | .--------------. | | |feedback | | feedback '-------|network |---------' voltage |I/O ratio= B | Vfb '--------------' I have added the "CRUD" signal between the amp output and the real output. You can add the CRUD signal anywhere INSIDE the feedback loop. Vao = A*Va Vout = A*Va + CRUD (eq.1a) but Va = Vin - Vfb (eq.2a) Va = Vin - (B * Vout) (eq.3a) Vao = A * (Vin - (B * Vout)) Vao = A*Vin - B*A*Vout (eq.4a) Vout = A*Vin - B*A*Vout + CRUD gather terms, Vout( 1+B*A) = A*Vin + CRUD Vout = Vin * A/(1+B*A) + CRUD/(1+B*A) (eq.5a) Compare eq.5 and eq.5a, you'll see that the value of "CRUD" has been reduced by the factor (1 + B*A), just like the gain. So... if you designed your amp to have an open loop gain 10 times what you need, then reduce it with feedback to what you want, the noise, distortion and unwanted behaviour is also reduced. Similarly, the frequency response is extended, but with some reservations. If you go to the frequency extremes, the open loop phase response is such that the phase change can get to 180 degrees, and that means your feedback becomes positive feedback, which makes everything worse, or the system goes nuts and oscillates. LOW TECH METHOD of OPTIMIZING FEEDBACK ........................................... You need an oscilloscope, it should be able to respond to at least 10-50 times your intended high frequency limit. Similarly, it should be able to respond to 1/10 to 1/50 of your low frequency limit (DC response is preferred). You need a signal generator that can give a quality square wave at 1000 Hz. Put a resistor load on the amp, correct power rating and value. Your feedback circuit (whose transfer function or "gain" is "beta" or B) will look something like this: to summing point Vfb ------,---/\/\/\/\/--------- to output | R1 \ / \ R2 / \ | gnd In this case B = R2/(R1 + R2) voltage divider Since the effective gain is A/(1 + B*A) A= amp openloop gain, if A is large compared to 1, you can approximate the equation to: effective gain = 1/B = (R1 + R2)/R2; the error is 100%/A Usually I keep R2 at a fixed value, and change R1. R2 is often a cathode resistor for tube amps, and is awkward to change since biasing is affected. In op-amps R2 is often to be matched to values on the non-inverting side. So R1 is the one usually changed. For gains of 10- 50 (typical), R1 will be 10-50 times the value of R2. Here's the problem - as you increase the feedback, the term (1 + BA) increases. This is the factor by which the bad stuff goes away. BUT, as that factor increases, the amp is more likely to go nuts at some very high or very low frequency. My first defense against nutty behaviour is to place a capacitor in parallel with R1, and pick the cutoff frequency (where |Xc|=1/(2*pi*f*c)=R) to a reasonable value. This is a tough choice. You have a nice circuit, possibly with a very good output transformer, that claims to go way beyond your usual limit (20KHz for audio). Let's say your openloop amp (with transformer?) can have a cutoff of 150KHz. I claim you should set the feedback cutoff to approximately 25-40 KHz. One way or another, if your goal is to reduce distortion without crazy amp behaviour, you'll be dropping your cutoff back to this value. Here's how I go about it. Put a 1 KHZ square wave in the input, watch the output on the 'scope. Choose a reasonable signal level,that gives 2-5%of full output. Start off with no capacitor in parallel with R1, and R1 about 50-100 times that value of R2. R1 is easiest to be a trimmer resistor or potentiometer (pot). Decrease R1, watch out for overshoot.Add a capacitor in parallel with R1, calculated to give the desired cutoff. The over shoot should disappear. Decrease R1 further, until overshoot shows up. You will need to increase the capacitor to the proper cutoff. You will reach a point where R1 is as low as it can go without lowering the cutoff freq. to silly values. Drop R1 to the point where there is quite a bit of overshoot. Now it's necessary to modify the phase response of the amplifier. In order to do that, we must compromise its open loop response. I will place a capacitor (optional variable resistor in series with it - stepping network) from the first gain stage output (collector or triode plate) to ground. This is called a "lag network". In the case of transistor amps, you can put a cap from collector to base on an intermediate stage. Select capacitors that make the overshoot go away, without excessive rounding off the square wave edges. You can play with the optional series resistor as well. This is a "dominant pole", and reduction of gain at high frequencies. To really understand what's happening here, you need to have a Bode plot (phase, and gain). Pretty soon, you should have optimal feedback for the amp. Now you need to tweak the system for stability. Remove the amp's main load resistor, and use a .1uF - 4.7uF low ESR (Equivalent Series Resistance) capacitor as load. Keep the leads short. The overshoot should get real bad, and maybe the system will oscillate. Avoid that! Things can overheat real quick. Fiddle with the lag network (or stepping network) until it stops oscillating, or reduce the ringing. It's not necessary to eliminate the ringing, 'cause you don't normally have a very capacitive load (unless you have electrostatic speakers). Increase R1, play with the values of capacitor in parallel with R1, change the capacitor and optional series resistor that were in the first stage output. You don't need to eliminate the overshoot under these conditions, just keep the system well away from oscillating. Try to figure out what frequency the system likes to oscillate at. Sometimes you can bypass cathode resistors or emitter resistors that are "degenerative", with capacitors such that cutoff (Fc=1/(2*pi*R*C)) is at the oscillating frequency. This may shift the phase of the amp towards something more stable. It may help to put a snubber network across the output. Typically its a resistor and capacitor in series. R is about 1-3 times the usual load, and C is a value that gives a cutoff 2-3 times the max. useful frequency (60-70 KHz). I used 10 ohms in series with 0.22uF. It helped a bit. Watch out for the power rating of the resistor! If the amp oscillates, the resistor may be cooked! Usually you must back off the feedback to keep the system stable with strange loads. Once you think you have it set up (normal load), momentarily overload the amp (set your signal generator to a much higher output) and then quickly drop the signal to normal. Watch the amp's output for low frequency junk (monitoring the DC current of the output devices is a sensitive check). Things should immediately settle down, without low or high frequency jiggle. If they don't... you have a problem with low frequency phase shift. You should have one stage with poor low frequency response (often a transformer), the rest should have better low frequency responses. This establishes a "dominant zero", which with feedback should give better stability. At the end of this exercise, you have an amp whose frequency response is about 25-40 KHz, with a maximum amount of feedback, without going nuts with a capacitive load. Maximizing feedback, without losing too much stability is the goal. Determine the ratio of openloop (no feedback) gain to closed loop (max feedback) gain. This will be the factor 1 + BA, and all the bad things you amp can do will be reduced by this factor. You weren't able to extend your frequency response, but the compromise allowed you to further reduce distortion, noise, hum and other extraneous signals generated by the amp. DETAILED, THEORETICAL METHOD .................................. Now you need to make a whole lot of measurements on your system. Most importantly, you need to measure the open loop (no feedback) response of your amplifier. You need the amplitude of the output, and the phase of the output as compared to the input. These will establish gain and phase response of the system. The frequency range (sine wave) must extend beyond where the gain drops to 1.0 . Include the frequencies where the gain is 0.5 or lower, and you should be fine. Some of these frequencies are difficult to measure, it's probably easiest to use an oscilloscope (one that responds down to DC). A 'scope isn't all that accurate, however! I prefer using a 'scope and I'll measure the input and output voltages using peak-to-peak, simply because it's easier on a 'scope. Be careful if you switch between RMS and p-p to adjust the values. At very high frequencies, when the output is very low, you may have trouble measuring in the presence of hum from the amp output. Switch your 'scope triggering to "line", and measure the thickness of the trace. The hum should be stationary on the display. Phase measurements can be tricky. You need a 2 channel 'scope, it must have DC response (otherwise phase will be affected at low frequencies). Your triggering should be external, and the 'scope triggering input should come from the "sync" or equivalent (square wave output) from the signal generator. Your scope should also allow you to view each channel separately, in addition to both at the same time ("chop" or "alternate"). You need to see the reference channel by itself every once in a while, to establish which way to time the signals. Here's the way I do it: Ch. A: reference (input) Ch. B: output - Ch. A and B both set to DC, both zero'd so the baselines overlap. - amplitudes set on both channels to fill at least half the screen. - timebase set to show 1 period (approx) - start at mid band frequency (500 Hz). - waveforms should cross baseline at about the same time, and should be similiar in shape. If one is opposite (one goes up when the other goes down), switch the invert button on one of the 'scope vertical channels. - As the phase changes, similiar events (+ going signal crossing the baseline) for the two waveforms will shift. You need to measure and record the event's TIME shift (measured compared to reference). ref .'. / \__ / .' '.measured / / \ \ / / / \ \ / / --/---/-------\---\-------*---/-------------- / / \ \ / / / / \ \ / / \ '. .' | | \ /-- | | '.' | | t0 t1 In the above ASCII drawing (yuchh!) I have shown the reference and measured signals. The +going measured signal crosses the axis at t1, the reference at t0. Record the value (t1 - t0) which will be a positive value in time corresponding to a negative phase angle. It's negative because at t0 the reference is at zero degrees. At that same time (t0) the measured angle is at some negative phase angle. Reference=sin(wt), measured=sin(wt+theta). If we assume t=0, then theta must be negative. It's confusing, especially when you look at the orientation of the two waveforms. Be very careful measuring phase when it gets close to 180 deg or more. It's EASY to lose track of which one is reference. With a 'scope (externally triggered), you should be able to select one channel at a time to make sure you know which is reference. To calculate the phase angle (degrees), phase=-360*(t1-t0)*frequency. Above the midband you should see + time (- phase), and below midband you should see - time (+ phase). First we need the open loop response of the amplifier system being tested. Here is data taken from a tube amplifier that I'm playing with: Table 1.1 Open loop response Vin(Volts-pp)= 0.0148 delta-t V out gain gain phase Freq us v-pp db degrees 1.5 -225000 0.02 1.4 2.6 121.5 2 -200000 0.038 2.6 8.2 144.0 2.5 -165000 0.076 5.1 14.2 148.5 3.5 -130000 0.17 11.5 21.2 163.8 5 -100000 0.36 24.3 27.7 180.0 7.5 -54000 0.8 54.1 34.7 145.8 10 -35000 1.2 81.1 38.2 126.0 20 -12000 2.4 162.2 44.2 86.4 35 -4800 3.5 236.5 47.5 60.5 50 -2600 4 270.3 48.6 46.8 100 -700 5 337.8 50.6 25.2 200 -160 5.4 364.9 51.2 11.5 350 0 5.4 364.9 51.2 0.0 500 40 5.4 364.9 51.2 -7.2 1000 60 5 337.8 50.6 -21.6 3000 50 3.4 229.7 47.2 -54.0 4000 44 2.9 195.9 45.8 -63.4 5000 40 2.4 162.2 44.2 -72.0 7500 32 1.65 111.5 40.9 -86.4 10000 27 1.28 86.5 38.7 -97.2 15000 20 0.8 54.1 34.7 -108.0 20000 16 0.6 40.5 32.2 -115.2 30000 12.5 0.36 24.3 27.7 -135.0 40000 11 0.25 16.9 24.6 -158.4 50000 9.2 0.17 11.5 21.2 -165.6 75000 8 0.08 5.4 14.7 -216.0 100000 7 0.032 2.2 6.7 -252.0 150000 5 0.013 0.9 -1.1 -270.0 200000 3 0.0044 0.3 -10.5 -216.0 250000 2.6 0.0036 0.2 -12.3 -234.0 300000 2.2 0.0024 0.2 -15.8 -237.6 "delta-t" is the time between reference and measured events (t0-t1), in microseconds. The phase shift is quite excessive! It has been compensated (a dominant pole has been placed in the plate load of the first stage). Without the compensation, the phase shift is not as bad. However, without the compensation, I cannot apply very much feedback before the system gets unstable. This open loop circuit also has positive feedback inside it (between 1st and 2nd stage). this has slight effect at high frequencies, but has considerable effect at the very low frequencies due to capacitive coupling within the positive feedback loop. The following chart shows the slope of the above open loop gain vs. frequency, to show possible areas of trouble. You can see that the slope is bad at 75KHz, where the slope is 19 db/octave. In fact, with capacitive loads (which aggravate the stability), that's the frequency where the system wants to oscillate. It get worse at higher frequencies, but at that point, the gain is very low. The attenuation in db/octave is: db/octave= (db@f1-db@f2) * log-base-2-of(f2/f1) Table 1.2 Open loop gain-phase data gain phase Freq db degrees octaves db/octave 1.5 2.62 121.50 0.42 -13.43 2 8.19 144.00 0.32 -18.70 2.5 14.21 148.50 0.49 -14.41 3.5 21.20 163.80 0.51 -12.67 5 27.72 180.00 0.58 -11.86 7.5 34.66 145.80 0.42 -8.49 10 38.18 126.00 1.00 -6.02 20 44.20 86.40 0.81 -4.06 35 47.48 60.48 0.51 -2.25 50 48.64 46.80 1.00 -1.94 100 50.57 25.20 1.00 -0.67 200 51.24 11.52 0.81 0.00 350 51.24 0.00 0.51 0.00 500 51.24 -7.20 1.00 0.67 1000 50.57 -21.60 1.58 2.11 3000 47.22 -54.00 0.42 3.33 4000 45.84 -63.36 0.32 5.11 5000 44.20 -72.00 0.58 5.56 7500 40.94 -86.40 0.42 5.31 10000 38.74 -97.20 0.58 6.98 15000 34.66 -108.00 0.42 6.02 20000 32.16 -115.20 0.58 7.59 30000 27.72 -135.00 0.42 7.63 40000 24.55 -158.40 0.32 10.41 50000 21.20 -165.60 0.58 11.19 75000 14.66 -216.00 0.42 19.18 100000 6.70 -252.00 0.58 13.38 150000 -1.13 -270.00 0.42 22.67 200000 -10.54 -216.00 0.32 5.41 250000 -12.28 -234.00 0.26 13.39 300000 -15.80 -237.60 In much of the design literature, you are supposed break the above data into segments, where each segment is a multiple of 6db/octave, corresponding to a pole or zero. From the above real data, this is not going to be a simple task. The only clear sloped area is from 4KHz to 15KHz, and that's from my lag network. To get a measure of how prone the system is to oscillation, you can make a "Nyquist diagram". That is the measure of B*A in magnitude and phase, plotted on a polar scale (or on rectangular complex scale). You can also (more easily) make a "Bode diagram". This is a measure of B*A and phase, each one plotted vs. frequency, on a conventional X-Y graph. It is possible to compute "BA" from open and closed loop responses, but the results are very rough at low signal levels due to measurement problems. Trying to use that data in calculations generates a lot of error. It's not worth the effort to do it that way, unless you have some automated systems. You can get "B*A" by the following method: Disconnect the feedback network from the amplifier output. The system will be running open loop now. Insert your test signal into the feedback wire you just disconnected! Now your test signal goes through the feedback network (B or beta), and THEN through the open loop amplifier ("A"). The gain will be B * A ! Measure the response (gain and phase shift). When you do this, the output will be inverted because of the polarity at the summing point. You can flip the polarity with most scope input controls. Here are the results from doing this measurement with the amp above: Table 1.3 "BA, or loopgain, or T, or return ratio)" for Bode diagram for Nyquist diagram gain gain phase BA-real BA-imaginary Freq db 1.5 0.350 -9.1 162.0 -0.33 0.11 2 0.625 -4.1 154.8 -0.57 0.27 2.5 0.925 -0.7 144.0 -0.75 0.54 3.5 1.500 3.5 126.0 -0.88 1.21 5 2.400 7.6 108.0 -0.74 2.28 7.5 3.750 11.5 90.5 -0.03 3.75 10 4.500 13.1 72.0 1.39 4.28 20 7.500 17.5 54.7 4.33 6.12 35 10.000 20.0 36.5 8.03 5.95 50 11.000 20.8 28.8 9.64 5.30 100 12.500 21.9 12.6 12.20 2.73 200 12.500 21.9 2.9 12.48 0.63 350 12.500 21.9 -5.0 12.45 -1.10 500 12.500 21.9 -7.2 12.40 -1.57 1000 11.000 20.8 -25.2 9.95 -4.68 3000 6.750 16.6 -48.6 4.46 -5.06 4000 5.500 14.8 -54.7 3.18 -4.49 5000 4.750 13.5 -57.6 2.55 -4.01 7500 3.500 10.9 -64.8 1.49 -3.17 10000 3.000 9.5 -68.4 1.10 -2.79 15000 2.100 6.4 -75.6 0.52 -2.03 20000 1.650 4.3 -82.8 0.21 -1.64 30000 1.200 1.6 -90.7 -0.02 -1.20 40000 0.900 -0.9 -97.9 -0.12 -0.89 50000 0.700 -3.1 -102.6 -0.15 -0.68 75000 0.450 -6.9 -121.5 -0.24 -0.38 100000 0.250 -12.0 -133.2 -0.17 -0.18 150000 0.150 -16.5 -124.2 -0.08 -0.12 200000 0.083 -21.7 -129.6 -0.05 -0.06 250000 0.075 -22.5 -144.0 -0.06 -0.04 300000 0.070 -23.1 -151.2 -0.06 -0.03 By plotting B*A in db vs. frequency, and phase vs. frequency you get a "Bode diagram". The stability can be determined directly from this data. Determine how far away you are in degrees from +- 180 deg at gain=1 or 0db gain. In my case, unity gain (0db) has a margin of 180-94 (86) degrees at around 35KHz. At low frequencies, unity gain has a margin of 180-140 (40) degrees at around 2Hz. You can plot (B*A) and phase as a polar plot. This is a "Nyquist diagram". The B*A vector is pinned at one end to the origin, it swings around according to the phase, and the other end of the vector draws a line. Few people have that kind of paper, so you can convert it to complex and plot it on a rectangular scale using complex numbers. You can use simply the classic methods to convert phasors to complex numbers and back: phasor pair : magnitude, phase complex pair : real, imaginary real=magnitude*cos(phase) imaginary=magnitude*sin(phase) or magnitude=sqrt(real^2 + imaginary^2) phase=ATAN(imaginary/real) The last two columns in Table 1.3 are the rectangular complex values used to draw the Nyquist diagram on an XY scale. The following is a crude representation of the B*A data for my amp, graphed directly from the B*A phasor data converted to complex: Nyquist diagram for table 1.3 y axis (imaginary) | a=10Hz | b=100 Hz |.--''--.._ c=1 KHz / a \b d=10KHz { \ _______\ |_________x axis (real) / | { / polar plot of gain(B*A)vs phase \ / or complex plot of the BA vector |'-..__..-'c | d | This plot shows direction how the B*A magnitude is pointing (rotating about 0,0). In this case, B*A is pointing at 0 degrees around 300Hz, 90 deg just below 10Hz, and -90 deg. at 30KHz. The shape you get is like a heart with a rounded bottom, but rotated counterclockwise so the rounded bottom is now pointing to the right. The dimple at what used to be the top of the heart is now on the left, and the dimple is at the origin (0,0) of the plot. This origin corresponds to the zero gain at very low and very high frequencies. A DC amplifier wouldn't have the top half (DC gain constant at all low freq. Stability and tendency to oscillate depend on the behaviour of the lobes that are on the left, above and below the x axis, and have negative x values. If the Nyquist diagram touches or encloses where x=-1, y=0, the system may oscillate. How close it gets to that point determines the degree of instability and overshoot. Some Nyquist diagrams have multiple loops in this area - determining stability in those cases is a bit more complex. You really need to dig up a textbook in that case. The tangent of the curves (or angles of the lines) in this area also determine stability. The tangents that are close to x=-1,y=0 determine phase margins. The trick is to keep the lines from sloping to +180 or - 180 where gain=1. Often you'll see a circle drawn with a radius=1 (that's gain=1). Where your graph cuts that circle, the angle between the phase at that intersection and the -X axis is the system "phase margin". It's probably easier to use the Bode plots to establish this value. It turns out that you can do a better job using Nyquist diagrams and plotting all the poles and zeros, than analyzing using Bode plots. The increased complexity, math, and weird mapping puts that method out of reach for most hobbyists and "weekend designers". LAG COMPENSATION: Let's say your system with feedback is on the verge of oscillating. You can compensate the system by prematurely rolling off the frequency response, and altering the system phase response. This is done as described in the "LOW TECH METHOD" section above, by putting a capacitor to ground in the collector/plate circuit, or adding a local feedback loop with a capacitor between a stage input and output. .--------------. C2 | Gain stage | | | | current |-----.----| |-------. | output | | | | | '--------------' | | LAG compensation \ \ network. "stepping / R2 / R3 (optional) network" if R3 is \ \ zero ohms not zero / / normally | | | | gnd gnd R3 is replaced by a short, normally. If you use R3 it's called a "stepping network". Without R3, the -3db point (where the phase angle would be -45 deg) is fc=1/(2*pi*R2*C2). At an octave above fc, the phase angle would be about -65 deg. With R3 in the circuit, you need to calculate the phase/amplitude response for the different values of R2,R3,C2. In the example I used here, I built the amp with the simple lag network. I tried the "stepping network", but it wouldn't cut the gain enough to get the desired phase margin. Here's the before and after response for a lag compensation: Table 1.4 open loop response uncompensated compensated difference Freq Phase Gain Phase Gain gain phase 1.5 135.0 0.0 135.0 0.0 0.0 0.0 2.5 148.5 13.6 148.5 13.6 0.0 0.0 3.5 170.1 21.0 170.1 21.0 0.0 0.0 5 194.4 28.0 194.4 28.0 0.0 0.0 7.5 118.8 34.0 118.8 34.0 0.0 0.0 10 108.0 36.9 108.0 36.9 0.0 0.0 20 57.6 41.2 57.6 41.2 0.0 0.0 35 37.8 42.6 37.8 42.6 0.0 0.0 50 27.0 42.9 27.0 42.9 0.0 0.0 100 14.4 43.5 14.4 43.5 0.0 0.0 200 7.2 43.8 7.2 43.8 0.0 0.0 350 2.5 44.1 0.0 44.1 0.0 -2.5 500 0.0 44.1 -2.2 44.1 0.0 -2.2 1000 -3.6 44.1 -10.8 44.1 0.0 -7.2 3000 -13.0 44.1 -32.4 43.2 -0.9 -19.4 4000 -17.3 43.8 -41.8 42.3 -1.5 -24.5 5000 -21.6 43.5 -48.6 41.6 -1.9 -27.0 7500 -32.4 42.9 -64.8 39.1 -3.8 -32.4 10000 -43.2 42.6 -79.2 37.5 -5.1 -36.0 15000 -59.4 41.2 -94.5 34.6 -6.6 -35.1 20000 -72.0 39.6 -108.0 32.0 -7.5 -36.0 30000 -97.2 36.9 -129.6 27.6 -9.3 -32.4 40000 -115.2 34.2 -146.9 24.6 -9.5 -31.7 50000 -129.6 32.0 -162.0 21.2 -10.8 -32.4 75000 -172.8 26.0 -194.4 14.0 -12.0 -21.6 100000 -187.2 18.8 -216.0 6.8 -12.0 -28.8 150000 -194.4 13.1 -216.0 -2.5 -15.6 -21.6 200000 -201.6 5.1 -216.0 -12.0 -17.1 -14.4 250000 -216.0 1.9 -234.0 -15.1 -17.1 -18.0 300000 -237.6 0.0 -259.2 -18.4 -18.4 -21.6 Looking at the uncompensated system, the upper -3db point is around 15KHz. The compensated system's upper -3db point is near 5KHz. At first glance this seems quite unproductive! Let's assume you have about 20db of feedback. That means subtract 20 db from the openloop gains. What I have done is simulate a feedback network that is a 10:1 voltage divider, or B (beta) equals 0.1 (- 20db). When the result = 0db, check the phase angle..... If it's close to, or greater than +- 180 deg., the system is unstable. In the uncompensated amp, the result is close to 0db at 75-100 KHz, and the phase is near +-180 deg. This system will oscillate. Doing the same calculation for the compensated amp, the resultant gain (subtract 20 db) is 0 db at 40 KHz, but you are 30-40 degrees away from oscillating (phase margin about 35 deg). Not really great, but much better than having the amp go nuts! The primary reason for using lag compensation is to reduce the loopgain (B * A) at high frequencies, thereby giving a better phase margin (keeps you away from the -180 deg. at B*A=1 or 0db). The "stepping network" is a less extreme version of the lag network, and is used where the phase response is important beyond where the gain has been cut. It drops the phase lag back to zero at frequencies much higher. In many amps, once you get above the point where B*A is 1 (or 0db), it's well out of the frequencies of interest, and phase response isn't much of an issue (you've already set the phase margin for where B*A=1). From my limited experience, the stepping network works well where you have large margin of stability, but when you are squeezing a system hard, the plain lag network works better. AMOUNT of FEEDBACK VS. STABILITY (BODE Method) Here's a table of the same open loop response, but limited to the extreme frequencies, and showing the effect of -10db, -20db, -30db feedback. All I did was add the db's corresponding to the beta's of -10db, -20db, -30db. -10db corresponds to beta=.31, 20db beta=0.1, 30db beta=0.03. The 10db corresponds to a LOT of feedback. Table 1.4A (B*A), or loopgain beta (voltage divider in feedback) none (huge db) -20db -10db -30db compensated compensated compensated compensated Freq Phase Gain Phase Gain Phase Gain Phase Gain 500 -2.2 44.1 -2.2 24.1 -2.2 34.1 -2.2 14.1 .................................................. ................. 20000 -108.0 32.0 -108.0 12.0 -108.0 22.0 -108.0 2.0 30000 -129.6 27.6 -129.6 7.6 -129.6 17.6 -129.6 -2.4 40000 -146.9 24.6 -146.9 4.6 -146.9 14.6 -146.9 -5.4 50000 -162.0 21.2 -162.0 1.2 -162.0 11.2 -162.0 -8.8 75000 -194.4 14.0 -194.4 -2.0 -194.4 4.0 -194.4 -16.0 100000 -216.0 6.8 -216.0 -15.2 -216.0 -3.2 -216.0 -23.2 150000 -216.0 -2.5 -216.0 -22.5 -216.0 -12.5 -216.0 -32.5 200000 -216.0 -12.0 -216.0 -32.0 -216.0 -22.0 -216.0 -42.0 250000 -234.0 -15.1 -234.0 -35.1 -234.0 -25.1 -234.0 -45.1 300000 -259.2 -18.4 -259.2 -38.4 -259.2 -28.4 -259.2 -48.4 By inspecting the above data, you can see that the -10db feedback will most likely oscillate since the phase angle exceeds 180 deg. where B*A is 0db (or 1). The system with -30db feedback won't oscillate and has a phase margin of about 70 degrees. You can see that in this case, the -20db feedback is as close as we can get before the system is unstable. You can apply more aggressive compensation (in the lag network). This will allow you to put more feedback, but there's a point where you roll off B*A in the audio range. That means you aren't getting the benefit of feedback any more. It will be good for low frequencies, but have little effect at higher audio ranges. My guess is that you shouldn't set the lag network to cut in much below 10KHz for audio work. You do get the frequency response back again (and better) with the closed loop response. Here's the closed loop response for the same amp whose open loop response appeared in table 1.1: Table 1.5 closed loop gain gain phase Freq db degrees 1.5 10.0 226.8 2 17.9 172.8 2.5 19.5 135.0 3.5 20.9 100.8 5 22.0 82.8 7.5 23.4 67.5 10 25.1 68.0 20 26.5 30.2 35 27.3 20.2 50 27.5 12.6 100 27.7 7.2 200 27.7 2.9 350 27.7 0.0 500 27.7 0.0 1000 27.7 -3.6 3000 27.7 -8.6 4000 27.7 -11.5 5000 27.7 -14.4 7500 27.5 -21.6 10000 27.3 -28.8 15000 26.9 -43.2 20000 26.0 -57.6 30000 24.5 -86.4 40000 22.7 -103.7 50000 21.3 -122.4 75000 16.4 -183.6 100000 9.1 -201.6 150000 0.0 -216.0 200000 -8.0 -230.4 250000 -10.0 -234.0 300000 -13.5 -237.6 The -3db points are at 9 Hz and 30 KHz. The upper -3db point would have been a lot higher, but I dropped it down to 30KHz to give a better phase margin and improve stability. So far feedback sounds pretty wonderful except for it's ability to cause instability. You can fiddle with an amplifiers phase/gain response to maximize the amount of feedback, but there is a limit if you want to keep the closed loop response to reasonable values. I set the -3db point by applying a compensation in the feedback circuit. the feedback compensation does two useful things: it sets a reasonable overall system bandwidth, and it shifts the phase change the other way by using a "lead network". LEAD NETWORK in FEEDBACK Here's the feedback network with C1 added, which provides lead compensation, and also sets the system bandwidth: | | C1 .-----| |------. | | | | to summing point | | Vfb ------,---/\/\/\/\/--'------- to output | R1 \ / My amp: \ R2 R2=400 ohms (approx.) / R1=10 Kohms \ C1=390 pF | gnd The above values (R1, R2) set the closed loop gain to approximately R1/R2, or about 25. It's not exact, it depends also on the open loop gain (A). R1 and C1 set the -3db point to be approximately 40KHz (f=1/(2*pi*R1*C1). The important aspect of this network is that it gives a "lead" (positive) phase angle, opposite to the normal lagging phase angle you normally get at the upper frequency limit. Because this circuit defines "beta", it will affect B*A (the loopgain). It will shift the phase THE OTHER WAY, and improve the phase response, and margin of stability. In other words, it will make the system less prone to oscillate. Here's the above circuit's amplitude and phase response: TABLE 1.5A Response of feedback network ("beta") freq db phase 1.5 -26.96 0.00 2 -26.96 0.00 2.5 -26.96 0.00 3.5 -26.96 0.00 5 -26.96 0.01 7.5 -26.96 0.01 10 -26.96 0.01 20 -26.96 0.03 35 -26.96 0.05 50 -26.96 0.07 100 -26.96 0.13 200 -26.96 0.27 350 -26.96 0.47 500 -26.96 0.67 1000 -26.95 1.34 3000 -26.93 4.02 4000 -26.92 5.35 5000 -26.89 6.67 7500 -26.81 9.94 10000 -26.70 13.14 15000 -26.41 19.24 20000 -26.02 24.85 30000 -25.09 34.43 40000 -24.04 41.91 50000 -22.99 47.63 75000 -20.57 56.73 100000 -18.56 61.52 150000 -15.46 65.41 200000 -13.18 66.06 250000 -11.42 65.35 300000 -10.00 63.99 Multiplying the "beta" response by the open loop response "A" we get the loopgain (B*A). Multiplication of phasors is product of magnitudes, and sums of angles. Because I used db for magnitudes, I add them. That's the nice feature of db's! Here's the result (B*A) including the lead network in the feedback (beta) circuit: TABLE 1.5B Freq B*A B*A Hz db phase 350 24.3 0.5 500 24.3 -6.5 1000 23.6 -20.3 3000 20.3 -50.0 4000 18.9 -58.0 5000 17.3 -65.3 7500 14.1 -76.5 10000 12.0 -84.1 15000 8.2 -88.8 20000 6.1 -90.4 30000 2.6 -100.6 40000 0.5 -116.5 50000 -1.8 -118.0 75000 -5.9 -159.3 100000 -11.9 -190.5 150000 -16.6 -204.6 200000 -23.7 -149.9 250000 -23.7 -168.6 300000 -25.8 -173.6 Now after using the lead network in the "beta" circuit, when you check B*A for stability, ie., how close to +-180 degrees at BA=1 (or 0db), you can see we have 80 or 90 degrees phase margin. That's probably enough to tolerate a purely capacitive load. FEEDBACK ISSUES Feedback DOES have a related dark side. We expect feedback to reduce the overall gain. It does this in most of the bandwidth, but at the upper and lower (if your amp is not DC coupled) extremes, it INCREASES the overall gain. This happens when you look at the Nyquist diagram, and your response curves get inside the little unity gain circle. This is very common! Seeing as Nyquist diagrams are a pain to produce, check the phase/gain data for the amplifier described above at the frequency extremes, and compare the closed loop (feedback) and open loop (no feedback) responses. The feedback amp has about about 8db more gain at low frequencies, and 2 db more gain than the open loop system! Table 1.6 Amplifier Gain (not B*A) and phase - no feedback vs. feedback open loop closed loop gain phase gain phase Freq db degrees db degrees 1.5 2.62 121.50 10.0 226.8 2 8.19 144.00 17.9 172.8 2.5 14.21 148.50 19.5 135.0 3.5 21.20 163.80 20.9 100.8 5 27.72 180.00 22.0 82.8 ....................................... 150000 -1.1 -270.0 0.0 -216.0 200000 -10.5 -216.0 -8.0 -230.4 250000 -12.3 -234.0 -10.0 -234.0 300000 -15.8 -237.6 -13.5 -237.6 This means (for audio work) that the system is giving excessive amplification for "rumble" or radio frequencies. If you didn't pay attention to this behaviour and filter out those frequencies on the input (prior to the feedback loop), you could get some nasty behaviour. This is also a mechanism whereby RF (radio frequencies) can get into your system from the output (speaker) wiring. In some situations you might need to filter BOTH the inputs and outputs at high frequencies! Normally the system gain at these extremes is pretty low, and should be quite a bit lower than midband gain. You should be aware of this problem if you are exposed to unusual signal conditions such as turntable rumble, radio frequency interference, EMI, crosstalk, clock frequency noise (CD player, or digital circuits), etc. AND your feedback forces a high gain at the extremities. As I mentioned above, there IS a limit to how much feedback you can apply. My guess is that it is difficult to have B*A much in excess of 20-25db for a tube amplifier. Semiconductor amps can allow considerable more, especially with high bandwidth FET's/transistors. I have heard of designs with 250db of feedback! Those would be low frequency extremely high accuracy systems. Feedback systems can be very naughty when the system is nonlinear, and especially if the system has low phase margins. Poorly designed class B amplifiers with too much bias current or badly designed current "crossover" on output devices can change "A" on the fly, and cause problems. Near saturation, some circuits will cause the system to momentarily oscillate as phase/gain characteristics change. I have had some otherwise well designed amplifiers show this kind of trouble. The maximum gain under dynamic conditions would need to be calculated or measured, and the stability margin would be calculated or measured based on the worst-case value for the changing open-loop gain ("A"). Feedback systems want instant feedback..... delays really mess up stability. Normally you don't think of delays other than phase shifts, but some circuits will have "slew rate limiting". That's where the system can only change the voltage at a fixed rate (slewing), no matter how hard it's driven. During this time, the circuits on either side of the "slew rate limiter" are out of touch with each other, and must wait until things settle down. This behaviour can haunt badly designed semiconductor amplifiers. The delay time can be converted to phase at the frequency where stability margin is calculated, and considered as an additional lag. It will reduce the stability margin. The last 2 problems should prompt the designer to back way off on the amount of feedback! -Paul |
#3
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feedback & stability (long)
On Thu, 03 Jul 2008 13:15:11 GMT, "Alex" wrote:
Paul, if you wrote this article yourself, I really admire your technical writing tallent! Thank you.... I try to write these things, and project myself as a reader who would question " what the %$#! is that all about!". Have been an instructor for a number of years helps quite a bit. Perhaps the same transient response observation method can be used on the low frequencies (LF). Apply 1Hz rectangular signal to the input and observe the output. In a good amplifier it shall respond by neat exponential "teeth" (alternating polarity) on each edge of the input signal, with some overshoot. If you see prolonged undulation, like a swell on the sea, it is a problem. Feedback widens LF bandwidth to a few Hz. It is useless, causing nothing but rumble overloading tubes and magnetizing the transformer for no purpose. In this case I usually deepen feedback at LF just below mechanical resonance of a speaker by connecting a series RC (say 47uF + 120ohm) in parallel to the first stage cathode resistor where the feedback is applied. Now while the transformer is rolling off, this circuit is rising loop gain up, compensating phase advance of the transformer. In this configuration, output impedance is maintained low at LF, helping damping speaker resonance better. The dominant null now can be an interstage cap (set to about 30...40Hz cut-off), not the transformer. (There is no need to select an enormous interstage capacitor.) At 3...4Hz open loop gain will gently cross 0dB with ample phase margin. Nice trick.... that's putting a lag compensation in the feedback network (and a small amount of lead in the forward system) to compensate the lead phase shift at low frequency in the open loop system. You describe an exact analogy to the high frequency lead compensation in the feedback network. For simplicity, I would think that the total impedance of the cathode circuit should be small compared to the plate circuit, so there isn't too much "degenerative" effect from the cathode circuit when the feedback network is disconnected from the output terminal. As you say, no one needs low frequency response that low, so it makes sense to roll off the open loop system at what appears to be a rather high LF cutoff frequency. Of course that dominant zero doesn't clobber the overall low frequency response, just like the high frequency stepping networks or lag networks don't directly dictate the system -3db point. The amp I built was a bit peculiar since it had positive feedback applied from the second cathode to the first cathode. When the positive feedback was DC coupled (through a resistor), it caused low frequency motor-boating. Within the local loop, I had the open loop section AC coupled, and the "beta" (feedback) section DC coupled. When I applied the positive feedback through a capacitor and a resistor, between the two cathodes, the instability disappeared, and the overall global low frequency response was much more stable. That still is a puzzle to me. I don't think the BODE diagrams and analysis works with nested feedback loops, at least not in a manner that I understand! I'll try your suggestion .... it's basically a low frequency version of the "stepping network". It will throw off the bias, but that shouldn't be difficult to fix, since I designed the circuit so a good part of the bias is from a "fixed bias" supply (but has a trimmer resistor to set it up). Thanks... -Paul |
#4
Posted to rec.audio.tubes
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feedback & stability (long)
"Paul G." wrote in message ............... The amp I built was a bit peculiar since it had positive feedback applied from the second cathode to the first cathode. When the positive feedback was DC coupled (through a resistor), it caused low frequency motor-boating. Within the local loop, I had the open loop section AC coupled, and the "beta" (feedback) section DC coupled. When I applied the positive feedback through a capacitor and a resistor, between the two cathodes, the instability disappeared, and the overall global low frequency response was much more stable. That still is a puzzle to me. I don't think the BODE diagrams and analysis works with nested feedback loops, at least not in a manner that I understand! Local positive feedback used to be known but not very commmon trick. You apply local positive feedback across the most linear portion of your amplifier. By that you win open loop gain without much sacrificing distortion, because the positively feedbacked section is very linear anyway. Then since you have won open loop gain, applying a general negative feedback will better nail down distortion of the most non-liear (final) stage of your amplifier. A (remote) example of this approach is bootstrappinig of a pentode gain stage load using a cathode follower. (I believe, Patrick calls it a "mu-follower"). Gain rises from 150 to 1500 easily without any significant distortion increase. But now you have 20dB more open loop gain (virtally for free!) for a general NFB to take advantage of. Another use of a positive feedback is reduction of the output impedance of an amplifier. To illustrate this case, imagine you have 2-stage amplifier, the second output stage being a single ended class "A" triode (e.g., EL84 in a triode connection). Imagine you throw a positive feedback from the cathode of EL84 (of course, automatic bias resistor is not bypassed by an electrolytic) via a resistor to the cathode of the first stage. At the same time, a NFB goes from the speaker to the same cathode of the first stage via another resistor. Since the PFB is current feedback, and the NFB is voltage feedback, and in a triode plate current equals cathode current, output impedance will be very low, almost nill for practical purposes. I'll try your suggestion .... it's basically a low frequency version of the "stepping network". It will throw off the bias, but that shouldn't be difficult to fix, since I designed the circuit so a good part of the bias is from a "fixed bias" supply (but has a trimmer resistor to set it up). Yes, R of this LF RC circuit shall be at least 10 times smaller than 1/gm of the first stage tube. If the feedback resistor from the speaker to the cathode is not enough to create a required automatic bias, you can always top it up by an additional resistor paralleled by a large electrolytic -- directly in series with the cathode pin of the valve. Regards, Alex Thanks... -Paul |
#5
Posted to rec.audio.tubes
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feedback & stability (long)
Alex wrote: "Paul G." wrote in message ............... The amp I built was a bit peculiar since it had positive feedback applied from the second cathode to the first cathode. When the positive feedback was DC coupled (through a resistor), it caused low frequency motor-boating. Within the local loop, I had the open loop section AC coupled, and the "beta" (feedback) section DC coupled. When I applied the positive feedback through a capacitor and a resistor, between the two cathodes, the instability disappeared, and the overall global low frequency response was much more stable. That still is a puzzle to me. I don't think the BODE diagrams and analysis works with nested feedback loops, at least not in a manner that I understand! Local positive feedback used to be known but not very commmon trick. You apply local positive feedback across the most linear portion of your amplifier. By that you win open loop gain without much sacrificing distortion, because the positively feedbacked section is very linear anyway. Then since you have won open loop gain, applying a general negative feedback will better nail down distortion of the most non-liear (final) stage of your amplifier. Examples of PFB around an internal section of the input/driver amp are in RDH4. But there isn't a commercially made I know which uses the RDH4 style of PFB. However Quad-II have the first EF86 driving the second which is PFB to double open loop gain. Its a poor implementation though if you analyse it. A (remote) example of this approach is bootstrappinig of a pentode gain stage load using a cathode follower. (I believe, Patrick calls it a "mu-follower"). Gain rises from 150 to 1500 easily without any significant distortion increase. But now you have 20dB more open loop gain (virtally for free!) for a general NFB to take advantage of. The distortion will change its spectra and maybe even reduce when RL goes high for a pentode because the load becomes close to a horizontal load line and the change in Ia is low, therefore gm change is low compared to having a low value load. Without much gm change in each wave cycle the thd is low. And yes, reducing the high OLG from say 1,500 to 10 with loop shunt FB will give lower THD than if you have a R load on the pentode to get OLG = 150. I've never tried any of this and had a listen. Maybe it sounds as well as it should. Another use of a positive feedback is reduction of the output impedance of an amplifier. To illustrate this case, imagine you have 2-stage amplifier, the second output stage being a single ended class "A" triode (e.g., EL84 in a triode connection). Imagine you throw a positive feedback from the cathode of EL84 (of course, automatic bias resistor is not bypassed by an electrolytic) via a resistor to the cathode of the first stage. At the same time, a NFB goes from the speaker to the same cathode of the first stage via another resistor. Since the PFB is current feedback, and the NFB is voltage feedback, and in a triode plate current equals cathode current, output impedance will be very low, almost nill for practical purposes. PCFB was used in Bogan amps for adjustable damping. Its possible to adjust the PCFB to get Rout = 0.0 ohms, and damping factor of infinity. Of even go further, increase PCFB and get a negative output resistance where the output voltage rises as the load becomes lower. Trouble is that distortion increases and BW is reduced with PCFB. And if RL is too low the whole thing oscillates very badly. So PCFB isn't used in any amp I know of today. But its a way to slightly lower Rout with mild NFB, but then the THD is higher than if more NFB was used alone. PCFB was sometimes derived using a small low impedance suited 1:1 transformer where the primary was cathode to cathode of an AB amp, with 10 ohms from each cathode to 0V. The sec with a CT is then connected k to k of the balanced driver amp with the CT taken to a CCS to 0V or a -ve supply. The nonlinear currents in each output tube are summed k to k and you get a sine wave current signal without the distortion at the pri, and the sec then sends this signal to the driver amp. The idea works well and the tranny doesn't need many turns, and can be quite small; a core 20mm x 15 mm with say 0.4mm wire used will do and the P and S windings cab be random wound with two wires together for quasi bifilar, but a layered neat tranny can be wound. BW of the tranny is usually very good, and needs to be. This way the Rout can be lowered to say 1ohm from say 5 ohms in a UL stage, and the 12dB of global NFB does the rest. But its running artifacts around and around and around, and I have never bothered to employ such a scheme in an amp to sell to someone. Patrick Turner. I'll try your suggestion .... it's basically a low frequency version of the "stepping network". It will throw off the bias, but that shouldn't be difficult to fix, since I designed the circuit so a good part of the bias is from a "fixed bias" supply (but has a trimmer resistor to set it up). Yes, R of this LF RC circuit shall be at least 10 times smaller than 1/gm of the first stage tube. If the feedback resistor from the speaker to the cathode is not enough to create a required automatic bias, you can always top it up by an additional resistor paralleled by a large electrolytic -- directly in series with the cathode pin of the valve. Regards, Alex Thanks... -Paul |
#6
Posted to rec.audio.tubes
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feedback & stability (long)
On Fri, 04 Jul 2008 08:24:42 GMT, Patrick Turner
wrote: .......snip.... The OPT might saturate at 20Hz at full power in many good amps. Saturation frequency falls about proportially to reducing voltage output, so that if Fsat was 20Hz for 20Vo, you'd have Fsat at 2Hz for 2Vo. IVo might be the average levels of listening, and since there isn't much music signal below 40Hz, unless its modern crap with atrifically created LF sound. So OPTs don't suffer from satauration with or without NFB or with bandwidth extending down to 2Hz, -3dB, which rarely is the case. This particular amplifier (the one I did all my testing on) is rated at about 15W (clean RMS output). It's single-ended, 2 - EL34's in parallel, and I have gapped the output transformers (.016 in. air gap). Anyhow, here's a table of the CLEAN power available as a function of frequency: Freq. power (Hz) (watts) 2 0.1 3.5 0.32 5 0.5 10 2.5 15 3.5 20 5.6 35 10 50 15 ................ 500 16 They would provide more power at lower frequencies if there were no air gap. On the other hand, the primary inductance wanders all over the place without airgaps. I tested about 8 output transformers (ungapped), 10 mA-DC would multiply the primary inductance (@0 ma) by a factor 0.2 - 0.5 . With a gap, I'd see the multiplication factor of 0.8 to 0.95. For a range of currents from 0 to 100 mA DC, ungapped would be a range of 3:1, gapped 1.3:1 . That translates to cleaner low frequencies. It could be argued that more bass could be had and cleaned up with feedback by using ungapped transformers. That sounds like an argument for another thread.... As you state, no amount of feedback can fix the problem of saturation, except when it starts, and the B-H curves still have a reasonable slope. On the other hand, you can get around some of the effects by driving the transformer from a low impedance. The output devices will need to supply MUCH more current than normal. This would suggest that triodes would be more tolerant of transformer saturation (the magnetizing inductance drops). The design for extra current would not sit well for most people, since the extra current reserve would be more likely used for a more marketable higher power output. A LF pole at 10Hz with 3rd order attenuation below 7Hz is common due to RC couplings plus the Lp shunting RL, so high levels of very llow F don't make it through the amp. Nevertheless, I use a stepped network to shelve the LF open loop response and thus effectively reduce the applied global NFB which might be 15dB at 1kHz to only 3 db at 7 Hz, so the amp never has to perform electronic gymnastics to process LF signals and impossible corections. LF saturation effects are left mainly uncorrected, because they cannot be corrected. Patrick Turner. This sounds like a good practice. There's little sense in trying to maintain a clean response to frequencies that are almost inaudible, extremely rare (except for silly sound), and compromise the midband. I would think the same applies to the high frequencies. High frequency problems would be instability (with feedback), cross-talk, wiring induction (low level circuitry picks up signals from magnetic fields of high current output stage), slew-rate limiting. -Paul |
#7
Posted to rec.audio.tubes
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feedback & stability (long)
"Paul G." wrote: On Fri, 04 Jul 2008 08:24:42 GMT, Patrick Turner wrote: ......snip.... The OPT might saturate at 20Hz at full power in many good amps. Saturation frequency falls about proportially to reducing voltage output, so that if Fsat was 20Hz for 20Vo, you'd have Fsat at 2Hz for 2Vo. IVo might be the average levels of listening, and since there isn't much music signal below 40Hz, unless its modern crap with atrifically created LF sound. So OPTs don't suffer from satauration with or without NFB or with bandwidth extending down to 2Hz, -3dB, which rarely is the case. This particular amplifier (the one I did all my testing on) is rated at about 15W (clean RMS output). It's single-ended, 2 - EL34's in parallel, and I have gapped the output transformers (.016 in. air gap). Anyhow, here's a table of the CLEAN power available as a function of frequency: Freq. power (Hz) (watts) 2 0.1 3.5 0.32 5 0.5 10 2.5 15 3.5 20 5.6 35 10 50 15 ............... 500 16 The cleanliness of of LF ouput as mentioned above could be a lot better and IS a lot better if the OPT and amp is designed just right. The cause of the distortion suddenly exceeding say 3% then rising exponentially is the core saturation, and/or the Lp being too low and in shunt with RL. If the load was say 2.5k, and if you set up with the amp producing 15 watts at 500Hz and then just turn down the F of the sig gen, there will be a point where the distortion suddenly increases, and from your figures above that is at about 40Hz, and you might find XLp (the reactance of the primary inductance) becomes equal to RL at 40Hz, and the load has become partially reactive with a net value of 0.707 x 2.5k = 1.77k. The amp cannot sustain the same high PO as RL value falls. So, if the XL was 2.5k, the Lp = 2,500 / ( 6.28 x 40 ) = 9.9H. To get great bass from SE, the XLp for a lower power amp as you have should be = RL at 15Hz, not 40Hz. At 3.5Hz, your XLp would be 6.28 x 9.9 x 3.5 = 217 ohms, which will shunt the output signal at a low voltage and PO level as you have measured. They would provide more power at lower frequencies if there were no air gap. Not really. if there was no gap, the OPT would saturate with dc so badly that at all F there would be severe distortion at a few watts. The gap lowers the distortion. It also lowers the inductance, and reduces the tendency for saturation to occur. So one should end up with 0.6Tesla of core field strength due to dc, and at 215Hz the Bmax should be 0.6Tesla maximum at full PO of 15 watts. The two causes of magnetization add to 1.2Tesla, and that is all the iron can take. So the trick with SE is to set the gap in the amp and with the wanted Idc and adjust the gap for lowest undistorted F at the 15 watt output voltage level. Unless this is done properly, the OPT won't be properly matched to the tubes. On the other hand, the primary inductance wanders all over the place without airgaps. The permability varies without a gap more than with a gap, but that's with no dc present. A PP tranny has widely varying iron µ with applied voltage and F. But the Lp is large, and the Lp does not shunt the load current much. With dc Lp becomes low with enough dc present to saturate the core with dc magnetisation, so µeffectively becomes low. The gapping makes the Lp rise with dc, but only to a maximum peak with the amount of dc, and then it falls as the gap is increased further. tested about 8 output transformers (ungapped), 10 mA-DC would multiply the primary inductance (@0 ma) by a factor 0.2 - 0.5 . ?? With a gap, I'd see the multiplication factor of 0.8 to 0.95. For a range of currents from 0 to 100 mA DC, ungapped would be a range of 3:1, gapped 1.3:1 . That translates to cleaner low frequencies. It could be argued that more bass could be had and cleaned up with feedback by using ungapped transformers. That sounds like an argument for another thread.... The only way to successfully remove air gaps from an OPT used with SE is to cap couple a PP tranny to the anode and supply the dc to the anode via chokes from the PS, or have a dc flow through the sec from a CCS that magentizes the iron in the opposite direction to the anode current. As you state, no amount of feedback can fix the problem of saturation, except when it starts, and the B-H curves still have a reasonable slope. Hence for good bass, you need to have the best tranny money can buy or your hand can wind. See my website educational pages for more. http://turneraudio.com.au/education+diy.html On the other hand, you can get around some of the effects by driving the transformer from a low impedance. Tubes don't have low enough impedance to prevent saturation from causing severe THD when the Bmax and dc B combine to make the total B 1.2T. At saturation, the impedance of the Lp coil becomes as low as the wire dc resistance for part of the wave cycles. the output devices will need to supply MUCH more current than normal. This would suggest that triodes would be more tolerant of transformer saturation (the magnetizing inductance drops). Triodes put up with Lp and sat better than anything else, but are still affected by sat. The design for extra current would not sit well for most people, since the extra current reserve would be more likely used for a more marketable higher power output. Use a better OPT than the one you have. If it saturates at 40Hz, you need twice the turns, or twice the core Afe. One or the other, but preferably both, and make sure BW is 15Hz to 50kHz, and that windig losses are under 10%. A LF pole at 10Hz with 3rd order attenuation below 7Hz is common due to RC couplings plus the Lp shunting RL, so high levels of very llow F don't make it through the amp. Nevertheless, I use a stepped network to shelve the LF open loop response and thus effectively reduce the applied global NFB which might be 15dB at 1kHz to only 3 db at 7 Hz, so the amp never has to perform electronic gymnastics to process LF signals and impossible corections. LF saturation effects are left mainly uncorrected, because they cannot be corrected. Patrick Turner. This sounds like a good practice. There's little sense in trying to maintain a clean response to frequencies that are almost inaudible, extremely rare (except for silly sound), and compromise the midband. I would think the same applies to the high frequencies. High frequency problems would be instability (with feedback), cross-talk, wiring induction (low level circuitry picks up signals from magnetic fields of high current output stage), slew-rate limiting. Good circuit design eliminates the problems. Patrick Turner. -Paul |
#8
Posted to rec.audio.tubes
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feedback & stability (long)
On Sat, 05 Jul 2008 06:38:06 GMT, Patrick Turner
wrote: ........snip!.............. This particular amplifier (the one I did all my testing on) is rated at about 15W (clean RMS output). It's single-ended, 2 - EL34's in parallel, and I have gapped the output transformers (.016 in. air gap). Anyhow, here's a table of the CLEAN power available as a function of frequency: Freq. power (Hz) (watts) 2 0.1 3.5 0.32 5 0.5 10 2.5 15 3.5 20 5.6 35 10 50 15 ............... 500 16 The cleanliness of of LF ouput as mentioned above could be a lot better and IS a lot better if the OPT and amp is designed just right. The cause of the distortion suddenly exceeding say 3% then rising exponentially is the core saturation, and/or the Lp being too low and in shunt with RL. If the load was say 2.5k, and if you set up with the amp producing 15 watts at 500Hz and then just turn down the F of the sig gen, there will be a point where the distortion suddenly increases, and from your figures above that is at about 40Hz, and you might find XLp (the reactance of the primary inductance) becomes equal to RL at 40Hz, and the load has become partially reactive with a net value of 0.707 x 2.5k = 1.77k. The amp cannot sustain the same high PO as RL value falls. So, if the XL was 2.5k, the Lp = 2,500 / ( 6.28 x 40 ) = 9.9H. To get great bass from SE, the XLp for a lower power amp as you have should be = RL at 15Hz, not 40Hz. At 3.5Hz, your XLp would be 6.28 x 9.9 x 3.5 = 217 ohms, which will shunt the output signal at a low voltage and PO level as you have measured. They would provide more power at lower frequencies if there were no air gap. Not really. if there was no gap, the OPT would saturate with dc so badly that at all F there would be severe distortion at a few watts. The gap lowers the distortion. It also lowers the inductance, and reduces the tendency for saturation to occur. So one should end up with 0.6Tesla of core field strength due to dc, and at 215Hz the Bmax should be 0.6Tesla maximum at full PO of 15 watts. The two causes of magnetization add to 1.2Tesla, and that is all the iron can take. So the trick with SE is to set the gap in the amp and with the wanted Idc and adjust the gap for lowest undistorted F at the 15 watt output voltage level. Unless this is done properly, the OPT won't be properly matched to the tubes. The lightbulb in my head blinked on momentarily, and I thought, just how does the clean power vary with quiescent plate current? It turns out, quite a bit, as Patrick suggests. So I repeated those measurements of clean power vs. frequency and quiescent plate current. Note that these are power levels, not voltage, so -3db point is actually half-power. The currents are per tube (there are two in parallel), so total current is double the shown values. CLEAN power (watts into 5ohms) as a function of freq. & plate current freq 20mA 40mA 50mA 60 mA 80mA 100mA (Hz) 10 0.4 1.4 2 3 3 4 20 1.4 4 7.6 7.6 10 11 30 2.5 5.6 10 12.6 15.6 15.6 50 4 12.6 15.6 15.6 15.6 15.6 500 5.6 18 18 18 18 18 1/2pwr 40Hz 25Hz 22Hz 18Hz 16Hz (approximately) The results are surprising, except for the lower power at a bias of 20 mA, which I expected because the operating point was so far off. I did not expect the lower frequency outputs to be so dependent on bias currents, even though the mid-band power wasn't greatly affected. My only idea, is that the transformer also has a set of characteristic curves, and needs to be biased properly for the correct "operating point". That doesn't seem quite right, as I would have expected the ouputs to be dirtier at the high bias currents because the tranformer would be closer to saturation. Any explanation???? ........snip....... Good circuit design eliminates the problems. Patrick Turner. - right on the button! -Paul |
#9
Posted to rec.audio.tubes
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feedback & stability (long)
On Sun, 06 Jul 2008 01:28:16 -0500, flipper wrote:
.......snip!....... CLEAN power (watts into 5ohms) as a function of freq. & plate current freq 20mA 40mA 50mA 60 mA 80mA 100mA (Hz) 10 0.4 1.4 2 3 3 4 20 1.4 4 7.6 7.6 10 11 30 2.5 5.6 10 12.6 15.6 15.6 50 4 12.6 15.6 15.6 15.6 15.6 500 5.6 18 18 18 18 18 1/2pwr 40Hz 25Hz 22Hz 18Hz 16Hz (approximately) The results are surprising, except for the lower power at a bias of 20 mA, which I expected because the operating point was so far off. I did not expect the lower frequency outputs to be so dependent on bias currents, even though the mid-band power wasn't greatly affected. My only idea, is that the transformer also has a set of characteristic curves, and needs to be biased properly for the correct "operating point". That doesn't seem quite right, as I would have expected the ouputs to be dirtier at the high bias currents because the tranformer would be closer to saturation. Any explanation???? It's for the same reason you got a change at 500Hz going from 20mA to 40mA. For any given Eb and Rl there is a maximum current you can pull across the load before plate goes to (near) zero. Any more idle current than needed for that maximum current draw gets you little more than heat because you can't 'use' the available 'extra' current swing. I.E. I=E/R At some point, though, as frequency falls the parallel inductance begins to significantly affect (lower) the effective impedance so you either 'can' or 'need to', depending on your point of view, pull more current across the lower load (with/for the same Eb swing). .....snip!..... It helps to have a nights sleep before trying to answer a problem..... This is a slightly different approach to the same answer: I think it's the effect of the loadline (load plotted on characteristic plate curves). At the midband, the load is pretty much a straight line. At the frequency extremes it becomes elliptical, and probably almost circular. At midband, the tube current cuts off only when the grid voltage is at its most negative. If you stay away from that voltage, the plate currents stays on. However with an elliptical loadline, you can dip into the cutoff region, even if you are well above midband cutoff current. The more circular the loadline, the more likely you'll hit cutoff. That means at very low frequencies (more circular loadline), the worse this problem is. The solution is to simply bias the output stage higher, thereby lifting the elliptical loadline away from the Ip=0 axis. The above measurements bear this out. The other measurement that helps confirm this, is that the "dirty waveform" consists of a gently rounded notch on the output sinewave, just after (scope: on the right, going up) the most negative part of the sine wave. That would correspond to the bottom part of the elliptical loadline where the anode current may get cut off. That assumes that you move around the ellipse in a clockwise motion... it's probably counterclockwise for capacitive loads. Nobody ever mentioned that in any theory I learned! I suspect the above to be the primary cause of this effect, saturation is only losing 20-40% of the inductance at the plate currents being used. (I have measured inductance vs. DC current for this transformer.) I am always amazed at how much OTHER electronics theory pops up when you are trying to focus on one aspect of a design. -Paul |
#10
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feedback & stability (long)
"Paul G." wrote: On Sat, 05 Jul 2008 06:38:06 GMT, Patrick Turner wrote: .......snip!.............. This particular amplifier (the one I did all my testing on) is rated at about 15W (clean RMS output). It's single-ended, 2 - EL34's in parallel, and I have gapped the output transformers (.016 in. air gap). Anyhow, here's a table of the CLEAN power available as a function of frequency: Freq. power (Hz) (watts) 2 0.1 3.5 0.32 5 0.5 10 2.5 15 3.5 20 5.6 35 10 50 15 ............... 500 16 The cleanliness of of LF ouput as mentioned above could be a lot better and IS a lot better if the OPT and amp is designed just right. The cause of the distortion suddenly exceeding say 3% then rising exponentially is the core saturation, and/or the Lp being too low and in shunt with RL. If the load was say 2.5k, and if you set up with the amp producing 15 watts at 500Hz and then just turn down the F of the sig gen, there will be a point where the distortion suddenly increases, and from your figures above that is at about 40Hz, and you might find XLp (the reactance of the primary inductance) becomes equal to RL at 40Hz, and the load has become partially reactive with a net value of 0.707 x 2.5k = 1.77k. The amp cannot sustain the same high PO as RL value falls. So, if the XL was 2.5k, the Lp = 2,500 / ( 6.28 x 40 ) = 9.9H. To get great bass from SE, the XLp for a lower power amp as you have should be = RL at 15Hz, not 40Hz. At 3.5Hz, your XLp would be 6.28 x 9.9 x 3.5 = 217 ohms, which will shunt the output signal at a low voltage and PO level as you have measured. They would provide more power at lower frequencies if there were no air gap. Not really. if there was no gap, the OPT would saturate with dc so badly that at all F there would be severe distortion at a few watts. The gap lowers the distortion. It also lowers the inductance, and reduces the tendency for saturation to occur. So one should end up with 0.6Tesla of core field strength due to dc, and at 215Hz the Bmax should be 0.6Tesla maximum at full PO of 15 watts. The two causes of magnetization add to 1.2Tesla, and that is all the iron can take. So the trick with SE is to set the gap in the amp and with the wanted Idc and adjust the gap for lowest undistorted F at the 15 watt output voltage level. Unless this is done properly, the OPT won't be properly matched to the tubes. The lightbulb in my head blinked on momentarily, and I thought, just how does the clean power vary with quiescent plate current? It turns out, quite a bit, as Patrick suggests. So I repeated those measurements of clean power vs. frequency and quiescent plate current. Note that these are power levels, not voltage, so -3db point is actually half-power. The currents are per tube (there are two in parallel), so total current is double the shown values. CLEAN power (watts into 5ohms) as a function of freq. & plate current freq 20mA 40mA 50mA 60 mA 80mA 100mA (Hz) 10 0.4 1.4 2 3 3 4 20 1.4 4 7.6 7.6 10 11 30 2.5 5.6 10 12.6 15.6 15.6 50 4 12.6 15.6 15.6 15.6 15.6 500 5.6 18 18 18 18 18 1/2pwr 40Hz 25Hz 22Hz 18Hz 16Hz (approximately) The results are surprising, except for the lower power at a bias of 20 mA, which I expected because the operating point was so far off. I did not expect the lower frequency outputs to be so dependent on bias currents, even though the mid-band power wasn't greatly affected. My only idea, is that the transformer also has a set of characteristic curves, and needs to be biased properly for the correct "operating point". That doesn't seem quite right, as I would have expected the ouputs to be dirtier at the high bias currents because the tranformer would be closer to saturation. Any explanation???? A gapped transformer has a fairly constant amount of inductance because the air gap inceases the magnetic length to much more than the iron length, so permeability is more constant, since air is linear, unlike the iron. The lower the idle current, the less PO you get. And the lower the F, the lower the load, because Lp shunts RL. The maximum class A power into RL is where the idle current falls to zero and doubles within each wave cycle. But with Lp shunting RL, and where XLp is low, then there is considerable change of Ia through Lp, which doesn't make any load power, and also through RL. As F falls so XLp is much lower than RL, nearly all the Ia is through the Lp. This also flows through the tube, and it soon becomes a distorted current flow causing horrendous intermodulation Dn in the load where the power dilevered is low. .......snip....... Good circuit design eliminates the problems. Patrick Turner. - right on the button! Use as much iron as you can afford, and as many primary turns. If you want clean bass, there is no other way with tubes. http://turneraudio.com.au/4+4-set-stereo-2a3.html This 2 channel amp with 2A3 has large Hammond 4.5Kg trannies. Its the best 4 watts on the planet. Patrick Turner. -Paul |
#11
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feedback & stability (long)
"flipper" wrote in message ... This is a slightly different approach to the same answer: I think it's the effect of the loadline (load plotted on characteristic plate curves). At the midband, the load is pretty much a straight line. At the frequency extremes it becomes elliptical, and probably almost circular. At midband, the tube current cuts off only when the grid voltage is at its most negative. If you stay away from that voltage, the plate currents stays on. However with an elliptical loadline, you can dip into the cutoff region, even if you are well above midband cutoff current. The more circular the loadline, the more likely you'll hit cutoff. That means at very low frequencies (more circular loadline), the worse this problem is. The solution is to simply bias the output stage higher, thereby lifting the elliptical loadline away from the Ip=0 axis. The above measurements bear this out. The other measurement that helps confirm this, is that the "dirty waveform" consists of a gently rounded notch on the output sinewave, just after (scope: on the right, going up) the most negative part of the sine wave. That would correspond to the bottom part of the elliptical loadline where the anode current may get cut off. That assumes that you move around the ellipse in a clockwise motion... it's probably counterclockwise for capacitive loads. Nobody ever mentioned that in any theory I learned! There's no need to hypothesize elliptical load lines. The load impedance drops with F. Agree. At low frequencies the load "line" becomes a thin stretched almost vertically ellipse covering the whole current range, but only a narrow portion of voltage range around Va,supply. Alex |
#12
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feedback & stability (long)
On Mon, 07 Jul 2008 12:51:50 GMT, "Alex" wrote:
"flipper" wrote in message .. . This is a slightly different approach to the same answer: I think it's the effect of the loadline (load plotted on characteristic plate curves). At the midband, the load is pretty much a straight line. At the frequency extremes it becomes elliptical, and probably almost circular. At midband, the tube current cuts off only when the grid voltage is at its most negative. If you stay away from that voltage, the plate currents stays on. However with an elliptical loadline, you can dip into the cutoff region, even if you are well above midband cutoff current. The more circular the loadline, the more likely you'll hit cutoff. That means at very low frequencies (more circular loadline), the worse this problem is. The solution is to simply bias the output stage higher, thereby lifting the elliptical loadline away from the Ip=0 axis. The above measurements bear this out. The other measurement that helps confirm this, is that the "dirty waveform" consists of a gently rounded notch on the output sinewave, just after (scope: on the right, going up) the most negative part of the sine wave. That would correspond to the bottom part of the elliptical loadline where the anode current may get cut off. That assumes that you move around the ellipse in a clockwise motion... it's probably counterclockwise for capacitive loads. Nobody ever mentioned that in any theory I learned! There's no need to hypothesize elliptical load lines. The load impedance drops with F. Agree. At low frequencies the load "line" becomes a thin stretched almost vertically ellipse covering the whole current range, but only a narrow portion of voltage range around Va,supply. Alex This is a single-ended triode output circuit. There's no question that the impedance at the plates should drop quite low at low frequencies. The reactance of my transformers (9-10H) will drop to about 1000-1200 ohms at 20 Hz. That is in parallel with the nominal impedance reflected from the load (3000-5000 ohms), and in series with the tube plate impedances (two EL34 "triodes" in parallel) 800-900 ohms. The plate impedances were calculated graphically from the triode curves, at the nominal operating point. Working the math, I get an output at 20Hz that's roughly -3db down, it's phase angle across the output is about +32 degrees. The tubes "see" (at 20Hz) an equivalent load of 1102 ohms, angle 77 degrees, or equivalent series complex: 242 + 1075j ohms. That's not a thin ellipse, it's a real fat one. Plotting this on the plate characteristics, the minor axis of the ellipse is big enough to drive the loadline well below the Ip=0 axis, unless I bias the whole thing upward. The 'scope shows a unique form of distortion that confirms the above. The top half is clean, but the lower part of the cycle has a broad notch cut out of the upward going output, just after the most negative excursion of the sine wave. That would match the area where the almost circular loadline drops below the Ip=0 axis of the plate characteristics. That's not the kind of distortion you see from clipping or saturation. Saturation effects would be expected at the upper part of the waveform, and symmetrical about the vertical axis. This is a single ended amp! The rules change a lot when you go push-pull! I don't think you can discount the effect of the highly reactive loadline. It's not going to be a thin ellipse at these low (or high) frequencies. If what I am proposing is correct, it suggests that an amp (SET) that is optimized for max. power, lowest plate current will give sub-optimal performance at low frequencies. The easy solution is to increase the idling current (I assume class A operation), probably adding about 50% more current. That should give a decent margin for the elliptical/circular loadline. It probably may not be all that practical, since that runs the tubes a LOT hotter. Again, this is a single ended amplifier..... I'm pretty sure push-pull doesn't exhibit the same behaviour. This would be an easy way to improve the bass of a single-ended amp, provided you have enough margin in heat dissipation. -Paul |
#13
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feedback & stability (long)
"Paul G." wrote: On Mon, 07 Jul 2008 12:51:50 GMT, "Alex" wrote: "flipper" wrote in message .. . This is a slightly different approach to the same answer: I think it's the effect of the loadline (load plotted on characteristic plate curves). At the midband, the load is pretty much a straight line. At the frequency extremes it becomes elliptical, and probably almost circular. At midband, the tube current cuts off only when the grid voltage is at its most negative. If you stay away from that voltage, the plate currents stays on. However with an elliptical loadline, you can dip into the cutoff region, even if you are well above midband cutoff current. The more circular the loadline, the more likely you'll hit cutoff. That means at very low frequencies (more circular loadline), the worse this problem is. The solution is to simply bias the output stage higher, thereby lifting the elliptical loadline away from the Ip=0 axis. The above measurements bear this out. The other measurement that helps confirm this, is that the "dirty waveform" consists of a gently rounded notch on the output sinewave, just after (scope: on the right, going up) the most negative part of the sine wave. That would correspond to the bottom part of the elliptical loadline where the anode current may get cut off. That assumes that you move around the ellipse in a clockwise motion... it's probably counterclockwise for capacitive loads. Nobody ever mentioned that in any theory I learned! There's no need to hypothesize elliptical load lines. The load impedance drops with F. Agree. At low frequencies the load "line" becomes a thin stretched almost vertically ellipse covering the whole current range, but only a narrow portion of voltage range around Va,supply. Alex This is a single-ended triode output circuit. There's no question that the impedance at the plates should drop quite low at low frequencies. The reactance of my transformers (9-10H) will drop to about 1000-1200 ohms at 20 Hz. That is in parallel with the nominal impedance reflected from the load (3000-5000 ohms), and in series with the tube plate impedances (two EL34 "triodes" in parallel) 800-900 ohms. The plate impedances were calculated graphically from the triode curves, at the nominal operating point. Working the math, I get an output at 20Hz that's roughly -3db down, it's phase angle across the output is about +32 degrees. The tubes "see" (at 20Hz) an equivalent load of 1102 ohms, angle 77 degrees, or equivalent series complex: 242 + 1075j ohms. That's not a thin ellipse, it's a real fat one. Plotting this on the plate characteristics, the minor axis of the ellipse is big enough to drive the loadline well below the Ip=0 axis, unless I bias the whole thing upward. You have got this exactly right. When there is 45D of phase shift, the elipse is half way between a circle caused by 90D and a straight line with 0D shift. a typical voltage source feeding an R through an L will give a "fat" elipse for the -3dB pole when there is 45D of phase shift, and the ratio of distance across the elipse and the length of the elipse can be used to calculate phase shift. At 45D the XL = RL . The 'scope shows a unique form of distortion that confirms the above. The top half is clean, but the lower part of the cycle has a broad notch cut out of the upward going output, just after the most negative excursion of the sine wave. That would match the area where the almost circular loadline drops below the Ip=0 axis of the plate characteristics. That's not the kind of distortion you see from clipping or saturation. Saturation effects would be expected at the upper part of the waveform, and symmetrical about the vertical axis. This is a single ended amp! The rules change a lot when you go push-pull! Many folks wouldn't know the difference between saturation Dn and inductive load Dn. Both often occur near the same F as F is reduced with power set for clipping levels at 1kHz. To more clearly see when Fsat occurs with triodes, set the level for just under clipping at 1kHz and loaded, and note that level. Then remove the load, and adjust the input voltage so the unloaded VO = the former loaded VO. The VO won't be distorted until the inductance begins to shunt Ra, or saturation occurs. When there is too little Lp, the are too few OPT pri turns or not a big enough core, or the wrong gap. And the OPT will saturate at too a high an F. The gapped core effective permability, µe, should be known by measuring the Lp at 1/2 full VO and then working out the µe. From this the max Bdc, or dc caused magnetic field stength can be calcuated. Bdc should never exceed 0.8Tesla for GOSS cores. The Ac Bmax is worked out using VO and the right formula which does not include µe, and ac Bmax should never exceed 0.3Tesla in any OPT at 50 Hz, a conveneient test FF because mains signals can be used to test. At 20Hz, Bac should never exceed 0.8 Tesla. If its 0.3T at 50, then its 0.6T at 25Hz, and about 0.8T at 20Hz. The dc and ac magnetization add to make a maxima and this should never become more than 1.6T for GOSS, and maybe 1.2T for poorer grade iron. The inductance should ideally have a reactance = Ra at below 20Hz. Gain without a load should then be -3Db at 20Hz. If one wants to have -3dB at 10Hz, Lp must be doubled, or Np turns x 1.41, or twice the irn Afe used. There is a point reached where the weight and size of an OPT rise exponentially to accomodate the extra lower octaves. The iron in an SE amp OPT is ideally under 1/2 fully magnetized by dc. The usual levels of ac for listening will ideally cause *negligible* variations in core magnetization, and thus generate low distortion. I don't think you can discount the effect of the highly reactive loadline. It's not going to be a thin ellipse at these low (or high) frequencies. If what I am proposing is correct, it suggests that an amp (SET) that is optimized for max. power, lowest plate current will give sub-optimal performance at low frequencies. There are additional things to consider. The easy solution is to increase the idling current (I assume class A operation), probably adding about 50% more current. But only if its safe. You should never idle any power tube at more than 2/3 the data rated maximum Pda. EL34 might take 28 watts, but idling at 21 is a lot safer. 300B might take 42 watts briefly, but safer to idle them at 30W or less. Changing the Ea and Ia to suit a load is possible, and if there isn't enough Lp, then Ea has to be reduced and Ia increased so that Ra is as low as possible, RL will be low, and Va will also be low so that the shunting effect of LP and the Fsat occurs at as low an F as possible. That should give a decent margin for the elliptical/circular loadline. It probably may not be all that practical, since that runs the tubes a LOT hotter. Again, this is a single ended amplifier..... I'm pretty sure push-pull doesn't exhibit the same behaviour. PP are badly affected by the same defect of design; having too little Lp. But because PP OPT are not gapped and the µe is maybe 20 times higher, the Lp will also be 20 times greater. Then in PP amps the Fsat becomes the determining factor for the turns and core size required. This would be an easy way to improve the bass of a single-ended amp, provided you have enough margin in heat dissipation. -Paul It all depends....... Amp design is about juggling many things. Patrick Turner. |
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