9. More Compensation

[Music] oh [Applause] [Music] [Applause] [Music] hi last time we had looked at an introduction to compensation of feedback systems in other words how do we modify Loop transmission of a feedback system in order to improve its performance possibly increase its speed of response or increase sensitivity improve the stability of the system those sorts of modifications that we very frequently want to make to to feedback systems the simplest kind of modification we can make is to change the low frequency magnitude of the loop Transmission in the notation we've been using the a fnot product and we've looked at how that influences the root Locust diagram for example and and last time looked at how that might improve the stability of the system today I'd like to look at somewhat more complicated modifications of the loop Transmission in particular ones that change the Dynamics associated with the loop transmission the simplest thing that we can do is to add a dominant Pole to the system suppose we start with A system that has a bod plot as shown here's the AF product as a function of frequency the magnitude and the angle in our usual body plot coordinates and as I've drawn things the system is actually unstable uh this is the unity magnitude point so this would be the crossover frequency for the system the frequency at which the magnitude of the AF product goes through unity and we notice that by that frequency the angle associated with the AF product has gone through minus 180° consequently this is a system that's unstable as shown now suppose I'd like to modify that system and in fact try to increase its phase margin possibly to 45° if I added one more Pole to the system at a sufficiently low frequency what I could do is get the magnitude to roll off and in fact get down to Unity before I acquired enough phase shift to cause instability let's see how we might do that I know that if I add a single pole well below crossover by the time I get out to the crossover frequency the phase shift associated with that pole will be very nearly minus 90° suppose I look at the original uncom uncompensated bod plot and find the frequency where the phase shift of the original uncompensated system is minus 45 degrees I recognize that if I added an additional 90 degrees of negative phase shift I'd end up with the angle at this frequency being minus 135° in other words if I could force crossover to occur at this frequency by adding a single pole to the loop transmission at a sufficiently low frequency I would end up with a system that has 45 degrees of phase margin so let's see if we can do that all we do assuming the physical constraints of the system allow that is to somehow add energy storage to the system and let's suppose as we indicated we'd like to cross over at this frequency and that tells us that we'd like to have a single pole roll off like so in the vicinity of crossover if I've accomplished that by adding a single pole at a sufficiently low frequency why we'd end up with a loop transmission magnitude that looks something like this after compensation out in this region we actually have to follow the original curve uh on top of the single pole roll off so possibly we'd get a a magnitude that looks something like that the influence on the angle associated with the AF product is simply to add an additional 90 de of phase shift 45 of it occurs at the location of the pole and then we ASM totically Trail off to an additional 90° of negative phase shift so the compensated angle might look something like so here we've acquired our additional 90 de of phase shift and we now go and continue parallel to the original angle curve but displaced by minus 90° so our new angle curve might look something like that and now we've in fact accomplished our objective of achieving 45° of phase margin for the compensated system the price we've paid however is a very significant one in terms of the bandwidth of the resultant system originally our crossover frequency was out here we've made a very very major change in crossover frequency we've lowered crossover frequency considerably by putting in our dominant pole starting the roll off at a much lower frequency uh than we originally did we've also lost desensitivity over a very wide range of frequencies we notice of course here's the original magnitude curve we've lowered the AF product at all frequencies Beyond this frequency and so again compared to the original uncompensated system we've lost desensitivity and and the Shaded area is a measure of the amount of desensitivity we've lost as a function of frequency say the disadvantage of that sort of compensation is simply the greatly narrowing effect that it has on the bandwidth of the system but there's one important class of of systems where that sort of of approach is a very very good one to to compensation in particular suppose our objective is to build a regulator that is a system where the intent is to keep an output variable fixed this is in contrast to the usual following kind of feedback system where we try to command an output to follow a Time varying input and under those conditions when we're trying to track a Time varying input the bandwidth of the system is of course very important the closed loop bandwidth however when we're trying to get the output of a system to be time invariant a regulator where we want the output to to remain fixed in for all time adding energy storage to the output creating a dominant pole by basically adding energy storage to the output really improves the stiffness of the output if you will it improves the ability of the output to resist disturbances as an example of that consider what happens if we take an ordinary laboratory power supply and we go searching around and find the largest capacitor we can find and roll that up to the power supply and just clamp it onto the output terminals we may have some trouble starting the supply when we first turn it on it may take a while to charge up the large capacitor that we've added to the output of the system but once we finally get the capacitor charged we're in very good shape the capacitor at the output of the feedback system or the regulator in this case resists disturbances that come for example from changes in load current from changes in primary unregulated voltage and so forth and really helps improve the resistance of the system to those kinds of of disturbances so if our intent is to build a regulator where what we want to do is keep the output of the system time in variant then adding a dominant pole providing we put the energy storage associated with the pole at the output of the system at the variable where we're trying that we're trying to keep constant why we get an improvement in performance I do an example in the book of an electronic regulator and show how that works what I'd like to talk about here as an alternative is a speed control system so let's look at that a little bit here we have a system where we assume that what we're trying to do is keep an output velocity which I indicate as aega response fixed we might command some speed but we assume that this command is is really a fixed quantity or at most incred incredibly slowly time varying so we put in a command and our hope is that eventually the system reaches a speed that's about equal to that command in the system I've shown notice that we have Unity feedback in this system and from then on we're supposed to keep or the intent of the feedback system is to keep the output velocity fixed and the disturbances that we'll consider are torque disturbances that are applied to the mechanical system our system consists of an amplifier driving a motor we measure the velocity of the motor with some sort of a device they're called tachometers which allows us to feed back information concerning the output velocity We compare that with a set point or commanded velocity amplify the difference and of course apply that voltage to the motor we assume that the total inertia of the load this includes the Armature inertia of the motor uh any rotating load that we have we assume that the that total inertia is J and as I say the objective of the system is to keep this velocity fixed as we apply torque disturbances to the system we have to start out modeling the motor and there's a large and important class of Motors that are very well modeled by this simple a model if we have a motor that has a fixed magnetic field either as a consequence of a winding excited by fixed current or as a consequence of a permanent magnet why we're very often able to model that kind of a motor in this way here we have a resistance which is the Armature resistance of the motor and there's a back voltage generator whose voltage is some constant that has to do with the way the motor is constructed times the mechanical shaft velocity of the motor at the same time the torque that the motor supplies is simply equal to the Armature current multiplied by a constant and in fact if we're working in a set of units such as MKS units where the product of velocity times torque gives us power in the same units as the product of current times voltage these two KS have to be the same so uh this is this K and this K are in fact identical providing uh we're working in in MKS units well with that model for the motor let's go ahead and construct a block diagram for our speed control system we have our commanded value for Speed or shaft velocity and we have the response of the system out here and we subtract the two of them to develop an error signal we amplify that error signal and the resultant voltage is that that we apply to the motor but in order to complete our block diagram we have to find out what torque the motor supplies and we notice from our motor model that the torque is proportional to the Armature current well how do we find the Armature current the Armature current is equal to the voltage that we apply to the motor terminals minus the voltage of the back voltage generator that difference is the voltage across this resistor and if we divide that voltage by the value of the resistance we get the Armature current and we can do that on a block diagram basis by taking the voltage out of the amplifier subtracting the back voltage of the motor which we get by multiplying the motor velocity by K so we subtract that back voltage from the voltage supplied by the amplifier we D divide the resultant Difference by R and we get the Armature current once we have the Armature current we can get the the torque that the motor supplying electrically simply by multiplying that Armature current by the constant K so at this point in the block diagram we have the electrical torque being supplied by the motor or the torque that's a consequence of the energy conversion process associated with the motor we add to that the disturbance torque that's applied externally to the system and the sum of these two is the net torque applied to the inertia of associated with both the motor Armature and additional rotating inertia in the system the inertia J as we defined it earlier if we divide that net torque by J we get acceleration of the motor and load combination if we then integrate multiply by 1 / s we get the velocity of the motor load combination once we've gotten to that point in the block diagram of course we have the variable necessary to complete the various feedback paths we multiply by motor velocity by our constant K to get the back voltage associated with the motor similarly we bring back velocity information probably via some sort of a transducer as I mentioned earlier a tachometer to the input of the system well we can now look at the properties of this sort of a feedback system if I have a sufficiently large value for a we have a system that has unity feedback between the out put variable and our input summing point so we have a system uh which has a so-called major Loop that has an F or a feedback transfer function of one and in fact providing a is sufficiently large the gain from the input of the system to the output of the system is about equal to Unity the reciprocal of the feedback path and this tells us that for a large a the response speed should be just about equal to the commanded speed our real interest in this particular example however is how well the system rejects torque disturbances that are applied to the motor load combination so what we really want to do is look at the response the change in speed that occurs for a given torque disturbance we so we want to write the closed loop transfer function aega R over TD if we do that this expression results we find out that the dependence of output velocity on disturbing torque depends of course on various system parameters the multiplying factor out in front R over K * k + a we we can sort of rationalize why that ought to ought to be if we had a larger amplifier gain of course we'd get greater attenuation the system would be less sensitive to applied torque disturbances similarly if R is smaller the motor itself giv is greater current and hence greater or experiences greater Armature current and hence provides greater torque for a given voltage applied to the motor and so that's the reason that R appears as it does a smaller R leading to smaller disturbances similarly if the back voltage coefficient uh is larger the motor sort of intrinsically U tends to minimize disturbances applied to it velocity disturbances so this collection of constants out in front gives us the rejection if you will at at low frequencies this is hopefully a small number and then at some higher frequency we get a single pole associated with this transfer function rep reflecting the energy storage associated with the rotating inertia so our disturbance to controlled variable transfer function has some low frequency magnitude given by this expression and then a single pole the important point to note is that if we increase the inertia in this expression it doesn't change the low frequency rejection at all but what it does is lower the frequency at which the pole comes in and we can look at the effect on this transfer function of changing inertia in bod plot form and I show that over here here we plot the magnitude of the speed over the disturbance in bod plot coordinates as a function of frequency so what we're doing is measuring the sosal steady state rejection of disturbing torqus we're proposing an experiment where we go in and apply a sinusoidally varying torque to the Armature load combination and we look at how well the motor rejects that uh there's a DC value which was given in our earlier expression that's R over K * k + a so that's the low frequency value and then there's a corner frequency associated with the system and it involves a collection of constants but the important point is that that corner frequency is inversely proportional to the total inertia it's inversely proportional to J and so if we add an additional J we build up our system uh we adjust k for example uh in order to control our our DC sensitivity to the level that we'd like to have it um we then can by appropriate choice of J ensure that the system crosses over before higher order poles in the system transfer function become important so we can ensure that we have stability uh possibly as much as 90 degrees of phase margin by forcing Crossover at a sufficiently low frequency through appropriate choice of J and as we make J larger the system performance improves the rejection of the system gets better because an increasing J lowers the frequency of this pole for example we might get this sort of a transfer function when we increase J and so this then in our body plot is the direction of increasing inertia and we see that the rejection of the system remember we'd ideally like to have this transfer function zero we'd ideally like to have no change in speed for a given torque disturbance but we push the first pole down as we increase J and in fact in the limit uh we could increase the band of frequencies over which we got very high attenuation by using a sufficiently large rotating inertia so in this is one important class of systems Regulators where the objective is to reduce or ideally eliminate a disturbance why creating a dominant pole is probably the compensation Form of choice particularly if we can locate the dominant pole at the output if we can put our energy storage that creates the dominant pole at the output of the system our attempts to reduce disturbances are greatly aided by the inertia or in the case of the power supply the electrical power supply the energy storage and the capacitor that we attach to the output well suppose we have a system where we can't tolerate the bandwidth decrease that comes via cre a dominant Pole or that comes along with creating a dominant pole there are other modifications that we can make to the loop Transmission in order to improve the stability of the system and typically uh they retain more more desensitivity than does the option of creating a dominant pole and possibly retain more of the system bandwidth than would occur if we forced a dominant pole and let's look at the kinds of things we might like to do here again I've shown a bod plot for the AF product for some hypothetical feedback system this time I've show showed a stable one uh the magnitude is flat at DC uh possibly a single pole roll off in this region and then some higher order poles or additional Dynamics at higher frequencies an Associated angle might be one that was near 0 degrees at at low frequencies uh here's the 180° point or minus 180° point start out close to zero we get about 90° of phase shift reflecting this first pole and then at higher frequencies again reflecting the energy storage or the other modes of energy storage in the system the angle associated with the AF product begins to drop off eventually goes through minus 180° in this case uh I've assumed that the system is stable in other words the way I've drawn things we go through Unity magnitude before the angle gets to- 180° but I've also tried to show in the drawing that we have a very small amount of phase margin in this case we might not feel that the system has adequate stability let's see what we're able to do to improve that situation the U kinds of modifications we might like to make are the following suppose we could do something to increase the angle associated with a transfer function or with Loop transmission out in this region if we could somehow push up the angle and do that in a way that didn't increase crossover frequency here we've shown the crossover frequency of the system uh if we could do that in a way that didn't increase the crossover frequency so that we didn't get additional negative phase shift associated with the uncompensated system hurting is we could just push up this magnitude this angle but keep the magnitude the same out here that would certainly help if we could make this angle more positive of course we'd increase the phase margin of the system and thereby improve the stability so that's one thing that that we might like to do another thing that we might like to do is somehow push down the magnitude but not at all frequencies we spoke last time about simply lowering the a fnot product which of course pushes down the magnitude at all frequencies the disadvantage to that approach is that it lowers desensitivity at all frequencies suppose we could find a way of modifying the loop transmission so that we push down the magnitude in this vicinity so that we crossed over at a lower frequency without making a significant change in the angle if that were the case crossover frequency would lower we'd still have our original desensitivity at lower frequencies because the crossover frequency lowers we'd end up with a greater amount of phase margin so those those are the two things we might like to do either push up the angle portion of the bod plot in the vicinity of the crossover frequency without making a corresponding change in the crossover frequency or a corresponding change in the magnitude or push down the magnitude curve in the vicinity of crossover to lower crossover without making changes or significant changes at much lower frequencies and without modifying the angle associated with the transfer function well unfortunately our ability to do that limited the implications of physical realizability uh tell us that there's a relationship between the magnitude and the angle of a transfer function and in fact the kinds of things we like to do we really can't do with finite with elements that contain a finite number of of of elements uh consider for example a a transfer function that does uh the first operation one that gives us positive phase shift without corresponding magnitude changes such a transfer function is e to the Plus St but unfortunately e to the plus s to is the transfer function of an element that predicts in other words the impulse response of an an element whose transfer function is e to the Plus St is an Impulse that occurs exactly to seconds before we put in the original impulse and consequently obviously a physically unrealizable Network in fact if we had such a network there's far probably far better things to do with it than waste it on feedback systems we can get tomorrow stock market reports and that sort of thing and not waste it on building feedback systems so those kinds of things we we we're not really completely able to do we don't have the complete Freedom we'd like to have to make the kinds of changes that would give us the greatest Improvement in the stability of our feedback system but we do have some degrees of freedom and in fact the simple transfer functions that work in this direction are th those that include pole zero doublets a single pole and a single zero are the simplest forms of networks that go in this direction or the simplest forms of transfer functions that go in this Direction one of the transfer functions that we're interested in and that we can use to improve the performance of feedback systems is is called a lead transfer function and the output over the input for that sort of system as a function of s uh is one that combines a low frequency zero with a Pole located at a higher frequency in this representation we assume that Alpha is a number larger than one consequently we get a zero in the lead transfer function at a relatively lower frequency than the pole if I normalize the expression by bringing a one over Alpha out in front I get a an expression that can be realized with a simple passive Network and so very frequently a lead type transfer function is written as 1 over alpha alpha to S +1 over to S + one the lead type transfer function as the name name implies gives us positive or leading phase shift actually at all frequencies but with the greatest positive phase shift occur uring between the zero and the pole if we'd like to have something to look at we can mechanize a lead type transfer function with a simple Network shown here two resistors and a single capacitor if we Define 1/ Alpha as being equal to the attenuation ratio R2 over r1+ R2 why the low frequency gain of this network is of course uh simply one over Alpha at low frequencies the capacitor has no influence on the performance of the network at high frequencies the gain becomes one and so we have a transfer function that must have a zero uh that causes the gain to increase from its low frequency value we have a single pole that eventually causes the gain to flatten out out if you simply write the network Expressions you find out that you get this form uh the zero at a relatively lower frequency than the pole for the V out over Vin transfer function where Alpha and the time constant are as defined here so here's a a simple electrical Network that provides an output over input transfer function which has lead type characteristics we can look at both the magnitude and the angle associated with that sort of lead transfer function here's the magnitude uh for several different values of alpha here we have an alpha equals 20 so we start out at low frequencies with a magnitude of 12th 05 in this case we're normaliz to a common pole location for all of the networks the pole location being one unit on the frequency axis uh in the case of alpha equals 20 our zero would occur a factor of 20 below that at 0.005 units and we then get basically a single zero rise in this region then we get the pole associated with the lead type Network similarly if we look at the lead characteristics for Alpha equals 10 why we get this characteristic a tenth at low frequencies a zero location at a tenth of unit on the frequency axis and similarly for Alpha equals 5 so this then is the magnitude associated with a lead Network and we show parametrically the behavior for various values of alpha we can look at the angle at the same time or for the same kind of transfer function and here is the angle associated with the lead type trans ER function we notice that even for a relatively small value of alpha we get a positive phase shift whose maximum value actually occurs at the geometric mean between the zero and the pole location remember that all of our networks were normalized to Unity for the location of the pole so all of our networks have the pole at this frequency at in the case of an alpha equals 5 the zero occurs a factor of Five Below that at this frequency we notice the maximum positive phase shift out of our Network at the geometric mean of those two frequencies and we get somewhere something over 40° of phase shift at the maximum a maximum when Alpha equals 5 uh similarly for Alpha equal 10 Alpha equals 20 we get up to something like 65 degrees of Maximum phase shift again occurring at the geometric mean of the zero and the pole in the case of of an alpha equals 20 here's the Z location for Alpha equals 20 at 05 on our frequency scale again here's the pole location the maximum positive phase shift from our lead Network occurs at the geometric mean of those two frequencies the important thing to that that we exploit in this sort of a transfer function or at least one of the things that we frequently exploit with this sort of a transfer function to improve the performance of a feedback system is that we begin to get positive phase shift before we have make an appreciable change in the magnitude characteristics if we look at our angle curve we notice that there's appreciable positive phase shift out of all three of the characteristics shown at the location of the zero here's the zero location for Alpha equals 20 and we actually have more than 40 degrees of positive phase shift at that frequency here's the zero location for Alpha equals 10 once again we have just about 40° of positive phase shift at that frequency here's the zero location for Alpha equals 5 and we have somewhere on the order of 35° of positive phase shift in that case so we begin to get a a very significant angular contrib contribution remember that we are interested in of times and getting phase margins and systems on the order of 45 degrees and so 45 degrees of positive phase shift can make a very important Improvement in the performance of a system here we have transfer characteristics that give us that sort of phase shift at the location of the zero well let's look back at the magnitude characteristics for a moment and again let's examine the magnitude characteristics at the location of the zero we notice that at the location of the zero there's only about a factor of 1.4 increase in the magnitude here again is the zero location for Alpha equals 20 the magnitude is increased from its low frequency value by about a factor of 1.4 at the zero location similarly here here so the important part about the lead transfer function or at least one of the properties of the lead type transfer function that we're able to exploit is the fact that the phase shift becomes positive by a very significant amount at least in terms of improving the performance of a feedback system before there's much change in the magnitude characteristics and we can use that and the way we use it is as follows here's our magnitude characteristics for our uncompensated system let's locate the zero at least one of the things we can do and of course there are many ways of using these kinds of transfer functions to improve performance but one of the things we might do is locate the zero of the lead Network somewhere in the vicinity of crossover for the uncompensated system we might locate it somewhere out in here maybe like so so our magnitude characteristics would begin to go up in this region what we depend on is the fact that in this region before the magnitude has increased appreciably that would lead us to much higher crossover frequencies in in this vicinity we get appreciable positive phase shift out of our lead type transfer function and so the hope would be that the modification made to the angle is something like so eventually it drops off again but our hope is that we can get enough positive phase shift in the vicinity of crossover notice here crossovers increase just a little bit but the hope is that we've gotten appreciable positive phase shift in the vicinity of crossover so that we improve the phase margin of our system consequently its relative stability so that's one kind of modification we might like to make to the loop transmission of our feedback system a second possibility is to add a transfer function to the loop transmission that's called a lag type transfer function and here's the expression for a a lag transfer function again the output over the input is a combination this time of a pole at a relatively lower frequency than the zero we have this sort of a pole zero diagram for a lag kind of transfer function the pole occurs at a frequency one over Alpha to the zero at a frequency one over to again assuming Alpha is a number larger than one we get a Pole located at a lower frequency than the zero let's look at the characteristics of that sort of a network or first let's look at at how we might Implement that kind of a network and then let's look at its characteristics um here's the kind of a network that will give us a lag type transfer function uh we have again two resistors a single capacitor at sufficiently low frequencies where the capacitor is an open circuit uh the output over the input is simply one whereas at higher frequencies we get an attenuation R2 over R1 + R2 the net result is that we get an input to Output transfer function of the form I I showed on the Blackboard to s+1 over Alpha to s+ one Again by doing a little bit of arithmetic we're able to identify Alpha as being the reciprocal of the attenuation ratio uh Alpha is r1+ R2 over over R2 and to is simply R2 C for this particular Network topology as the name implies a lag Network gives us lagging or negative phase shift at all frequencies and let's look at the phase shift characteristics of a lag Network um for a lead Network we said that we have this sort of characteristic positive phase shift at all frequencies well the lag Network basically gives us characteristics like that it's a little bit confusing in this form we have a little problem with the axes but it emphasizes the Symmetry uh between a lead and a lag Network and is actually the correct form if we simply modify the the axes here is the actual angle characteristic for a lag Network again for several different values of of Alpha we have negative phase shift at at all frequencies of course running from 0 degrees down and so forth uh we have an for Alpha equals 5 we hit a maximum negative phase shift of of again about minus 42 or 43 degrees the same amount as we got in the lead Network only the opposite sign and so forth so this is a completely identic curve except flipped over to that that we got for the lead network uh these are all normalized uh to a common pole location and uh we get the characteristics as we've we've shown them let's look at the corresponding magnitude characteristics and here are the magnitude characteristics uh for the lag net nwor again all normalized in this case to a pole location of one the zero comes out at progressively higher frequencies for increasing Alpha the way we exploit this network or at least one way that we can do it is as follows we try to use the lag Network to push down the magnitude over a range of frequencies in other words we locate the mag the lag type transfer function locate the pole and the zero somewhere below crossover and we recognize because of the attenuation characteristics associated with the lag type transfer function that will push the magnitude curve down but not at DC it doesn't have the same effect as as simply lowering the a fnot product it pushes the magnitude down at frequencies beyond the pole associated with the lag type transfer function so the modification that we might make in the case of lag is to possibly put the pole in here the curve would drop down we then add the zero and we eventually end up with our compensated curve going parallel to the original curve what happens to the angle under those conditions well here's the pole and here's the zero associated with our lag transfer function and we recall from the plot of the angle characteristics of the lag function and the lead function for that matter that the most profound effect on the angle occurs over a frequency range between the two singularities between the pole and the zero associated with the transfer function so here in the case of of a lag Network put in possibly in this range we'd anticipate that we'd put a negative lump in the phase characteristics we do something like this but but the hope is that we locate the lag Network so that the angle has very nearly returned to its uncompensated value before crossover notice that with the lag Network we've lowered crossover back to this frequency and if we can locate the lag Network at a low enough frequency we can ensure that the residual negative phase shift associated with the lag network has become very small and as I've shown in this Construction we end up improving the phase margin of the system from its original uncompensated value to this value by simply pushing down the magnitude curve causing the crossover frequency to lower and without having if we've located things properly so we don't have a great deal of residual negative phase shift from the lag Network we improve the phase margin of the system in that way while we've lowered desensitivity over a range of frequencies here we haven't located lower desensitivity back in this region so we're able to use the lag Network to improve our desensi or maintain our desensitivity at lower frequencies and yet improve the stability of the system and so that's the way we typically use a lag type transfer function to improve the performance of a feedback system one final thing that's useful for design purposes in in these when we use these kinds of networks is the maximum phase shift that we get out of a network or a trans transfer function uh the maximum angle associated with either a lead or a lag transfer function the only difference is the sign of course we notice that for a lead type transfer function we get positive phase shift for a lag type we get Negative phase shift the maximum angle is simply the arc sign of alpha minus one over Alpha + one for either the lead or the lag type transfer function we also recognize that we have to have more degrees of freedom than simply adding these transfer fun functions very frequently we combine a lead or a lag type transfer function with a modification to the a fnot product and that combination is a very very powerful way to improve or to modify the performance of a feedback system next time I'd like to look at how we use these ideas to compensate an actual feedback system we'll do a demonstration where we take an amplifier and with given characteristics and see how we use these kinds of Concepts to improve the performance of the of the feedback system that includes that amplifier thank you