Understanding Closed Loop System Stability

So good morning, So good morning, last class we discussed about frequency response of CS amplifier, last class we discussed about frequency response of CS amplifier. we know how to find the poles of a system by doing exact analysis or by considering the Miller's theorem. We know how to find the bolts of a system by doing exact analysis or by considering the Miller's theorem. So today we will start the discussion of stability of a system, So today we will start the discussion of stability of a system. So whenever we say stability, so whenever we say stability we are talking about the stability of a closed loop system, we are talking about the stability of a closed loop system. we never talks about stability of an open loop system. We never talk about stability of an open loop system. So what is the need of stability and what will happen if the system is not stable and what are the methods to keep the system stable. So what is the need of stability? And what will happen if the system is not stable and what are the methods to keep the system stable. Those things we will be discussing in this topic. Those things we will be discussing in this topic stability and frequency compensation and this is actually a part of unit 3. Stability and frequency compensation and this is actually a part of unit 3. So So I hope you have some idea about feedback amplifiers through the course AMC in third semester. I hope you have some idea about feedback amplifiers through the course AMC in 3rd semester. So with that background I will start. So with that background I will start. So we will look at the need of feedback also. So we look at the need of feedback also. So say you have an amplifier and all the amplifiers what we have discussed in this course can be an example for this amplifier. So say you have an amplifier and all the amplifiers what we have discussed in this course can be an example for this amplifier. So assume that you have an amplifier with a open loop gain of A and if I want to use this amplifier. So assume that you have an amplifier with the open loop gain of k. And if I want to use this amplifier. So what is the basic problem of using an open loop amplifier? What is the basic problem of using an open loop amplifier? Or why do we need feedback systems? Or why do we need feedback systems? Output? Output? Very high? Very high? Very high gain? Very high gain? Yeah, Yeah, if we have a very high gain, if you have a very high gain, yeah, yeah, that is one thing. that is one thing. The other thing is, The other thing is, for example, for example, we look at a case where we need a very small gain. we look at a case where we need a very small gain. I think we have discussed this even in the case of OPAMs. I think we have discussed this even in the case of opamps. So whenever I have a circuit this is your CS amplifier circuit and the gain of the CS amplifier is minus gmRD and now if I am looking for an accurate gain of say 10. So whenever I have a circuit, this is your CS amplifier circuit and the gain of the CS amplifier is minus GMRD. And now if I am looking for an accurate gain of say 10, So I am looking for an accurate gain of 10. so I am looking for an accurate gain of 10, What is the challenge here? what is the challenge here? I know that both this gm and this RDR especially this gm there is a lot of variation. I know that both the GMRD and CS amplifier have the same gain. and this Rd or especially this gm there is a lot of variation the gm variation will be with respect to mu and co same term with respect to temperature as well and if you look at the gain of a cs amplifier you can say that almost 20% variation will The gm variation will be with respect to mu and cox in turn. with respect to temperature as well. And if you look at the gain of a CS amplifier, you can say that almost 20% variation will be there if you design for a gain of say 10. be there if you design for a gain of say 10 so if you are designing for a gain of 10 all these variations can altogether vary your gain around 20% now what is the solution for this So if you are designing for a gain of 10, all these variations can altogether vary your gain around 20%. Now what is the solution for this? The solution is actually The solution is to if I want a gain of set 10 accurate gain the solution is to employ a feedback in this open loop system and make the system a feedback a feedback system or to make the system a closed loop system and then work for this accurate gain of 10. The solution is actually, the solution is to, if I want a gain of set 10, accurate gain, the solution is to employ a feedback in this open loop system and make the system a feedback system or to make the system a closed loop system and then work for this accurate gain of 10. That is the idea of feedback. That is the idea of feedback. So if you look at any general feedback system or this can be considered as a basic feedback system. So if you look at any general feedback system or this can be considered as a basic feedback system. So the main part will be your amplifier which is marked as A where A is the open loop gain of the amplifier. So the main Main part will be your amplifier which is marked as A where A is the open loop gain of the amplifier. Now, Now so so Vin is the input of this amplifier and Vout is the output of this amplifier. Vin is the input of this amplifier and Vout is the output of this amplifier. So what we will do, So what we will do in order to employ a feedback we will sense a portion of this output. in order to employ a feedback, we will sense a portion of this output. How will we sense? How will we sense? through another circuit so sensing again we can send either voltage or current those details we are not looking at here so through a network through some network we call this network as feedback network through this network we will get a portion of the output voltage and this is actually fed back to the input now when i feed back to the input i Through another circuit, so sensing again we can send either voltage or current, those details we are not looking at here. So through a network, through some network we call this network as feedback network. Through this network we will get a portion of this output voltage and this is actually fed back to the input. Now when I feed back to the input, I need some mechanism over here. need some mechanism over here i can call this mechanism as some compare mechanism so or i need some I can call this mechanism as some compare mechanism. So or I need some. addition or some mixing at the input side so when I take a portion of output signal and when I feed it back if I am calling it as Vs now depending on how I am mixing this I will understand Addition or some mixing at the input side. So when I take a portion of output signal and when I feed it back, what I am calling it as Vs. Now depending on how I am mixing this, I will understand whether this is a positive feedback or a negative feedback system. whether this is a positive feedback or a negative feedback system right so if it is the feedback signal if I mark this feedback signal as say some Vf if that is negative. Right. So if it is the feedback signal, if I mark this feedback signal as Vs, then that is... subtracted from your input signal we call this as a negative feedback system and whenever you look at a negative feedback system you can see one sensing mechanism at the output and one comparison mechanism at the input Subtracted from your input signal we call this as a negative feedback system and whenever you look at a negative feedback system you can see one sensing mechanism at the output and one comparison mechanism at the input. This is a disc part from input to output. And this is your feedback network and this path from input to output this path is known as feed forward path or the main path and this is your feedback path. This part is known as feed forward and this is a feed path part. Now why you need feedback? Why you need feed path? The need of feedback is because I have an open loop amplifier whose gain is actually poorly controlled. The need of feed path is because I have an open loop amplifier whose gain is actually poorly controlled. I have a I have a... poorly controlled open loop gain or I have a gain which is varying. So this gives the need of a feedback system. So this gives the need of a feedback system. Now so what is A here? Now, so what is A here? A gain of your amplifier. A gain of your amplifier. Most of the time we call this as open loop gain. We call this as open loop gain. Beta is your feedback factor and what is the closed loop gain here or I can write gain with the feedback. This is the This is the This is the This is the This is the This is the This is the I will mark as A f. What is gain with feedback if this is a negative feedback system. Yeah it is A by 1 plus A beta. So after defining this we will identify this A beta. So after defining this we will. Identify this A beta. What is A beta called? What is this A beta called? A beta is actually the, A beta is actually the feedback factor and A beta is actually the gain of the loop or we call this as loop gain. beta is the feedback factor and A beta is actually the gain of the loop or we call this as loop gain. So now what happened actually when I did this feedback? So now what happened actually when I did this feedback Say we will take one example Assume that the open loop gain is What happened when I did this feedback? Say we'll take one one example assume that the open loop gain is 10 10 and say your beta is 0.1 what is your AF? And say your theta is 0.1 What is the gain? 5. That means what happened when I That means what happened when I applied feedback? Again reduced and if you look at that we can see that by applying this feedback I am actually deliberately reducing gain. and if you look at that we can see that by applying this feedback I am actually deliberately reducing gain. So the gain with the feedback is actually less compared to your open loop gain. So the gain with the feedback is actually less compared to your open loop gain. But we always talk about high gain right. We always talk about high gain, So. So what then tell me what is the advantage of applying a feedback? right? So, then tell me what is the advantage of applying a feedback? Can you remember some advantages of negative feedback? Can you remember some advantages of negative feedback? What are the advantages? This is gain control or I can say that the why we introduce feedback is because you have an amplifier whose gain is poorly controlled and what I am doing is by providing a feedback mechanism to the main amplifier I am actually desensitizing the gain of the amplifier Gain. gain control or I can say that the see why we introduce feedback is because you have an amplifier whose gain is poorly controlled and what I am doing is by providing a feedback mechanism to the main amplifier I am actually desensitizing the gain of the amplifier which is poorly controlled. which is poorly controlled. So I can say that desensitize the gain of the main amplifier So I can say that desensitize the gain of the. The bike can't be controlled by the chain. Why you need to desensitize because that gain is a poorly controlled gain apart from that okay reduces noise any Apart from that, other advantages yeah stability yeah correct yeah there will be a change in input and output impedance so depending on the type of feedback what we apply it has stability. change in input and output impedance so depending on the type of feedback what we apply we have change in input and output impedance and all those changes or whatever advantages we are getting over negative feedback is actually by a factor of 1 plus a theta and we call this factor as amount of feedback We have change in input and output impedance and all those changes or whatever advantages we are getting over negative feedback is actually by a factor of 1 plus a beta and we call this factor as amount of feedback. So negative feedback is advantages in all respects except for reduction in gain where the gain is reduced by 1 plus a beta but again that is an advantage because the whole idea of implementing negative feedback is actually to advantage in all respects except for reduction in gain where the gain is reduced by 1 plus a beta but again that is an advantage because the whole idea of implementing negative feedback is actually to get a gain which can be controlled in a better way. a get a gain which can be controlled in a better way right even though we are getting a lesser gain for the closed loop system we have a more controlled gain getting a lesser gain for the closed loop system we have a more controlled gain yeah one very important thing is bandwidth is also bandwidth Yeah, one very important thing is bandwidth is also bandwidth is also increased by this factor of 1 plus a beta. is bandwidth is 5Ab so from the previous discussion of the CS amplifier so what I can say here is if your AF is A by 1 plus A beta So from the previous discussion of the CS amplifier, so what I can say here is if your AF is a by 1 plus a beta and if your a which is the open loop gain. And if your A which is the open loop gain, Even if it is a very less value, even if it is a very less value, even if it is a poorly controlled gain, even if it is a poorly controlled gain, depending on the value of depending on the value of A, A, I can approximate. I can approximate. If your A beta is much much higher than 1, If your A beta is much much higher than 1, your closed loop gain can be approximated as your closed loop gain can be approximated as. 1 by beta. Or I can say the closed loop system purely depends on the feedback factor and if I am establishing this feedback factor using ratio of resistors. Or I can say the closed loop system purely depends on the feedback factor and if I am establishing this feedback factor using ratio of resistors. So one small example we can consider here is. So one small example we can consider here is. So, same example we have considered before also. So, this is your V in and this is your V out. If your A is infinity, A is infinity, your A is very large, if your A is very large, your A beta is also very much higher compared to 1 and what will be your close loop gain? your A beta is also very much higher compared to 1. And what will be your close loop gain? 1 plus R1 by R2 or here the gain purely depends on the ratio of resistors. Yeah, 1. Here the gain purely depends on the ratio of resistors. So, So even though we have a very high gain for this amplifier, even though we have a very high, we have gain for this amplifier and this gain will be definitely in terms of gm into r0 parallel r0 where I have chances of variation of this gm right is that fine. and this gain will be gm into r0 parallel r0. Chances of variation of the sphere Right The will So we will just see the same concept so write this question for Let's see the same concept So write this question So write this question So write this question So write this question So write this question So write this question a given amplifier with open loop gain A as 200 Load the closed loop response for feedback factor 0, Close Loop Response Kill 0.1 and 0.5. Pole Amplifier Assume that this is a single pole amplifier having a pole at omega peak at some frequency. So what is closed loop response? A f will be a by 1 plus a beta. So what is this beta? is a feedback factor. So what is meant by beta equal to 0? So what is beta equal to zero? No feedback. No feedback. That means the loop is open. That means the loop is open. And what is the range of this beta value? And what is the range of this beta value? What can be the range of beta value? What can be the range of beta value? Beta will be from zero to one, Beta will be from 0 to 1, right? So when you say beta is 0, right? So when you say beta is zero, the loop is open. the loop is open. And when you say beta is one, And when you say beta is 1, the entire thing is fed back or it is unity gain feedback. the entire thing is fed back or it is unity gain feedback loop, right? So beta, So beta So the range of beta is from the range of beta. So, when a beta is from 0 to 1, 0 to 1. Now can you draw the closed loop response. now can you draw the closed loop response? So, So when a beta is 0 that is basically an open loop amplifier. when a beta is 0, that is basically an open loop amplifier which is having a gain of 200 and say this is your omega 3. Which is having a gain of 200. And say this is your omega p. So this is when your beta is equal to 0. This is when your pizza is equal to zero. Now what is this? Now what is, yeah. What is the new AF when your beta is equal to 0.1? What is the new here? Correct, yeah. So your new AF is A by 1 plus A beta and that is reduced to a value 9.5. Yeah, new, uh... Let's see... What is the... So what is the feedback factor in this case? What is the... Yeah, around 21. So your gain actually reduced. So gain is drastically reduced. Gain actually reduced, so gain is drastically reduced So to another value 9.51 So to another value, say, this is 9.5. When your beta is equal to 0.1, what happens to bandwidth? So what happened to gain? So actually your gain is 200 that is reduced by a factor of 1 plus a beta. So actually your gain is 200 that is reduced by a factor of 1 plus a beta. At the same time your bandwidth which is omega p is increased by a factor of At the same time your bandwidth which is omega p is increased by a factor of plus a beta. 1 plus a beta. So it is like proportionate increase in that. So it is like proportionate increase in that. So your cutoff frequency or the bandwidth will not be at omega p that will be pushed towards right. So your cutoff frequency of the bandwidth will not be at omega p that will be pushed towards right. And actually it will from there it will actually drop down. there it will actually drop down. Now what will happen when your beta is equal to 0.5? Now what will happen when your beta is equal to 0.5 yeah your new yeah so when your for your new gain that is when Yeah, your new... So, when your... for your new gain, that is... When your beta is 0.5, your beta is 0.5 what is your closed loop gain 1.98 that means actually your what is your close to gain? N8. That means actually your gain still reduced to a very low value. Gain still reduced to a very low value. So this is almost 1.98. So this is almost 1.98. Your bandwidth also increased. Your bandwidth also increased. So you can see that as your gain reduces your bandwidth is actually shifting towards right. See that gain reduces, your bandwidth is actually shifting towards the high end. Or in your closed loop system your gain bandwidth product will actually remain constant. product will actually remain constant. So in your. This is your gain here with a bandit of this for the second when the gain reduced actually a bandit further increase. So this is your gain here with a banded of this. For the second when the gain reduces actually a banded will further increase. So that gain bandit product is likely to be equal. So that gain banded product is likely to be equal. Because there is only one pole in this case. There is only one pole in this case. Right. And decrease in gain and increase in bandwidth is actually proportional. And decrease in gain and increase in bandit is actually proportional. So if you plot in semi-log, So if you plot in semi-log you will also come to know that it is actually taking the same slope. you will also come to know that it is actually taking the same slope at almost the same point. Almost same point. Omega, which omega is unity gain frequency? Each omega is unity gain frequency. Omega, how omega, Omega, how omega, what is this unity gain frequency? what is this unity gain frequency? No, No, unity gain frequency is a frequency at which gain equals 1 or 0 dB. unity gain frequency is the frequency at which gain equals 1. or 0 dB that is unity gain frequency right so in this case if i am assuming this as my 0 dB line unity gain frequency actually will come at a very high volume so that basically determines the speed so if your unity gain frequency is very less you will not be able to use your transistor more That is unity gain frequency, right? So in this case, if I am assuming this as my 0 dB line. unit gain frequency actually will come at a very high value. So that basically determines the speed. So if your unit gain frequency is very less you will not be able to use your transistor more than that frequency. than that frequency So here the reduction in gain and an increase in bandwidth. Action in B And then we go back Okay, So so we have seen a negative feedback system, We have seen a negative feedback system. We discussed about the advantages of negative feedback system. we discussed about the advantages of negative feedback system. Now what is the problem in a negative feedback system? Now what is the problem in a negative feedback system? Or why we are actually discussing stability all of a sudden? Or why we are actually discussing stability all of a sudden? Or how will you generally study stability? Generally study stability. How will you say a circuit is stable or not? How will you say a circuit is stable or not? Small change in? Small change in... Yeah, Yeah, for a bounded input, for a bounded input, output should be bounded, output should be bounded. Correct. correct. So, So, whether those problems will happen in feedback system? whether those problems will happen in feedback system? Yeah, Yeah, so we are actually discussing the stability in negative feedback system. so we are actually discussing stability. negative feedback system you can state that if not designed properly if your feedback system is not designed properly the amplifiers can actually be disengaged So we can state If not designed properly, if a feedback system is not designed properly, the amplifiers can actually oscillate, amplifiers can actually oscillate. So that is the problem what we are discussing here. Amplifiers can actually oscillate. So in order to discuss that, So that is the problem what we are discussing here. I will take this loop gain as a by 1 plus a beta. So in order to discuss that, I will take this loop gain as a by 1 plus a theta. So why we are discussing stability is amplifiers with a proper negative feedback will actually give a lot of advantages. So while we are discussing stability is, amplifiers with a proper negative feedback will actually give a lot of advantages. But the problem is the amplifier tend to oscillate, But the problem is, if the amplifier tend to oscillate, then it is a problem, then it is a problem. right. Right. So, So... No, no, no, no, no. no, yeah, Yeah, we will come to that. we'll come to that. How it is oscillating, How it is oscillating, I will tell you. I'll tell you. So, So, if I'm... if I am, so what is the basic idea of an oscillator? So, what is the basic idea of an oscillator? Without any input, Without any input, the system is actually providing, the system is actually providing... So that, so that such situations can come even in the case of an amplifier. such situations can come even in the case of an amplifier. So that is the point when we say that an amplifier oscillates. So, that is the point when we say that an amplifier oscillates. We will discuss that. We'll discuss that. Now, Now, before discussing that, before discussing that. Now we have written AF as A by 1 plus A beta and we know that A is actually the open loop gain but from the previous discussion of the frequency response of CS amplifier we understood that this A is not a frequency independent term rather it is a frequency dependent term. now we have written AF as A by 1 plus A beta, and we... know that A is actually the open loop gain. But from the previous discussion of common frequency response of C-S amplifier, we understood that this A is not a frequency independent term, rather it is a frequency dependent term. So can I write A as A of S because here gain of an amplifier is actually a function of frequency that is why when you plot the frequency response you are even when you plot the open loop response. So can I write A as A? because here gain of an amplifier is actually a function of frequency that is why when you plot the frequency response here even when you plot the open loop response gain is actually varying with respect to frequency so i have written it as a function of frequency or gain is actually varying with respect to frequency. So I have written it as a function of frequency or I can write this as a function of j omega. i can write this as a function of k of s now what about this beta Now what about this beta? So at least in this case I am considering beta as a purely resistive network. So, at least in this case I am considering beta as a purely resistive network, So that is not providing any phase. so that is not providing any peace. But since here open loop gain is a function of frequency, But since here open loop gain is a function of frequency the term a of j omega into beta which we call this as loop gain. the term A of j omega into beta, which we call this as loop gain, is actually a function of frequency. is actually a function of frequency ok now So what is the problem of instability here? So I can write the combined the closed loop system is also a function of frequency and that is a of s divided by I can write the sum a of n. 1 plus a of s into beta. So can you tell me one possible condition for instability for this negative feedback system frequency. Can you tell me one possible condition for instability for this negative feedback system? Frequency? Yeah, Yeah, frequency variation is there at how it becomes a unstable condition, frequency. variation is there and how it becomes an unstable condition. What leads to an instability in this negative feedback system? what lead to an instability in this negative feedback system? Fouls lying on, Bones running on the... yeah. yeah, that is an S plane, that is an s plane we will come to that okay if a beta is minus one one minus one becomes zero so what is the what is the implication from that we will come to that. Okay, if A beta is minus 1, 1 minus 1 becomes 0. So what is the implication from that? Or can we say without input you will get some oscillations here. Or can we say without input we will get some oscillation? So here the problem is if at a certain frequency and I am calling that frequency as, So here the problem is if at a certain frequency and I am calling the frequency as this is very important, so this is very important, this is actually the key for studying the stability. this is actually the key for studying the stability so at a certain frequency say omega 1 if So at a certain frequency say omega 1, if a of j omega 1 A of j omega 1 into beta into beta into beta, if this is equal to minus 1, This is equal to minus right, 1. A of j omega 1 into b times, a of j omega 1 into beta, if this is equal to minus 1, this is equal to minus 1. What will happen to your first loop gain? what will happen to your closed loop gain? Your closed loop gain actually becomes infinite. Your first loop gain actually becomes infinite. Or I can better write this as if the magnitude of loop gain, Or I can better write this as, it would have to be, so Since your open loop gain is a function of frequency, Since your open loop gain is a function of frequency, your loop gain itself is a function of frequency and it is actually a complex term as well. your loop gain itself is a function of frequency and it is actually a complex term as well. So, So if it is a complex term, if it is a complex term, can I write it as magnitude and phase? can I write it as magnitude and phase? Right. Right. So, So I will write this as magnitude of A beta equal to 1 and angle of A beta. I will write this as magnitude of A beta equal to 1 and angle of A beta equal to minus 180 Minus one. So this is one instance where your negative feedback system which is a stable system will become unstable. Instance where your negative feedback system, which is a stable system, will become unstable. So when the magnitude of loop gain equal to 1 and at that particular frequency if the phase or the loop gain phase is equal to minus 180 degree your a beta term becomes minus 1 and that leads to an So when the magnitude of looping equal to one, and at that particular frequency, if the phase or the loop. to minus 180 degree your a theta term becomes minus 1 and that leads to an infinite gain of the system or you can actually see this in this way you have a system i will express it as y of s by x of s infinite gain of the system or you can actually see this in this way. So you have a system, I will express it as y of s by x of s which is equal to this a of j omega divided by 1 plus a of j omega into beta and that is a of j omega divided by which is equal to omega of z. 0. So that gives an understanding that without input, so your input is 0, This is the road to Torrington without input you have some output. That means what the system is oscillating when I say the system is oscillating the system is not stable. Does it make sense this condition? Right now so how can we ensure stability? Hello So what we have done we have considered a negative feedback system after discussing all the advantages of a negative feedback system we looked at the closed loop gain and we understood that in this closed loop gain this term a beta which is called as loop gain if Close loop gain, this term, a beta, which is called as loop gain, if the magnitude of loop gain is equal to 1 at a particular frequency, the magnitude of loop gain is equal to one at a particular frequency and at that frequency you and at that frequency, if the phase provided by the loop gain is minus 180, If the phase provided by the loop gain is minus 180, the system will actually oscillate. the system will actually oscillate. So that is a potential problem. So that is a potential problem. And now the question is how to ensure that the system will be stable. And now the question is how to ensure that the system will be stable. So what can be done for that? So what can be done for that? What can be done? So what is the problem here? Yeah, Yeah, the magnitude of loop gain is 1 and angle of loop gain is actually equal to minus 180. the mic is on me. So this is actually causing instability. So to avoid instability what I should do? So to avoid instability, what I should do? Make sure that these two conditions should not happen together. Make sure that these two conditions should not happen together, Right. right? So when the magnitude is 1 and when the face is minus 180, So when the magnitude is 1 and when the phase is minus 180 that leads to oscillation. that leads to oscillation. So to avoid instability, So to avoid instability, So we must ensure that these two conditions do not occur. so we must ensure that These two conditions These two conditions will definitely occur What we are trying to do is These two conditions should not occur at the same frequency These two conditions will definitely occur. What we are trying to do is these two conditions should not occur at same frequency. Okay, so we will continue the discussion. Maybe we will review certain concepts what you have already learned in your control systems. Maybe we will review certain concepts what you have already learned in your control systems. So, what are the tools or what are the methods to analyze stability of any system? So what are the methods to analyze stability of any system? Yeah. So what are the methods to analyze stability? So, what are the methods to analyze stability? Now what are the methods? What are the methods? What are the techniques? What are the techniques? Root lockers. Root lockers. Root lockers. Ok, root lockers is one. Root lockers is one. What is root lockers? What is root lockers? Determining the roots of a system and plotting it on a S plane. Determining the roots of a system and plotting it on a S plane. Ok, Okay. now then apart from that? Go to plot, how will you analyze stability there? How will you analyze? stability there? You have some idea right, you have some idea right? ok fine so the one method to analyze stability is actually root locus or by plotting the bolts of a system on an s plane and by looking at the location of bolts we can say whether the system is stable or not so okay fine. So the one method to analyze stability is actually root locus or by plotting the poles of a system on an S plane and by looking at the location of poles we can say whether the system is stable or not. So for any system. So for any system, To be stable where the poles should lie on S plane? stable should lie on S plane. Left half of S plane. So there you go. Oh very good. Poles should lie on left half of S plane. Should lie on left half of S plane. So if you plot that. your poles can come on LHS it can be poles on your this axis or it can be multiple like imaginary as well. LHs on your can be multiple flex imaginary as well. So when you have poles on LHS of S plane the system will be stable now what about marginally stable system? So when you have bolts on LHs of explained the system which is stable. Now what about marginally stable system? The poles lie on j omega axis and for an unstable system poles lie on right half of s plane. So if I plot these things this will be for a. marginally stable system and if poles are on RHS of S plane, their system is unstable. Similarly, how will be the time domain behavior? In the main behavior, The system is oscillating, if the system is having bolts on imaginary axis, if the system is having poles on imaginary axis, this is called oscillatory behavior. this is what your oscillatory behavior. If it is having bolts on the RHS, If it is having poles on the RHS how it will be? how it will be? Increasing. Yeah, actually it will go, and for your unstable system there will not be any oscillations it will actually come to it will stabilize it will take some time. it will keep on increasing and exponentially increase and for your unstable system there will not be any oscillations it will actually come to it will stabilize, it will take some time settling time will be there for any system. So this is one way if you know the roots of a system by plotting root blockers on x plane we can say that the system is stable. So. This is one way if you know the roots of a system by plotting root locus on a plane we can say that the system is safe. Now in that case can you consider a single pole amplifier and a single pole amplifier and a single pole amplifier and a single pole Now in that case can you consider a single pole amplifier and consider a single pole amplifier and try to plot the roots of open loop system as well as closed loop system. Try to plot the roots of open loop system as well as closed loop system and check whether the system is stable or not. Check whether the system is stable or not. So you can take the open loop transfer function as a naught divided by Again, take the question as A0 divided by 1 plus s by omega. S0. So this is your open loop. So this is your open loop transfer function or open loop gain. transfer function or open loop gain. So draw root locus in that plot the poles of the open loop system as well as closed loop system. So draw root locus in that plot the goals of the open loop system as well as closed loop system. So what is the pole location here? Code location here. For the open loop system where is the pole? For the open loop system, Open loop system where is the pole? pole is actually at minus omega naught. Open loop system, So this location is when your beta is actually equal to 0. this location is when your beta is actually equal to 0, there is no feedback. There is no feedback. So for your open loop system, For your open loop system, the pole is at minus omega now. the pole is at minus omega naught. Now what about closed loop system, Now what about closed loop system? how to write closed loop transfer function? How to write closed loop transfer function? Yeah, A of s is actually equal to A of s divided by 1 plus A of s into beta so that can be written as A naught by 1 plus s by omega naught. divided by 1 plus a naught by 1 plus s by omega naught into beta. One plus If you rearrange this yeah you can write this as you can rearrange in this way so that you can see here we are C here Open loop gain was a naught initially. In the closed loop system, that gain is actually reduced by 1 plus a naught beta. So, we have a new location. And open loop pole was at omega naught or the bandwidth was omega naught. The new location is somewhere here. By applying feedback, That is at minus omega naught into bandwidth is actually increased by 1 plus a beta. So what is the new location? Yeah, new location is somewhere here. That is at minus omega naught into 1 plus a beta. 1 plus a beta. Right. Right. So, So as beta increased, as beta increased... from 0 to 1, 0 to 1, here location of pole is actually moving from minus omega naught to minus omega naught to 1 plus a b into r. here location of pole is actually moving from minus omega naught to minus omega naught into 1 plus a theta. So, So by this, by this, what we can say, what we can say for any system to be stable, for any system to be stable, poles will be on left half of s plane. poles will be on left half of s plane. So, So here poles stay on left half of s plane even after feedback when the system is stable. here poles stay on left half of s plane even after feedback. So, the system is stable. Or the other way we can say the instability condition will not come in this case. Or the other way we can say, the instability condition will not come in this case. Why? Why? We have only one pole. We have only one pole. if you have a single pole, If we have a single pole, what is the maximum phase shift possible? what is the maximum facial possible? What is the maximum phase shift? What is the maximum patient? So what is the face? So what is the phase? Can you write the phase for this open loop system? Can you write the face for this open loop system? What is the numerator phase? What is the numerator face? What is the denominator phase? What is the denominator face? This is j omega, S is j omega right? right? When you have a plus i b how to write the face? When you have a plus IP how to write the face? Plan inverse of b by. Tan inverse of d by so what is space here numerator is 0 minus denominator is tan inverse of omega by omega naught that is a space right. So what is face here? Generator is 0 minus denominator is plan inverse of omega by omega. So omega is actually, So omega is actually omega varies here right. omega varies here. So by looking at the location of force we get the resistance is stable. So by looking at the location of poles we can say the system is stable. Similar thing you can do with drawing the Bode plots. Similar thing you can do with drawing the Bode plots. So can you plot magnitude and phase for this? So can you plot magnitude and phase for this. Body plot is actually drawn for your loop gain, Body Plot is actually a brown correa loop gain right? right? A beta. A beta So plot the magnitude and this is angle. So how will you plot? So at 0 frequency you have The pure frequency you have A beta or you can express this as 20 log of......will be there. A beta or you can express this as 20 log of A naught into beta that will be there and what happens when it encounters the first pole as per as per the body rules yeah so if you have pole at omega naught the And what happens when it encounters the first pole as per the body rules? Yeah, so if you have pole at the omega naught, the gain will reduce... gain will reduce by minus 20 dB per decade. Now how to draw the face for this? Let me draw the face for this. The face expression is minus tan inverse omega by omega naught. The face expression is minus tan inverse omega by omega naught. So as your omega varies from 0 to infinity face also will actually increase from 0 to a value right. So as your omega varies from 0 to infinity, face also will actually increase from 0 to a value. So initially the face is So initially the face is 0 when you what is the face when your omega equal to omega naught what is tan inverse 1 yeah. 0. What is the face when your omega is equal to omega naught? What is tan inverse 1? So this is actually a minus 45 degree So this is actually a minus 45 degree. And what is the maximum phase at omega equal to infinity minus 95 by 2 right. So, your maximum phase at omega equal to infinity minus 95 by 2, right? So, your maximum phase is actually asymptotically if you touch minus 90 degree. So your maximum phase is actually asymptotically it will touch minus 90 degree. So, we define... So we defined the problem of instability before what is the problem? problem of instability before what is the problem yeah in the loop gain when the magnitude of loop gain is equal to 1 if your face is minus 180 degree you have a potential instability problem so whether we have that problem Yeah in the loop gain when the magnitude of loop gain is equal to 1 if your phase is minus 90 degree you have a. potential instability problem. So whether we have that problem now, so this is the point where your magnitude of Lucan equal to one and at that particular frequency your phase will never touch minus 180. So this is the point where your magnitude of Lucan equal to 1 and at that particular frequency your face will never touch minus 190. That means there is no problem or I can say a single fold system is actually inherently stable. That means there is no problem or I can say a single pole system is actually inherently stable, It is not conditionally stable, it is not conditionally stable, it is inherently stable. it is inherently stable. So we have analyzed stability by looking at the pole locations as well as bode diagrams, So we have analyzed stability by Lucan. We are looking at the pole locations as well as the border diagrams, base as well as target. phase as well as magnitude. Any doubts? Any doubts? So similarly if you analyze a single two pole system Similarly, if we analyze a single pole, a two pole system, So I can take a transfer function as A of s is equal to the open loop gain since there are two poles I am marking it as omega p1 and one as omega p2. The structure action. We are not increasing the capacitance all the capacitance are internal capacitance so that depends on the structure actually. Then your C-S amplifier you have seen. In this sample you have seen maximum pole is 2 with the one zero. maximum pole is 2 with 1 0 right yeah that is why we analyze that first so this is a two-pole system now the task is check whether this system is stable by root locus as well as border diagrams you That is why we analyze that first. So this is a two pole system. Now the task is check whether this system is stable by root lockers as well as border diagrams. We have to solve that. have to solve that so if you look at the open loop response So if you look at the open loop response. So this is your j omega sigma. j omega sigma so the four locations are one net minus omega p1 another is omega p2 this is the four location and here beta is zero So the four locations are one at minus omega p1 another at minus omega p4. So this is the four location when your beta is 0. So what will happen when your beta increases from 0 to 1? What will happen when your beta increases from 0 to 1? So you have to write actually the closed root transfer function. This is actually the closed loop transfer function. So you will get a second order polynomial. So you will get a second order polynomial. You have to solve that. You have to solve that. You will get two roots for that. You will get two roots for that. And in that root there will be a beta dependency. And in that root there will be a beta dependency. And if you vary beta from 0 to 1 you will get that. And if you vary beta from 0 to 1 you will get that. So what will happen? So what will happen? Can you tell me what is the root locus rules? Can you tell me what is the root locus rules? When you have two poles. When you have two poles. When beta increases actually this will move towards left and this will move towards right. When beta increases actually this will move towards left and this will move towards right. Both meet at one point and from there actually it will split. Both meet at one point and from there actually it will split. So this will be the locus of the roots. So this will be the locus of the roots. So this we are just refreshing we are not going to do something with this. We are just refreshing, we are not going to do something with this. We are checking whether the root stays on the left half of S, We are checking whether the roots stays on the left half of S plane that is all. like that side. So these two roots will meet and split and it will actually move in this way. So these two roots will meet and split and it will actually move in this way. So whether the system is stable? So whether the system is stable? Yes. Yes. Can you do the same thing with the Bode plot? Can you do the same thing with the Bode plot? Draw the magnitude and phase and check whether the system is stable or not. Draw the magnitude and base and check whether the system is stable or not. Starts at 20 log of K0 in This is actually magnitude of A beta. This is angle of A beta. Now this starts at 20 log of a naught into beta. Bogota There is one pole which is at omega p1. There your gain drops by 20 dB per decade. Then you have second pole which is at omega p2. There again your gain falls by minus 20 dB. So that means this is minus 40 dB per decade. What the hell? And at some point, your gain will be equal to zero dp or one. Now what about phase how to write phase for this numerator phase is zero minus denominator you have two poles minus tan inverse omega by omega p one minus tan inverse omega by omega p two. So how to draw So at omega p1 what will be a, considering omega p2 as large what will be the phase at omega p1, minus 45, close to minus 45, it is something like this. What will be the maximum phase? So each can contribute to minus 90. Yeah, so at infinite frequency actually it can touch minus 180 degree. Touch minus 180. So the maximum phase of this is. So the maximum pace of this is also is minus 180. this also is minus 180 but that will happen only at infinite frequency. So as see in the root locus since the roots are on LHS and by looking at this bode diagrams when your magnitude is 1 your phase is never minus 180 that means the system is stable. This is never minus 180, that means the system is stable. So what will happen when there are 3 poles with 3 poles, So what will happen when there are three poles? if there are 3 poles how will be the root locus say you have one pole at minus omega p1 another at minus omega p2 and 3 poles, how will be the root of this? one more at minus omega p3. And these are the pole locations where with the beta equal to 0. Say you have one pole at minus negative p1, Now what will happen when there is a feedback and beta varies from 0 to 1? another at negative p1. What will happen? The farthest pole actually will move still further. This pole will move further. These two will come closer. These two will come closer. They will meet. They will meet. And from here actually, And from here actually it will take this locus. Locus. This will be the loop. And at some frequency these poles will actually enter right hand side. That means Whenever there is a system with 3 poles there are chances that the system becomes unstable. Whenever there is a system with three poles, there are chances that the system becomes unstable. The same thing we can notice in the case of by drawing the Bode diagram. The same thing we can notice in the case of, by drawing the border diagram, where since there are three poles, There since there are 3 poles, minus 90 plus minus 90 plus minus 90 is minus 90. minus 90 plus minus 90 plus minus 90 is minus 270. So, So at any finite frequency the gain can be minus 180. at any finite frequency the gain can be minus one h. Does it make some sense by comparing this root locus and Bode diagram and we are checking the potential instability problem of at. A beta when your magnitude is 1 your angle A beta equal to minus 180. when your magnitude is 1, your angle A beta is equal to minus 180. We are checking whether these two things are happening at a particular frequency omega 1. We are checking whether these two things are happening at particular frequency omega 1. That is what we are checking. That is what we are checking. Any doubts? Now we will have a closer look at this Bode diagrams.

we never talks about stability of an open loop system. We never talk about stability of an open loop system. So what is the need of stability and what will happen if the system is not stable and what are the methods to keep the system stable.

So what is the need of stability? And what will happen if the system is not stable and what are the methods to keep the system stable. Those things we will be discussing in this topic. Those things we will be discussing in this topic stability and frequency compensation and this is actually a part of unit 3. Stability and frequency compensation and this is actually a part of unit 3. So So I hope you have some idea about feedback amplifiers through the course AMC in third semester. I hope you have some idea about feedback amplifiers through the course AMC in 3rd semester.

So with that background I will start. So with that background I will start. So we will look at the need of feedback also. So we look at the need of feedback also.

So say you have an amplifier and all the amplifiers what we have discussed in this course can be an example for this amplifier. So say you have an amplifier and all the amplifiers what we have discussed in this course can be an example for this amplifier. So assume that you have an amplifier with a open loop gain of A and if I want to use this amplifier. So assume that you have an amplifier with the open loop gain of k.

And if I want to use this amplifier. So what is the basic problem of using an open loop amplifier? What is the basic problem of using an open loop amplifier? Or why do we need feedback systems? Or why do we need feedback systems?

Output? Output? Very high? Very high? Very high gain?

Very high gain? Yeah, Yeah, if we have a very high gain, if you have a very high gain, yeah, yeah, that is one thing. that is one thing. The other thing is, The other thing is, for example, for example, we look at a case where we need a very small gain. we look at a case where we need a very small gain.

I think we have discussed this even in the case of OPAMs. I think we have discussed this even in the case of opamps. So whenever I have a circuit this is your CS amplifier circuit and the gain of the CS amplifier is minus gmRD and now if I am looking for an accurate gain of say 10. So whenever I have a circuit, this is your CS amplifier circuit and the gain of the CS amplifier is minus GMRD.

And now if I am looking for an accurate gain of say 10, So I am looking for an accurate gain of 10. so I am looking for an accurate gain of 10, What is the challenge here? what is the challenge here? I know that both this gm and this RDR especially this gm there is a lot of variation.

I know that both the GMRD and CS amplifier have the same gain. and this Rd or especially this gm there is a lot of variation the gm variation will be with respect to mu and co same term with respect to temperature as well and if you look at the gain of a cs amplifier you can say that almost 20% variation will The gm variation will be with respect to mu and cox in turn. with respect to temperature as well. And if you look at the gain of a CS amplifier, you can say that almost 20% variation will be there if you design for a gain of say 10. be there if you design for a gain of say 10 so if you are designing for a gain of 10 all these variations can altogether vary your gain around 20% now what is the solution for this So if you are designing for a gain of 10, all these variations can altogether vary your gain around 20%. Now what is the solution for this?

The solution is actually The solution is to if I want a gain of set 10 accurate gain the solution is to employ a feedback in this open loop system and make the system a feedback a feedback system or to make the system a closed loop system and then work for this accurate gain of 10. The solution is actually, the solution is to, if I want a gain of set 10, accurate gain, the solution is to employ a feedback in this open loop system and make the system a feedback system or to make the system a closed loop system and then work for this accurate gain of 10. That is the idea of feedback. That is the idea of feedback. So if you look at any general feedback system or this can be considered as a basic feedback system.

So if you look at any general feedback system or this can be considered as a basic feedback system. So the main part will be your amplifier which is marked as A where A is the open loop gain of the amplifier. So the main Main part will be your amplifier which is marked as A where A is the open loop gain of the amplifier.

Now, Now so so Vin is the input of this amplifier and Vout is the output of this amplifier. Vin is the input of this amplifier and Vout is the output of this amplifier. So what we will do, So what we will do in order to employ a feedback we will sense a portion of this output. in order to employ a feedback, we will sense a portion of this output.

How will we sense? How will we sense? through another circuit so sensing again we can send either voltage or current those details we are not looking at here so through a network through some network we call this network as feedback network through this network we will get a portion of the output voltage and this is actually fed back to the input now when i feed back to the input i Through another circuit, so sensing again we can send either voltage or current, those details we are not looking at here.

So through a network, through some network we call this network as feedback network. Through this network we will get a portion of this output voltage and this is actually fed back to the input. Now when I feed back to the input, I need some mechanism over here. need some mechanism over here i can call this mechanism as some compare mechanism so or i need some I can call this mechanism as some compare mechanism.

So or I need some. addition or some mixing at the input side so when I take a portion of output signal and when I feed it back if I am calling it as Vs now depending on how I am mixing this I will understand Addition or some mixing at the input side. So when I take a portion of output signal and when I feed it back, what I am calling it as Vs. Now depending on how I am mixing this, I will understand whether this is a positive feedback or a negative feedback system.

whether this is a positive feedback or a negative feedback system right so if it is the feedback signal if I mark this feedback signal as say some Vf if that is negative. Right. So if it is the feedback signal, if I mark this feedback signal as Vs, then that is... subtracted from your input signal we call this as a negative feedback system and whenever you look at a negative feedback system you can see one sensing mechanism at the output and one comparison mechanism at the input Subtracted from your input signal we call this as a negative feedback system and whenever you look at a negative feedback system you can see one sensing mechanism at the output and one comparison mechanism at the input.

This is a disc part from input to output. And this is your feedback network and this path from input to output this path is known as feed forward path or the main path and this is your feedback path. This part is known as feed forward and this is a feed path part.

Now why you need feedback? Why you need feed path? The need of feedback is because I have an open loop amplifier whose gain is actually poorly controlled.

The need of feed path is because I have an open loop amplifier whose gain is actually poorly controlled. I have a I have a... poorly controlled open loop gain or I have a gain which is varying.

So this gives the need of a feedback system. So this gives the need of a feedback system. Now so what is A here?

Now, so what is A here? A gain of your amplifier. A gain of your amplifier. Most of the time we call this as open loop gain.

We call this as open loop gain. Beta is your feedback factor and what is the closed loop gain here or I can write gain with the feedback. This is the This is the This is the This is the This is the This is the This is the I will mark as A f.

What is gain with feedback if this is a negative feedback system. Yeah it is A by 1 plus A beta. So after defining this we will identify this A beta. So after defining this we will.

Identify this A beta. What is A beta called? What is this A beta called?

A beta is actually the, A beta is actually the feedback factor and A beta is actually the gain of the loop or we call this as loop gain. beta is the feedback factor and A beta is actually the gain of the loop or we call this as loop gain. So now what happened actually when I did this feedback? So now what happened actually when I did this feedback Say we will take one example Assume that the open loop gain is What happened when I did this feedback? Say we'll take one one example assume that the open loop gain is 10 10 and say your beta is 0.1 what is your AF?

And say your theta is 0.1 What is the gain? 5. That means what happened when I That means what happened when I applied feedback? Again reduced and if you look at that we can see that by applying this feedback I am actually deliberately reducing gain. and if you look at that we can see that by applying this feedback I am actually deliberately reducing gain. So the gain with the feedback is actually less compared to your open loop gain.

So the gain with the feedback is actually less compared to your open loop gain. But we always talk about high gain right. We always talk about high gain, So. So what then tell me what is the advantage of applying a feedback?

right? So, then tell me what is the advantage of applying a feedback? Can you remember some advantages of negative feedback? Can you remember some advantages of negative feedback? What are the advantages?

This is gain control or I can say that the why we introduce feedback is because you have an amplifier whose gain is poorly controlled and what I am doing is by providing a feedback mechanism to the main amplifier I am actually desensitizing the gain of the amplifier Gain. gain control or I can say that the see why we introduce feedback is because you have an amplifier whose gain is poorly controlled and what I am doing is by providing a feedback mechanism to the main amplifier I am actually desensitizing the gain of the amplifier which is poorly controlled. which is poorly controlled.

So I can say that desensitize the gain of the main amplifier So I can say that desensitize the gain of the. The bike can't be controlled by the chain. Why you need to desensitize because that gain is a poorly controlled gain apart from that okay reduces noise any Apart from that, other advantages yeah stability yeah correct yeah there will be a change in input and output impedance so depending on the type of feedback what we apply it has stability. change in input and output impedance so depending on the type of feedback what we apply we have change in input and output impedance and all those changes or whatever advantages we are getting over negative feedback is actually by a factor of 1 plus a theta and we call this factor as amount of feedback We have change in input and output impedance and all those changes or whatever advantages we are getting over negative feedback is actually by a factor of 1 plus a beta and we call this factor as amount of feedback.

So negative feedback is advantages in all respects except for reduction in gain where the gain is reduced by 1 plus a beta but again that is an advantage because the whole idea of implementing negative feedback is actually to advantage in all respects except for reduction in gain where the gain is reduced by 1 plus a beta but again that is an advantage because the whole idea of implementing negative feedback is actually to get a gain which can be controlled in a better way. a get a gain which can be controlled in a better way right even though we are getting a lesser gain for the closed loop system we have a more controlled gain getting a lesser gain for the closed loop system we have a more controlled gain yeah one very important thing is bandwidth is also bandwidth Yeah, one very important thing is bandwidth is also bandwidth is also increased by this factor of 1 plus a beta. is bandwidth is 5Ab so from the previous discussion of the CS amplifier so what I can say here is if your AF is A by 1 plus A beta So from the previous discussion of the CS amplifier, so what I can say here is if your AF is a by 1 plus a beta and if your a which is the open loop gain. And if your A which is the open loop gain, Even if it is a very less value, even if it is a very less value, even if it is a poorly controlled gain, even if it is a poorly controlled gain, depending on the value of depending on the value of A, A, I can approximate.

I can approximate. If your A beta is much much higher than 1, If your A beta is much much higher than 1, your closed loop gain can be approximated as your closed loop gain can be approximated as. 1 by beta.

Or I can say the closed loop system purely depends on the feedback factor and if I am establishing this feedback factor using ratio of resistors. Or I can say the closed loop system purely depends on the feedback factor and if I am establishing this feedback factor using ratio of resistors. So one small example we can consider here is.

So one small example we can consider here is. So, same example we have considered before also. So, this is your V in and this is your V out.

If your A is infinity, A is infinity, your A is very large, if your A is very large, your A beta is also very much higher compared to 1 and what will be your close loop gain? your A beta is also very much higher compared to 1. And what will be your close loop gain? 1 plus R1 by R2 or here the gain purely depends on the ratio of resistors. Yeah, 1. Here the gain purely depends on the ratio of resistors.

So, So even though we have a very high gain for this amplifier, even though we have a very high, we have gain for this amplifier and this gain will be definitely in terms of gm into r0 parallel r0 where I have chances of variation of this gm right is that fine. and this gain will be gm into r0 parallel r0. Chances of variation of the sphere Right The will So we will just see the same concept so write this question for Let's see the same concept So write this question So write this question So write this question So write this question So write this question So write this question a given amplifier with open loop gain A as 200 Load the closed loop response for feedback factor 0, Close Loop Response Kill 0.1 and 0.5.

Pole Amplifier Assume that this is a single pole amplifier having a pole at omega peak at some frequency. So what is closed loop response? A f will be a by 1 plus a beta. So what is this beta? is a feedback factor.

So what is meant by beta equal to 0? So what is beta equal to zero? No feedback. No feedback. That means the loop is open.

That means the loop is open. And what is the range of this beta value? And what is the range of this beta value? What can be the range of beta value? What can be the range of beta value?

Beta will be from zero to one, Beta will be from 0 to 1, right? So when you say beta is 0, right? So when you say beta is zero, the loop is open.

the loop is open. And when you say beta is one, And when you say beta is 1, the entire thing is fed back or it is unity gain feedback. the entire thing is fed back or it is unity gain feedback loop, right? So beta, So beta So the range of beta is from the range of beta.

So, when a beta is from 0 to 1, 0 to 1. Now can you draw the closed loop response. now can you draw the closed loop response? So, So when a beta is 0 that is basically an open loop amplifier.

when a beta is 0, that is basically an open loop amplifier which is having a gain of 200 and say this is your omega 3. Which is having a gain of 200. And say this is your omega p. So this is when your beta is equal to 0. This is when your pizza is equal to zero. Now what is this? Now what is, yeah.

What is the new AF when your beta is equal to 0.1? What is the new here? Correct, yeah. So your new AF is A by 1 plus A beta and that is reduced to a value 9.5.

Yeah, new, uh... Let's see... What is the... So what is the feedback factor in this case? What is the...

Yeah, around 21. So your gain actually reduced. So gain is drastically reduced. Gain actually reduced, so gain is drastically reduced So to another value 9.51 So to another value, say, this is 9.5. When your beta is equal to 0.1, what happens to bandwidth?

So what happened to gain? So actually your gain is 200 that is reduced by a factor of 1 plus a beta. So actually your gain is 200 that is reduced by a factor of 1 plus a beta. At the same time your bandwidth which is omega p is increased by a factor of At the same time your bandwidth which is omega p is increased by a factor of plus a beta.

1 plus a beta. So it is like proportionate increase in that. So it is like proportionate increase in that. So your cutoff frequency or the bandwidth will not be at omega p that will be pushed towards right. So your cutoff frequency of the bandwidth will not be at omega p that will be pushed towards right.

And actually it will from there it will actually drop down. there it will actually drop down. Now what will happen when your beta is equal to 0.5?

Now what will happen when your beta is equal to 0.5 yeah your new yeah so when your for your new gain that is when Yeah, your new... So, when your... for your new gain, that is... When your beta is 0.5, your beta is 0.5 what is your closed loop gain 1.98 that means actually your what is your close to gain?

N8. That means actually your gain still reduced to a very low value. Gain still reduced to a very low value.

So this is almost 1.98. So this is almost 1.98. Your bandwidth also increased.

Your bandwidth also increased. So you can see that as your gain reduces your bandwidth is actually shifting towards right. See that gain reduces, your bandwidth is actually shifting towards the high end. Or in your closed loop system your gain bandwidth product will actually remain constant. product will actually remain constant.

So in your. This is your gain here with a bandit of this for the second when the gain reduced actually a bandit further increase. So this is your gain here with a banded of this. For the second when the gain reduces actually a banded will further increase.

So that gain bandit product is likely to be equal. So that gain banded product is likely to be equal. Because there is only one pole in this case. There is only one pole in this case. Right.

And decrease in gain and increase in bandwidth is actually proportional. And decrease in gain and increase in bandit is actually proportional. So if you plot in semi-log, So if you plot in semi-log you will also come to know that it is actually taking the same slope.

you will also come to know that it is actually taking the same slope at almost the same point. Almost same point. Omega, which omega is unity gain frequency? Each omega is unity gain frequency.

Omega, how omega, Omega, how omega, what is this unity gain frequency? what is this unity gain frequency? No, No, unity gain frequency is a frequency at which gain equals 1 or 0 dB. unity gain frequency is the frequency at which gain equals 1. or 0 dB that is unity gain frequency right so in this case if i am assuming this as my 0 dB line unity gain frequency actually will come at a very high volume so that basically determines the speed so if your unity gain frequency is very less you will not be able to use your transistor more That is unity gain frequency, right? So in this case, if I am assuming this as my 0 dB line.

unit gain frequency actually will come at a very high value. So that basically determines the speed. So if your unit gain frequency is very less you will not be able to use your transistor more than that frequency. than that frequency So here the reduction in gain and an increase in bandwidth. Action in B And then we go back Okay, So so we have seen a negative feedback system, We have seen a negative feedback system.

We discussed about the advantages of negative feedback system. we discussed about the advantages of negative feedback system. Now what is the problem in a negative feedback system? Now what is the problem in a negative feedback system?

Or why we are actually discussing stability all of a sudden? Or why we are actually discussing stability all of a sudden? Or how will you generally study stability?

Generally study stability. How will you say a circuit is stable or not? How will you say a circuit is stable or not? Small change in? Small change in...

Yeah, Yeah, for a bounded input, for a bounded input, output should be bounded, output should be bounded. Correct. correct. So, So, whether those problems will happen in feedback system?

whether those problems will happen in feedback system? Yeah, Yeah, so we are actually discussing the stability in negative feedback system. so we are actually discussing stability. negative feedback system you can state that if not designed properly if your feedback system is not designed properly the amplifiers can actually be disengaged So we can state If not designed properly, if a feedback system is not designed properly, the amplifiers can actually oscillate, amplifiers can actually oscillate. So that is the problem what we are discussing here.

Amplifiers can actually oscillate. So in order to discuss that, So that is the problem what we are discussing here. I will take this loop gain as a by 1 plus a beta.

So in order to discuss that, I will take this loop gain as a by 1 plus a theta. So why we are discussing stability is amplifiers with a proper negative feedback will actually give a lot of advantages. So while we are discussing stability is, amplifiers with a proper negative feedback will actually give a lot of advantages. But the problem is the amplifier tend to oscillate, But the problem is, if the amplifier tend to oscillate, then it is a problem, then it is a problem.

right. Right. So, So... No, no, no, no, no. no, yeah, Yeah, we will come to that.

we'll come to that. How it is oscillating, How it is oscillating, I will tell you. I'll tell you. So, So, if I'm... if I am, so what is the basic idea of an oscillator?

So, what is the basic idea of an oscillator? Without any input, Without any input, the system is actually providing, the system is actually providing... So that, so that such situations can come even in the case of an amplifier.

such situations can come even in the case of an amplifier. So that is the point when we say that an amplifier oscillates. So, that is the point when we say that an amplifier oscillates.

We will discuss that. We'll discuss that. Now, Now, before discussing that, before discussing that.

Now we have written AF as A by 1 plus A beta and we know that A is actually the open loop gain but from the previous discussion of the frequency response of CS amplifier we understood that this A is not a frequency independent term rather it is a frequency dependent term. now we have written AF as A by 1 plus A beta, and we... know that A is actually the open loop gain.

But from the previous discussion of common frequency response of C-S amplifier, we understood that this A is not a frequency independent term, rather it is a frequency dependent term. So can I write A as A of S because here gain of an amplifier is actually a function of frequency that is why when you plot the frequency response you are even when you plot the open loop response. So can I write A as A? because here gain of an amplifier is actually a function of frequency that is why when you plot the frequency response here even when you plot the open loop response gain is actually varying with respect to frequency so i have written it as a function of frequency or gain is actually varying with respect to frequency.

So I have written it as a function of frequency or I can write this as a function of j omega. i can write this as a function of k of s now what about this beta Now what about this beta? So at least in this case I am considering beta as a purely resistive network.

So, at least in this case I am considering beta as a purely resistive network, So that is not providing any phase. so that is not providing any peace. But since here open loop gain is a function of frequency, But since here open loop gain is a function of frequency the term a of j omega into beta which we call this as loop gain.

the term A of j omega into beta, which we call this as loop gain, is actually a function of frequency. is actually a function of frequency ok now So what is the problem of instability here? So I can write the combined the closed loop system is also a function of frequency and that is a of s divided by I can write the sum a of n.

1 plus a of s into beta. So can you tell me one possible condition for instability for this negative feedback system frequency. Can you tell me one possible condition for instability for this negative feedback system? Frequency?

Yeah, Yeah, frequency variation is there at how it becomes a unstable condition, frequency. variation is there and how it becomes an unstable condition. What leads to an instability in this negative feedback system?

what lead to an instability in this negative feedback system? Fouls lying on, Bones running on the... yeah.

yeah, that is an S plane, that is an s plane we will come to that okay if a beta is minus one one minus one becomes zero so what is the what is the implication from that we will come to that. Okay, if A beta is minus 1, 1 minus 1 becomes 0. So what is the implication from that? Or can we say without input you will get some oscillations here. Or can we say without input we will get some oscillation?

So here the problem is if at a certain frequency and I am calling that frequency as, So here the problem is if at a certain frequency and I am calling the frequency as this is very important, so this is very important, this is actually the key for studying the stability. this is actually the key for studying the stability so at a certain frequency say omega 1 if So at a certain frequency say omega 1, if a of j omega 1 A of j omega 1 into beta into beta into beta, if this is equal to minus 1, This is equal to minus right, 1. A of j omega 1 into b times, a of j omega 1 into beta, if this is equal to minus 1, this is equal to minus 1. What will happen to your first loop gain? what will happen to your closed loop gain? Your closed loop gain actually becomes infinite. Your first loop gain actually becomes infinite.

Or I can better write this as if the magnitude of loop gain, Or I can better write this as, it would have to be, so Since your open loop gain is a function of frequency, Since your open loop gain is a function of frequency, your loop gain itself is a function of frequency and it is actually a complex term as well. your loop gain itself is a function of frequency and it is actually a complex term as well. So, So if it is a complex term, if it is a complex term, can I write it as magnitude and phase?

can I write it as magnitude and phase? Right. Right. So, So I will write this as magnitude of A beta equal to 1 and angle of A beta.

I will write this as magnitude of A beta equal to 1 and angle of A beta equal to minus 180 Minus one. So this is one instance where your negative feedback system which is a stable system will become unstable. Instance where your negative feedback system, which is a stable system, will become unstable.

So when the magnitude of loop gain equal to 1 and at that particular frequency if the phase or the loop gain phase is equal to minus 180 degree your a beta term becomes minus 1 and that leads to an So when the magnitude of looping equal to one, and at that particular frequency, if the phase or the loop. to minus 180 degree your a theta term becomes minus 1 and that leads to an infinite gain of the system or you can actually see this in this way you have a system i will express it as y of s by x of s infinite gain of the system or you can actually see this in this way. So you have a system, I will express it as y of s by x of s which is equal to this a of j omega divided by 1 plus a of j omega into beta and that is a of j omega divided by which is equal to omega of z. 0. So that gives an understanding that without input, so your input is 0, This is the road to Torrington without input you have some output. That means what the system is oscillating when I say the system is oscillating the system is not stable.

Does it make sense this condition? Right now so how can we ensure stability? Hello So what we have done we have considered a negative feedback system after discussing all the advantages of a negative feedback system we looked at the closed loop gain and we understood that in this closed loop gain this term a beta which is called as loop gain if Close loop gain, this term, a beta, which is called as loop gain, if the magnitude of loop gain is equal to 1 at a particular frequency, the magnitude of loop gain is equal to one at a particular frequency and at that frequency you and at that frequency, if the phase provided by the loop gain is minus 180, If the phase provided by the loop gain is minus 180, the system will actually oscillate. the system will actually oscillate. So that is a potential problem.

So that is a potential problem. And now the question is how to ensure that the system will be stable. And now the question is how to ensure that the system will be stable. So what can be done for that?

So what can be done for that? What can be done? So what is the problem here? Yeah, Yeah, the magnitude of loop gain is 1 and angle of loop gain is actually equal to minus 180. the mic is on me. So this is actually causing instability.

So to avoid instability what I should do? So to avoid instability, what I should do? Make sure that these two conditions should not happen together.

Make sure that these two conditions should not happen together, Right. right? So when the magnitude is 1 and when the face is minus 180, So when the magnitude is 1 and when the phase is minus 180 that leads to oscillation.

that leads to oscillation. So to avoid instability, So to avoid instability, So we must ensure that these two conditions do not occur. so we must ensure that These two conditions These two conditions will definitely occur What we are trying to do is These two conditions should not occur at the same frequency These two conditions will definitely occur. What we are trying to do is these two conditions should not occur at same frequency.

Okay, so we will continue the discussion. Maybe we will review certain concepts what you have already learned in your control systems. Maybe we will review certain concepts what you have already learned in your control systems. So, what are the tools or what are the methods to analyze stability of any system?

So what are the methods to analyze stability of any system? Yeah. So what are the methods to analyze stability?

So, what are the methods to analyze stability? Now what are the methods? What are the methods?

What are the techniques? What are the techniques? Root lockers.

Root lockers. Root lockers. Ok, root lockers is one. Root lockers is one. What is root lockers?

What is root lockers? Determining the roots of a system and plotting it on a S plane. Determining the roots of a system and plotting it on a S plane.

Ok, Okay. now then apart from that? Go to plot, how will you analyze stability there? How will you analyze?

stability there? You have some idea right, you have some idea right? ok fine so the one method to analyze stability is actually root locus or by plotting the bolts of a system on an s plane and by looking at the location of bolts we can say whether the system is stable or not so okay fine.

So the one method to analyze stability is actually root locus or by plotting the poles of a system on an S plane and by looking at the location of poles we can say whether the system is stable or not. So for any system. So for any system, To be stable where the poles should lie on S plane?

stable should lie on S plane. Left half of S plane. So there you go.

Oh very good. Poles should lie on left half of S plane. Should lie on left half of S plane. So if you plot that.

your poles can come on LHS it can be poles on your this axis or it can be multiple like imaginary as well. LHs on your can be multiple flex imaginary as well. So when you have poles on LHS of S plane the system will be stable now what about marginally stable system?

So when you have bolts on LHs of explained the system which is stable. Now what about marginally stable system? The poles lie on j omega axis and for an unstable system poles lie on right half of s plane.

So if I plot these things this will be for a. marginally stable system and if poles are on RHS of S plane, their system is unstable. Similarly, how will be the time domain behavior? In the main behavior, The system is oscillating, if the system is having bolts on imaginary axis, if the system is having poles on imaginary axis, this is called oscillatory behavior. this is what your oscillatory behavior.

If it is having bolts on the RHS, If it is having poles on the RHS how it will be? how it will be? Increasing.

Yeah, actually it will go, and for your unstable system there will not be any oscillations it will actually come to it will stabilize it will take some time. it will keep on increasing and exponentially increase and for your unstable system there will not be any oscillations it will actually come to it will stabilize, it will take some time settling time will be there for any system. So this is one way if you know the roots of a system by plotting root blockers on x plane we can say that the system is stable. So.

This is one way if you know the roots of a system by plotting root locus on a plane we can say that the system is safe. Now in that case can you consider a single pole amplifier and a single pole amplifier and a single pole amplifier and a single pole Now in that case can you consider a single pole amplifier and consider a single pole amplifier and try to plot the roots of open loop system as well as closed loop system. Try to plot the roots of open loop system as well as closed loop system and check whether the system is stable or not.

Check whether the system is stable or not. So you can take the open loop transfer function as a naught divided by Again, take the question as A0 divided by 1 plus s by omega. S0. So this is your open loop. So this is your open loop transfer function or open loop gain.

transfer function or open loop gain. So draw root locus in that plot the poles of the open loop system as well as closed loop system. So draw root locus in that plot the goals of the open loop system as well as closed loop system. So what is the pole location here?

Code location here. For the open loop system where is the pole? For the open loop system, Open loop system where is the pole? pole is actually at minus omega naught. Open loop system, So this location is when your beta is actually equal to 0. this location is when your beta is actually equal to 0, there is no feedback.

There is no feedback. So for your open loop system, For your open loop system, the pole is at minus omega now. the pole is at minus omega naught. Now what about closed loop system, Now what about closed loop system?

how to write closed loop transfer function? How to write closed loop transfer function? Yeah, A of s is actually equal to A of s divided by 1 plus A of s into beta so that can be written as A naught by 1 plus s by omega naught. divided by 1 plus a naught by 1 plus s by omega naught into beta.

One plus If you rearrange this yeah you can write this as you can rearrange in this way so that you can see here we are C here Open loop gain was a naught initially. In the closed loop system, that gain is actually reduced by 1 plus a naught beta. So, we have a new location.

And open loop pole was at omega naught or the bandwidth was omega naught. The new location is somewhere here. By applying feedback, That is at minus omega naught into bandwidth is actually increased by 1 plus a beta. So what is the new location?

Yeah, new location is somewhere here. That is at minus omega naught into 1 plus a beta. 1 plus a beta. Right.

Right. So, So as beta increased, as beta increased... from 0 to 1, 0 to 1, here location of pole is actually moving from minus omega naught to minus omega naught to 1 plus a b into r.

here location of pole is actually moving from minus omega naught to minus omega naught into 1 plus a theta. So, So by this, by this, what we can say, what we can say for any system to be stable, for any system to be stable, poles will be on left half of s plane. poles will be on left half of s plane. So, So here poles stay on left half of s plane even after feedback when the system is stable.

here poles stay on left half of s plane even after feedback. So, the system is stable. Or the other way we can say the instability condition will not come in this case. Or the other way we can say, the instability condition will not come in this case. Why?

Why? We have only one pole. We have only one pole.

if you have a single pole, If we have a single pole, what is the maximum phase shift possible? what is the maximum facial possible? What is the maximum phase shift?

What is the maximum patient? So what is the face? So what is the phase?

Can you write the phase for this open loop system? Can you write the face for this open loop system? What is the numerator phase?

What is the numerator face? What is the denominator phase? What is the denominator face? This is j omega, S is j omega right? right?

When you have a plus i b how to write the face? When you have a plus IP how to write the face? Plan inverse of b by. Tan inverse of d by so what is space here numerator is 0 minus denominator is tan inverse of omega by omega naught that is a space right. So what is face here?

Generator is 0 minus denominator is plan inverse of omega by omega. So omega is actually, So omega is actually omega varies here right. omega varies here.

So by looking at the location of force we get the resistance is stable. So by looking at the location of poles we can say the system is stable. Similar thing you can do with drawing the Bode plots. Similar thing you can do with drawing the Bode plots.

So can you plot magnitude and phase for this? So can you plot magnitude and phase for this. Body plot is actually drawn for your loop gain, Body Plot is actually a brown correa loop gain right?

right? A beta. A beta So plot the magnitude and this is angle.

So how will you plot? So at 0 frequency you have The pure frequency you have A beta or you can express this as 20 log of......will be there. A beta or you can express this as 20 log of A naught into beta that will be there and what happens when it encounters the first pole as per as per the body rules yeah so if you have pole at omega naught the And what happens when it encounters the first pole as per the body rules? Yeah, so if you have pole at the omega naught, the gain will reduce... gain will reduce by minus 20 dB per decade.

Now how to draw the face for this? Let me draw the face for this. The face expression is minus tan inverse omega by omega naught.

The face expression is minus tan inverse omega by omega naught. So as your omega varies from 0 to infinity face also will actually increase from 0 to a value right. So as your omega varies from 0 to infinity, face also will actually increase from 0 to a value. So initially the face is So initially the face is 0 when you what is the face when your omega equal to omega naught what is tan inverse 1 yeah.

0. What is the face when your omega is equal to omega naught? What is tan inverse 1? So this is actually a minus 45 degree So this is actually a minus 45 degree.

And what is the maximum phase at omega equal to infinity minus 95 by 2 right. So, your maximum phase at omega equal to infinity minus 95 by 2, right? So, your maximum phase is actually asymptotically if you touch minus 90 degree. So your maximum phase is actually asymptotically it will touch minus 90 degree. So, we define...

So we defined the problem of instability before what is the problem? problem of instability before what is the problem yeah in the loop gain when the magnitude of loop gain is equal to 1 if your face is minus 180 degree you have a potential instability problem so whether we have that problem Yeah in the loop gain when the magnitude of loop gain is equal to 1 if your phase is minus 90 degree you have a. potential instability problem. So whether we have that problem now, so this is the point where your magnitude of Lucan equal to one and at that particular frequency your phase will never touch minus 180. So this is the point where your magnitude of Lucan equal to 1 and at that particular frequency your face will never touch minus 190. That means there is no problem or I can say a single fold system is actually inherently stable.

That means there is no problem or I can say a single pole system is actually inherently stable, It is not conditionally stable, it is not conditionally stable, it is inherently stable. it is inherently stable. So we have analyzed stability by looking at the pole locations as well as bode diagrams, So we have analyzed stability by Lucan. We are looking at the pole locations as well as the border diagrams, base as well as target.

phase as well as magnitude. Any doubts? Any doubts? So similarly if you analyze a single two pole system Similarly, if we analyze a single pole, a two pole system, So I can take a transfer function as A of s is equal to the open loop gain since there are two poles I am marking it as omega p1 and one as omega p2. The structure action.

We are not increasing the capacitance all the capacitance are internal capacitance so that depends on the structure actually. Then your C-S amplifier you have seen. In this sample you have seen maximum pole is 2 with the one zero.

maximum pole is 2 with 1 0 right yeah that is why we analyze that first so this is a two-pole system now the task is check whether this system is stable by root locus as well as border diagrams you That is why we analyze that first. So this is a two pole system. Now the task is check whether this system is stable by root lockers as well as border diagrams. We have to solve that.

have to solve that so if you look at the open loop response So if you look at the open loop response. So this is your j omega sigma. j omega sigma so the four locations are one net minus omega p1 another is omega p2 this is the four location and here beta is zero So the four locations are one at minus omega p1 another at minus omega p4. So this is the four location when your beta is 0. So what will happen when your beta increases from 0 to 1?

What will happen when your beta increases from 0 to 1? So you have to write actually the closed root transfer function. This is actually the closed loop transfer function. So you will get a second order polynomial.

So you will get a second order polynomial. You have to solve that. You have to solve that.

You will get two roots for that. You will get two roots for that. And in that root there will be a beta dependency. And in that root there will be a beta dependency.

And if you vary beta from 0 to 1 you will get that. And if you vary beta from 0 to 1 you will get that. So what will happen? So what will happen?

Can you tell me what is the root locus rules? Can you tell me what is the root locus rules? When you have two poles.

When you have two poles. When beta increases actually this will move towards left and this will move towards right. When beta increases actually this will move towards left and this will move towards right. Both meet at one point and from there actually it will split. Both meet at one point and from there actually it will split.

So this will be the locus of the roots. So this will be the locus of the roots. So this we are just refreshing we are not going to do something with this. We are just refreshing, we are not going to do something with this. We are checking whether the root stays on the left half of S, We are checking whether the roots stays on the left half of S plane that is all.

like that side. So these two roots will meet and split and it will actually move in this way. So these two roots will meet and split and it will actually move in this way.

So whether the system is stable? So whether the system is stable? Yes. Yes.

Can you do the same thing with the Bode plot? Can you do the same thing with the Bode plot? Draw the magnitude and phase and check whether the system is stable or not.

Draw the magnitude and base and check whether the system is stable or not. Starts at 20 log of K0 in This is actually magnitude of A beta. This is angle of A beta.

Now this starts at 20 log of a naught into beta. Bogota There is one pole which is at omega p1. There your gain drops by 20 dB per decade. Then you have second pole which is at omega p2.

There again your gain falls by minus 20 dB. So that means this is minus 40 dB per decade. What the hell? And at some point, your gain will be equal to zero dp or one.

Now what about phase how to write phase for this numerator phase is zero minus denominator you have two poles minus tan inverse omega by omega p one minus tan inverse omega by omega p two. So how to draw So at omega p1 what will be a, considering omega p2 as large what will be the phase at omega p1, minus 45, close to minus 45, it is something like this. What will be the maximum phase? So each can contribute to minus 90. Yeah, so at infinite frequency actually it can touch minus 180 degree. Touch minus 180. So the maximum phase of this is.

So the maximum pace of this is also is minus 180. this also is minus 180 but that will happen only at infinite frequency. So as see in the root locus since the roots are on LHS and by looking at this bode diagrams when your magnitude is 1 your phase is never minus 180 that means the system is stable. This is never minus 180, that means the system is stable. So what will happen when there are 3 poles with 3 poles, So what will happen when there are three poles? if there are 3 poles how will be the root locus say you have one pole at minus omega p1 another at minus omega p2 and 3 poles, how will be the root of this?

one more at minus omega p3. And these are the pole locations where with the beta equal to 0. Say you have one pole at minus negative p1, Now what will happen when there is a feedback and beta varies from 0 to 1? another at negative p1. What will happen?

The farthest pole actually will move still further. This pole will move further. These two will come closer.

These two will come closer. They will meet. They will meet. And from here actually, And from here actually it will take this locus. Locus.

This will be the loop. And at some frequency these poles will actually enter right hand side. That means Whenever there is a system with 3 poles there are chances that the system becomes unstable. Whenever there is a system with three poles, there are chances that the system becomes unstable. The same thing we can notice in the case of by drawing the Bode diagram.

The same thing we can notice in the case of, by drawing the border diagram, where since there are three poles, There since there are 3 poles, minus 90 plus minus 90 plus minus 90 is minus 90. minus 90 plus minus 90 plus minus 90 is minus 270. So, So at any finite frequency the gain can be minus 180. at any finite frequency the gain can be minus one h. Does it make some sense by comparing this root locus and Bode diagram and we are checking the potential instability problem of at. A beta when your magnitude is 1 your angle A beta equal to minus 180. when your magnitude is 1, your angle A beta is equal to minus 180. We are checking whether these two things are happening at a particular frequency omega 1. We are checking whether these two things are happening at particular frequency omega 1. That is what we are checking.

That is what we are checking. Any doubts? Now we will have a closer look at this Bode diagrams.

Transcript for:Understanding Closed Loop System Stability

Transcript for:
Understanding Closed Loop System Stability