 Today we will be discussing feedback in systems or feedback systems it is one of the most important topics in signal processing we will see this presently in the previous lecture we had seen applications of multipliers comparators and macro models of BJTs and FETs. System of voltage buffer current buffer trans conductance amplifier and trans resistance amplifiers using nullator-norator concept and then corresponding equivalent models of the transistors and replacing these nullator-norators with transistor has been done to achieve these near ideal amplifiers voltage buffer means unity gain voltage amplifier current buffer means unity gain current amplifier trans conductance amplifier and trans resistance amplifier with specific trans conductance and trans resistance all realized using BJT or FET the topologies of these may be looking like for example using MOSFET this is the MOSFET and this was voltage follower this being a nullator for the MOSFET VI was completely appearing as VI at the output and then as a current buffer it was looking like this common gate this is common collector it was called common gate so this was the nullator coming in shunt and therefore it was whatever current was inputted okay was going into the other side that means this source current is same as drain current okay there is no current flow in this this is nullator so the voltage was 0 so it was appearing as a shorted input and this current II so I naught equals II current buffer you could put any load here similarly you could put any load here so and this is a voltage control voltage source with unity gain current control current source with unity gain and then we discussed the other two topologies G so VI was applied here G times VI was appearing here so a trans conductance amplifier and then a trans resistance amplifier so so these were the ahh stages that we had understood in the last class now you will see that these are also actually considered as feedback structures these are all feedback structures so we have synthesized this feedback structure not discussing feedback at all but using the concept of nullator and nullator and then today we are going to learn more about system level concepts of feedback a dynamic system with feedback is what we are going to discuss today we have XI as input and it is having us summing stage where these two variables XI and feedback parameter XF get added so this is a summer where we have the output equal to XI plus XF at this point so this is the feedback parameter is the input parameter this is what is called a gain block or this is a block which transfers the input to some form of output right and the relationship is governed by G1 okay so in a linear system this is simply multiplying its input by G1 and appearing as output so again the output here is fed back through this block that is why this is called the feedback part this is the forward part and G2 is the parameter relating input to output in the feedback part so XF is equal to X naught into G2 so this now becomes X naught equals XI plus XF into G1 okay XF itself is okay X naught into G2 as shown here that into G1 so from this we can get X naught okay coefficient as X naught into 1 minus G1 G2 okay equals XI into G1 which gives us X naught by XI relationship between output and input of this system as okay G1 by 1 minus G1 G2 is an important relationship okay and XF is the feedback parameter X naught is the output parameter XCUS is okay the error okay so we will see this that if G2 is negative it is called negative feedback okay so that this sign when it is coming here as minus and getting added then it is coming as XI minus this factor okay G2 is negative this is negative feedback and G2 is positive it is positive feedback if G1 is positive or G1 into G2 should be negative for negative feedback G1 into G2 is called the loop gain okay of the system so G1 into G2 equal to the loop gain if this loop gain is negative it is called negative feedback if the loop gain is positive it is called positive feedback okay so these are the set of equations which we have already derived XF by XI becomes G1 G2 by 1 minus G1 G2 XCUS is XI by 1 minus G1 G2 okay GF is X naught by XI is G1 by 1 minus G1 G2 or minus 1 by G2 divided by 1 minus 1 by G1 G2 so this is an important concept that XI and XF are related by this okay XCUS is XI divided by 1 minus G1 G2 so if G1 G2 is negative let us say G1 G2 is equal to minus some positive number it is going to be only a number okay dimensionless quantity okay so this is going to be XI by 1 plus N when N is very large okay this okay error this is then called error because it is going to appear as XI minus okay this factor okay even though you are summing it because of G2 being negative you can therefore see that this is the error it computes the error so the error becomes goes to 0 in a system or this one XF divided by XI okay becomes minus N by 1 plus N or this becomes minus 1 so if this is minus okay then it is negative feedback and therefore output okay XF okay here the feedback output portion of output is going to be following input in magnitude and if it is summing thing it is of opposite phase that okay design inversion is going to be there if this is instead of plus a differencing thing then this is going to be just one okay so please make a note of this conventional control system feedback okay always shows this summer as a differencing thing so this minus sign of the negative feedback is automatically taken into consideration so out this XF feedback thing follows the input okay whereas in this if it is a summer XI is going to be making XF follow it in magnitude but opposite in sign so that the error always gets computed here and error goes to 0 or the system has this concept that we have already introduced to you okay earlier so it is just that it is coming now again right so this becomes a negative feedback system where XF follows XI in magnitude and if it is a summer it is minus if it is a differencing component it is exactly following in magnitude and sign okay so this is what is explained in the GL has to be negative if magnitude of loop gain is much greater than 1 then X naught by XI okay is minus 1 by G2 okay that means it is insensitive to G1 is independent of G1 okay and XF by XI is equal to minus 1 that is what we had shown earlier and XES goes to 0 or actually it becomes a nullator at the input so whatever variable we have considered as inputting G1 okay that itself goes to 0 that is the primary thing about negative feedback what is the consequence of that because of XES going to 0 XF okay is going to be minus XI so that the summer actually has at its output the error okay and X naught by XI which is independent of okay G1 and minus 1 by G2 itself value G2 in the feedback results in the forward transmission from output input becoming 1 over G2 this is the inversion property this is another important conclusion so if G2 is the forward function of the feedback block when we put it in the feedback the forward transfer parameter of the system becomes 1 over G2 this is the inversion this is following action follower action output follows the input in magnitude and nullator at the input these are the 3 important conclusions that we can arrive at and application of almost all feedback systems today is lying in these 3 basic concepts being followed. So let us consider a positive feedback system a positive feedback system on the other hand as GL positive okay and let us say it lies between 1 over G1 G2 lies between 0 and 1 then X naught by XI becomes highly sensitive this is the to the variations in loop gain or the variation in G1 causing variation in loop gain whereas in the earlier case X naught by XI was independent of G1 it is insensitive whereas in this case X naught by XI becomes highly sensitive to variations in here so this kind of feedback is never to be used okay in our linear system design okay and if this G1 into G2 or the loop gain is positive and then if loop gain is positive it is positive feedback and it is greater than or equal to 1 then it is regenerative feedback what does it mean then output is just going to infinity at GL equal to 1 it already goes to infinity that is unstable there is no finite relationship between output and input output just happily goes independent of input okay to infinity means positive saturation or negative saturation okay so that is the limit of practical output possible. So in the case of amplifiers and all it is the supply voltage so when GL is positive and 1 over GL is greater than or equal to 1 that system cannot be discussed further in terms of equations as output is going to be either high or low so it is an interface component at that point of time this amplifier is working as a comparator and it is giving a digital indication of okay what the state of input is is it greater than some value or less than some value okay so X naught by XI going to infinity a practical dynamic system goes into saturation where the gain is 0 there is no relationship between output and input which is linear any longer. So sensitivity in negative feedback we are going to emphasize this in terms of equations oh this is an important parameter to be defined for understanding the advantage of negative feedback. So let us define sensitivity of GF with feedback gain with feedback for G1 sensitivity of the response to changes in G1 this defined mathematically as change in GF divided by GF change in G1 okay for change in GF cause for a change in G1 this is the nominal value of G1 at which this happens so delta GF by GF divided by delta G by G delta G1 by G1 so the percentage change in GF due to percentage change in G1 so that is mathematically you can derive that because we know the relationship between GF and G1 as it is G1 by 1 minus G1 G2 so you simply differentiate it GF this is GF so you just delta GF by delta G1 okay by not and then multiply by G1 divided by GF you can show that it is equal to 1 by 1 minus G1 G2. So if G1 G2 is negative like we just said let G1 G2 be minus N so this is 1 by 1 plus N as N goes to infinity this tends to 0 that means it becomes insensitive okay sensitivity in negative feedback systems if you now find out the sensitivity of GF to G2 obviously it just depends only on G2 so that is established here very clearly by again differentiating this GF delta GF by delta G2 multiplying by G2 by DF then it becomes minus 1 directly dependent upon G2 alone any variation in G2 causes okay any variation in G2 causes inverse variation in GF okay that means actually it is demonstrating the inverting property of the feedback system also the sensitivity is inversely proportional to G2 now that is demonstrated in practice by designing a voltage amplifier with a gain of 10 for example. So you can see that this is G1 block is a differential amplifier so we have VI here and V naught into the attenuation here so that is actually we can now say that 1 by 10 attenuation 1K and 9K so 1 by 1 plus 9 is 1 by 10 with a negative sign because you are summing it up with an inverse. So what you get here is a negative feedback system because G1 into G2 is going to be G1 into minus 1 by 10 this is G1 into G2 so if this is much greater than okay if this G1 into G2 magnitude okay is much greater than 1 then we get this fact that V naught over VI is going to be 10 1 over this minus 1 over G2 that is just 10. So this is demonstrated by this that is going to be demonstrated by changing G1 from let us say 1000 to 100 and showing that output remains okay same for a given input. So this error goes to 0 it becomes a null later that we have demonstrated earlier also right. So now inversion property is demonstrated 1 over 10 put in the feedback path makes it become amplifier with gain 10 voltage amplifier with gain 10 attenuator in the feedback path and it is insensitive to G1 that is demonstrated by changing G1 for example let us first take G1 as 1000 then the gain becomes 9.9 10 by 1 plus 1 over 1000 into 1 by 10 so that is 9.9. So we have sensitivity of GF to G1 as 1 by 11 okay for G1 equal to 100 and G2 equal to minus 1 over 10 GF is 9.09 it is still very close to 10 right but deviating slightly because the loop gain is reduced by a factor of 10 the loop gain is reduced by a factor of 10 okay even so the feedback amplifier gain is remaining pretty close to the design value of 10. This is demonstrating the insensitivity so voltage amplifier is simulated with G1 equal to 1000 and G2 equal to minus 1 over 10 G1 equal to 100 and G2 equal to minus 1 over 10 again and you do not find any difference this is the input and that is the output it remains almost the same except for a small reduction when the loop gain is lower. Sensitivity and negative feedback is again demonstrated by designing a current amplifier so this is a current amplifier using the same op amp here okay and you can see that the 9K and 1K form the current division here this current is divided between this 9K and 1K so the current going into this feedback is going to be 1 by 1 plus 9 which is one tenth again. So X naught is the current now that X naught by 10 is fed back and current feedback occurs at a single node whereas voltage feedback occurs between two nodes okay so using a single node we can feedback current so this input current now is XI minus X naught by 10 so the error current and when G this gain current I mean this is a trans conductance this is an amplifier let us say G when it goes to infinity then this whole system behaves as a current amplifier with gain equal to 10. So you can see this load is coming in series with whatever current that is flowing okay. So gain becomes 9.52 if G1 is 200 for example we have taken a different value of G1 G2 is equal to minus 1 over 10 in this case that remains the same so and it is nearly 10 okay it is 9.52 now sensitivity is 1 by 21 and we have designed the current amplifier for different gains earlier it was 250 now 1100 irrigation is very little you can see this is the input current that is the output current and these do not remain and this remain the same almost literally no difference exists between this and this. So before we sort of close this we can just say that using a transistor for example the current amplifier same thing this is the active device instead of the op amp okay the current gain is nearly infinity here input current and output current so I give the full output feedback here so this is II and I naught is going to be equal to II if this GM of this trans conductor this nothing but the trans conductor this is going towards infinity that means even in a MOSFET this becomes a current follower this automatically if you put the load here right even then this is valid so arbitrary load so this current feedback remains the same okay and therefore output current follows the input current this a current follower or a current buffer with gain equal to minus 1 and this is same as the common gate topology which we have since just earlier so this is common between input and output the gate so it is a feedback topology that we have discussed that can be even designed using a diode this is called MOSFET connected as a diode if RL is 0 right similarly the bipolar transistor circuit will look the same in the current feedback this II is very nearly equal to II naught but opposite in sign so if RL is 0 it is become looking like a diode connected transistor both of them so that is a feedback topology okay now we come to the other important application but using instead of a linear block in the feedback okay we are using a non-linear block but what is it we are using a multiplier okay VC has to be kept positive to ensure okay loop gain so now we have the same G1 here this is my G1 block and G2 block is nothing but okay sort of minus okay VC okay V naught by 10 so as long as VC is positive this is negative feedback if VC is negative it becomes positive feedback so you have to be careful here so for this to work in negative feedback mode VC has to be positive that is important if you do that ultimately we get this important relationship VI is always going to be equal to V naught and this is VC by 10 or V naught is 10 VI by VC so it becomes an inverse function what is it if it is a multiplier in the feedback path output in a take and put it in this feedback path multiplier it becomes V naught over VI becomes a dividing action 10 VI by VC VC is multiplying this voltage V naught whereas here it is coming as 10 into VI by VC so it is a divider so this illustrates the property of inversion that is if you have a block that is doing some function we can always get inverse of the function okay if you put it in the feedback path that is illustrated here we have got the attenuator converting it to an amplifier we have a multiplier converting it to a divider okay and now we will see a squareer put in the feedback path will make it obviously a square root so this is V naught and it is connected to both the inputs of the multiplier so the output of the multiplier is V naught squared by 10 so VI becomes equal to V naught squared by 10 this error voltage goes to 0 none later so V naught is equal to root of 10 times VI so we have got this square rooting operation perform okay without any problem square rooting because V naught squared V naught squared by 10 is equal to VI from which we get V naught as square root of 10 times VI so now you can actually now put the multiplier with this as BP sin omega t okay and this as VC so the output of the multiplier is going to be VC VPI sin omega t so it is going to be let us see VC VPI by 10 sin omega t this is the output so now this is the VPO so it is squared here so VPO squared by 10 okay because this is VPO sin omega t so VPO squared by 10 into sin squared omega t is what appears here which gives you DC if you put a low pass filter the voltage here will be VPO squared by 20 because sin squared omega t is 1 minus cos 2 omega t by 20 10 so that will be 20 so you have a DC voltage here and this is got redopt by the low pass filter so we have here a DC voltage of VPO squared by 20 becoming equal to V reference so that is what is demonstrated here so V reference is equal to VPO squared by 20 or VPO equal to square root of 20 into V reference so it remains constant this VPO the peak value of output voltage here at the output of the multiplier remains constant at square root of 20 times V reference so it is a square rooting thing because we have put a squaring thing okay in the feedback path but it is done for an AC voltage right so that means this is converting AC to DC here this is AC to DC converter okay and this is DC to AC converter AC to DC converter here this is DC to AC converter here right so that amplitude is going to remain constant at square root of 20 into V reference this is an important concept that we must indicate okay here so if I keep on changing this it is of no consequence in the feedback okay this will always make it make this one appear as equal to this okay that means output amplitude of this DC is remaining constant at V reference so consequently VPO of this remains constant at square root of 20 into V reference that means this whole system now becomes an AGC system automatic gain control system okay AGC or A BC or an AC regulator so if this line voltage keeps on changing here fluctuating here it is of no consequence this loop will make sure that the output voltage remains constant at square root of 20 into V reference sin omega t so this is an important signal processing concept this is the front end of most of our RF systems IF systems okay where the received signal antenna signal keeps on changing okay however inside the receiver system you have a means of controlling this output amplitude to remain fairly constant okay and that is necessary input stage or a front end of all receivers today television radio as well as your cell phones all these contain this kind of AGC system in order to maintain the received signal amplitude of RF okay remain at a fairly decent level constant okay and this kind of system everybody should know how to build so is also an important concept to be used in AC regulators in power supplies right the 50 hertz power supply if it keeps on fluctuating in the line it can be maintained constant using a similar feedback automatic AC voltage regulator here it is a DC voltage regulator always so for this okay this is a DC voltage regulator at this point and it is automatically becoming an AC voltage regulator at this point because of the feedback so immersion property and also is used in AC regulator and DC regulator is demonstrated in the negative feedback this will be discussed in detail further okay so this is simulated this AGC scheme there is a minor difference instead of comparing it with this voltage I can also convert this this way that I can ground this and connect a resistance here and apply the same resistance and inversion of this then it becomes a current comparator this we have discussed earlier okay you can ground this and convert it into a current by using the same resistor okay and filter it using the capacitor at capacitor can be put across this okay to increase the effect of this will show later that if you put a capacitor between output and input of an amplifier 8 times that value of capacitor gets simulated at the input so instead of using a capacitor at the ground you can put a lower capacitor between output and input and see a much larger capacitor at the input okay this will demonstrate later however even if you put a large capacitor across the same effect is same okay so this is the modification that we can do this can be grounded and this can be connected to minus the same equation gets valid so that is what is simulated in the next circuit. So you see I have maintained V reference equal to 0.8 so VPO becomes square root of 0.8 into 20 so that is 16 which is 4 square root of 16 is 4 so output amplitude of this multiplier is maintained at 4 volts irrespective of the input amplitude input amplitude I have changed from 10 to 8 okay to 5 and then 4 okay so the output amplitude remains constant at 4 at all points of time. So this is the beauty of this control system which is called AGC or AVC. Now another application of this negative feedback and inversion is to RMS generator you can square using multiplier and average using low pass filter and then you can put the square in the feedback path and get a square rooting action here this is the square rooting action. So this is a square okay this is a square root so this if you now apply a periodic waveform of any shape okay this signal processing block straight away determines this true RMS value of the waveform which is an important measure in most of our electronic applications through RMS indicator. Now we come to an important point which we have already discussed that we have a feedback system with G1 and G2 forming a loop okay and we are now going to discuss what happens to XF okay and XI and X0 in this feedback system which is a special feedback system okay this we will call it as encoder decoder feedback system feedback system with G1 as encoder G2 is a decoder that means it has input and output variables interchanged input of one becomes output of the other output of the other one becomes input of this first one. So G1 is called an encoder G2 is called a decoder so output parameter need not be the same as input parameter that is the difference. So this perform a lot number of communication functions encoding and decoding is depending upon the channel is the characteristic feature of most of the communication channels and signal processing functions needed in communication. So for example right if you whatever happens here is that inverse function is related obtained as output okay and follower function is obtained at the input okay let us consider the examples of this current voltage converter and voltage to current converter these are the two encoder and decoder combinations here trans resistance amplifier and trans conductance amplifier. So the parameter at this input is current parameter at this point is voltage. So you can give current as feedback okay so you call this output current and this is input current output current always follows input so this is a current follower the same loop if you put this here and this one here okay that is what it is trans conductance amplifier here trans becomes a voltage follower CD difference so this is a current follower where it takes the error current here as the input okay error and output here is voltage okay and it is a current to voltage converter with G2 coming into picture as the transfer parameter okay this is GM that is 1 over GM is the inversion factor here so GM is the forward that is feedback factor so 1 over GM is the forward gain of this stage which is V naught by VI so by putting a trans conductor in the feedback part of an amplifier we are realizing a trans resistor amplifier GM into RF is the loop gain it has to be much greater than 1 in this case the same loop with this point as input and the loop continue here is going to be acting as a voltage follower so this is the advantage of encoder decoder combination both voltage follower action and current follower action can be done by the same loop with the same loop gain GM into RF once it is much greater than 1 it can either act as a voltage to current converter or voltage follower or current follower and current to voltage converter AC to DC converter and DC to AC converter AC voltage follower here or the other way about this AC voltage follower and if you give the input at this point it is a DC voltage follower this is the beauty of the thing so voltage DC can be converted to AC with a certain factor which is 1 over G2 again okay or you can have that AC can be converted to DC by putting DC to AC in the feedback part okay this we have already demonstrated as a AGC scheme where we have maintained the DC input to this whole system error constant okay and it becomes a AC voltage follower okay so AC voltage follower or a DC voltage follower right and then AGC scheme adopts this kind of DC voltage follower being adopted as AC voltage follower so if you maintain this DC constant this DC will be constant and therefore this AC will be constant okay it will be related to this by this factor transfer parameter okay same thing can be done with phase followers Phi I is the input Phi naught is the output you have a phase 2 voltage converter and voltage to phase converter here so this full phase follower it becomes at this input and if you use the same loop with voltage as the input it becomes a voltage follower this is the voltage follower that means if you maintain voltage constant the phase will be constant here okay so if you may change this voltage the phase will change accordingly so it can be used for phase modulation okay direct phase modulation next finally if you have a V2F converter here and F2 V converter here F2 V converter is a phase detector that we have already seen in multiplier application V2F converter is nothing but a VCO it is a voltage follower so if voltage is maintained constant the frequency will be maintained constant and if you have the frequency as input here voltage becomes output here it can be converted from frequency to voltage here and the conversion factor is inverse of this which is voltage to frequency converter which is a VCO right so this is an important thing that we have to understand this is the VCO voltage to frequency converter this is called a phase detector so the phase difference is converted into a DC voltage that is by using a multiplier and a low pass filter here so multiplier and a low pass filter here with the VCO gives what is commonly called as frequency lock loop so if this frequency changes this changes accordingly right so this is the advantage of this concept and that is done in frequency lock loop here we have a VCO which can be nothing but an FM generator so if you apply DC here you get a fixed frequency here if this DC changes this frequency changes proportionately so if it is a linear VCO as demonstrated here it is a linear VCO 10,000 hertz is the quiescent frequency plus K times VC okay so that is what is done here K times VC so this output frequency is directly proportional to the control voltage so this is a multiplier with low pass filter forming a phase detector we have demonstrated the ability of this phase detector earlier it can be a linear phase detector these 2 are square waves so this is called the sensitivity of the VCO K 1000 hertz per VC right so what happens if this frequency is same as this frequency then we have seen that this is VP sin omega t let us say this is VP sin omega t plus phi then this will be VP VP dash cos phi so if it is not to change okay from a value which is different from this VC quiescent then this cos phi should be 0 so that means if you want the phase to be locked at 90 degree right the reference input to this should be 0 so this control voltage will get automatically adjusted as 0 when this frequency is same as this that is what we are going to demonstrate in this frequency locked loop if this is the same as the quiescent frequency which is for VC equal to 0 that means if I applied 10,000 hertz here the phase shift should be automatically that into cos phi cos phi should go to 0 because this should remain 0 so when I have not applied any frequency here also the same thing should happen so it is when I am applying nothing here nothing comes here when I applied 10,000 hertz input this should automatically adjust itself as 0 if it is frequency lock that means if thus output frequency is same as the frequency error is 0 output follows the input that is demonstrated here I am applying the same frequency as the free running frequency you call it 10,000 hertz then K that means T equal to 0.1 millisecond then the phase shift between output and input gets automatically adjusted at 90 degrees as I change the frequency of the input the phase will change now to some other value so that there is a finite VC in this case VC goes positive you can see okay so the frequency is higher so the control voltage goes positive to make the frequency of the output get locked to the same frequency as the input but cos phi is going to be finite that is how this whole thing is normally called as phase lock loop this is how it works phase keeps on changing from the quiescent phase of 90 degree away from 90 next one I am going to a lower frequency again it is locked the frequency of the output is the same as the frequency of the input I made the purposely different in magnitude so that it is highlighted and the control voltage goes negative you can see. So you can see understand this basic concept of the so called phase lock loop also very simply as a frequency follower and frequency to voltage converter so it can be used both as frequency follower or frequency to voltage converter which is nothing but FM detector if the input frequency is changing the output voltage will change according to the modulating component of the input frequency that is FM if it is an FSK frequency shift key two frequencies are used so the output which is corresponding to the input of the VCO is going to be changing at two levels indicating the digital data of FSK so it is FM detector VCO is an FM generator and if VCO is put in the feedback path it becomes an FM detector same thing can be extended through A to D converters and D to A converters they are coders and decoders put in the feedback we get a voltage analog follower VI following followed by V naught V naught is exactly equal to VI at this point you can give digital input and convert it into a digital output follower digital input you give digital output of the system is going to follow the digital input you get an analog replica of the whole thing at this point so if you put an A to D converter in the feedback path you can make a make the whole thing become a D to A converter this is how these feedback works inversion works right this is also used in what is called sigma delta modulators right so this is the basic principle that we use a D to A converter and A to D converter right in loop okay this is called the sigma actually if this is plus and the sign is attributed to this then this becomes sigma delta okay modulator okay sigma delta modulator this is A to D and this is D to A okay so these are the most important applications of feedback in most of the integrated circuit systems also at system level we have discussed it thoroughly and concluding negative feedback makes the output less sensitive to or insensitive to variations of parameters associated with active devices in the forward path negative feedback makes the output sensitive to parameter variations associated with feedback path positive feedback can make the output very sensitive to parameters associations associated with the loop encoder decoder feedback systems enable us to realize a wide range of analog signal processing functions so this is the conclusion that the last one was the conclusion that we made.