 Today in lecture 14 will be continuing with the understanding of dynamic behavior of feedback system. In the last lecture we had considered the static characteristic how the saturation non-linearity effects the dynamic range of operation of the feedback system and how distortion gets reduced by 1 plus loop gain and how noise gets reduced by 1 plus loop gain how the frequency response improves by 1 plus loop gain that means the corner frequency gets shifted the time constants get reduced okay or the system becomes a high speed system okay or wide bandwidth system with feedback. So the effect of desensitization okay is seen clearly in all aspects of negative feedback today will continue with the discussion of the dynamics of negative feedback system. Let us consider a simple circuit which involves just one MOSFET or one bipolar transistor it does not make any difference. So let us consider the MOSFET now this is the structure which is in negative feedback say you consider that this is a current amplifier or this is a trans conductor stage which converts voltage to current so we have the GM of the transistor coming into picture. So converting the input voltage to change in input voltage to change in output current and this current is fully fed back so this is a current feedback arrangement the error current now flows into the amplifier. So since the current gain is infinity the feedback results in current follower delta I naught by delta I I is equal to 1 delta I naught delta I I equal to 1 because the current gain okay or the whole thing is infinity so this is actually equal to the loop gain divided by 1 plus loop gain what is the loop gain so if you have delta V occurring between gate and source the current delta I naught is going to be GM times delta V and that current change in current is going to flow to the incremental resistance across the drain and the source which is called the conductance is called GDS. So one of our GDS okay into GM is the loop gain magnitude is minus because if this positive that making a current flow in this direction will cause inverted voltage to appear here so it is negative feedback loop gain is negative so it is negative feedback and this magnitude is GM by GDS. So one can therefore show that this is equal to 1 by 1 plus 1 over loop gain which is GDS by GM. So if GM is made very large that is the case with idealization of the device one can therefore have a current follower delta I naught by delta I equal to okay 1 this is equal to 1. So we get this as a current follower here now this is valid within the active region of the N channel enhancement MOSFET for this to be in the active region this voltage here has to be positive with respect to this and greater than threshold voltage VGS and then only current will flow in this direction and that current is equal to this current I naught is equal to K by 2 into VI minus VT whole square this we had already discussed earlier so if this is the VI this is the non-linear relationship so with this full current feedback now what happens the input current becomes the independent variable because input current is equal to output current zero current flows through this. So this actually generates an out voltage across this V naught okay which is square root of 2II by K plus VT this is the VGS that gets developed so that becomes the dependent variable now. So you can see the inversion property that is got by using a square law in feedback path okay you get inverse property for the voltage developed okay with respect to the input current. So and that inversion action is seen because the loop gain is very high okay and at every operating point this is valid that GDS by GM becomes extremely small and output change follows the input change. Now in order to make this work let us say the current in this should be in a polar should be only in this direction that means it should start with minimum of zero and it can go on up to the maximum current it can tolerate. So if you are aware to design such current follower for optimal operating region then one would have an offset current of ID max by 2 biasing the MOSFET so that is what is done in this. That means we have an ID max by 2 DC current flowing through this in order to have a current swing which is bipolar on either direction this way or that way it can flow to an extent of ID max by 2. So this is the way to bias the active devices in order to make them operate for signals which are bipolar in nature. Now it MOSFET as a current follower as a first order system let us now consider G10 is still equal to GM and G2 equals now minus 1 over GDS earlier we had taken but now because of the existence of a capacitance here which is essentially this capacitance plus the capacitance to substrate this is the source this is the gate. So it is CGS plus we will call this CDS and therefore we have a total of CGS plus CDS occurring at this point across the so called RDS in the equivalent circuit. So we have VI this is delta VGS developing GM times delta VGS and across this we have GDS and this is fed back here so the total capacitance is CDS plus CDS. So that is what is written we have the G2 equal to minus 1 by GDS plus SCGS plus CDS frequency dependent. So the loop gain now becomes frequency dependent a first order system minus GM by GDS plus S into CGS plus CDS. So GF is delta I naught by delta I which is DC loop gain divided by 1 plus DC loop gain which is minus 1 by 1 plus GDS plus SCGS CDS divided by GM this can be ignored because GM is very large so you can approximate this as minus 1 by 1 plus SCGS plus CDS by GM which you can say is equal to minus 1 by 1 plus S by gain bandwidth product which we have earlier defined for a first order system okay. So if it is an op-amp for example it is DDC gain A naught okay into it is for corner frequency omega 1 so that is the gain banded product. Here it is GM divided by total capacitance CGS plus CDS. So this is approximately equal to omega 3 okay equal to GM by CGS plus CDS even though normally it is omega t is defined for CGS plus CDG here CDG is shorted out okay. So it has only the effect of CDS drain to substrate in an IC. Here the other system that we have discussed for its lock range and other things automatic gain control system. So here again if you consider this frequency dependence of gain here and let us say this is A naught DC gain then you can consider this as grounded for AC because it is remaining constant only changes occur here and that changes some DC change here okay this change here causes a DC change here okay this is AC to DC converter this is DC to DC converter let us say. So we have here this forming attenuator from here R R and then this is the point and we have a capacitance if you connect a capacitor directly across it okay it might have to be done with large value to capacitor instead the capacitor is simulated by connecting it between input and output of an amplifier then you get almost A naught into C okay as the effective capacitor A naught into C plus okay C. So that is the effective capacitance that comes here okay. So that is roughly approximated C into A naught when A naught is very large right. So from here to here the gain is A naught so that is the equivalent circuit that is used to determine the loop gain and that loop gain is G1 is minus A naught into VP divided by 20 because that factor of half due to R and R okay and then VP by 10 is the VP by 10 is the sensitivity from here to here because it is VP O squared by 20 differentiate this for small signal so this is going to be 2 VP O by 20 or the transfer function or the sensitivity factor is VP O by 10. So this combined with this will give you a G1 of minus A naught by 20 into VP divided by 1 plus SC A naught R by 2 and G2 is VP O by 20 for this block this is the block that has an output of VP O okay which is equal to VC 10 VC by 10 into VP. So is gain factor okay where this is the input and this is the output so that gain factor is VP O by 10 G2 so loop gain is this into this so 1 by 1 plus loop gain is what is written here 1 by 1 plus this which is 1 by 1 plus S by GB so GB is equivalent to VP O divided by 100 C into R by comparing this. So again we get a control over what the gain may be product or the quality of the feedback system is for an automatic gain control so it is again a first order system with this value of GP that is tested out in simulation so apply a pulse here for example I just make this a pulse okay so what happens this should ultimately become equal to this value VP O square by 20 so but it takes time for it to reach that value and that is indicated by the gain bandwidth product so you can see that this is the input pulse so it has been given a sudden pulse like this then what happens to this right about 0.8 is the steady state value it has been changed by some amount okay say 400 millivolts okay around 0.8 so it is going to change to the new value okay exponentially okay the voltage is going to come to the requisite value for that V reference and therefore amplitude now becomes constant it is that value whatever it is so that is a step function given it responds in an exponential manner coming to steady state the same thing can be attempted in the so called frequency follower or the conventional phase lock loop where the input is a frequency and output is a frequency in the static characteristic we have seen that the dynamic range of this or the lock range of the system is given by the sensitivity of this when this is let us say VP 1 VP 1 so VP 2 sin omega t plus phi this is VP 1 sin omega t so the frequency of the input is going to be same as frequency of the output because a phase follower that we had shown already is a frequency follower delta phi naught by delta t delta phi I by delta t that means omega naught follows omega I once you establish that then it is a phase follower and for VC equal to 0 that is the quiescent frequency or the free running frequency in this case has been adjusted to be 10,000 hertz okay and if you apply 10,000 hertz at the input that is the static characteristics okay this will adjust itself to VP 1 VP 2 by 20 VP 1 VP 2 by 20 cos phi is the DC that gets generated so the sensitivity of this okay delta phi is minus VP 1 VP 2 by 20 sin phi so it is maximum at phi equal to 90 degrees right so it is maximum at phi equal to 90 degree okay and this is the way it is changing delta we have read by delta phi is equal to VP 1 VP 2 by 20 sin phi so you can see the thing varying okay is maximum okay at this point now this is this look now you can see that the look gain which is nothing but KPD you call this sensitivity factor as KPD of the phase detector this we will call it as KVCO so the look gain is KPD into KVCO in this case if you have an amplifier here it will enhance the look gain right in this case okay it is not there so KPD KVCO and since this is converting the frequency to phase it is acting as an integrator because we know that omega okay rate of change of phase is frequency so that means phi by S is equal to omega okay so that is phi into S is omega so or phi is equal to omega by S delta phi by delta t is S into phi is equal to omega so phi is equal to omega by S so it acts as an integrator so as far as phase is concerned so you can see here that the frequency dependent behaviour of this can be represented in this manner the loop gain is VP 1 VP 2 by 20 that is the KPD okay and that phi equal to 90 degrees it is maximum okay and therefore the sin phi is equal to 1 okay that divided by S into KVCO delta omega by delta VC is called KVCO. So we have this as the loop gain so using the same thing omega naught by omega I is equal to 1 by 1 plus 1 over loop gain so it becomes a first order system 1 by 1 plus S by GB without including the low pass filter it looks like a first order system so for VP 1 equal to VP 2 we have considered and equal to 10 volts and delta F by delta VC is equal to in this case taken as 1000 hertz per volt okay that is what we have taken GB is going to be equal to pi into 10 to power 5 radians per second okay. So we have this established as a first order system and we are now changing again the input frequency you can see this we have what is called an FSK frequency shift key in information so here if I apply a step starting at if I apply a step starting at the say 10,000 coefficient frequency so I apply a step change in frequency that is to this that can be done by change in this voltage K from 0 to let us say 1 volt then the frequency is going to change from 10,000 to 11,000 if I give a step of 1 volt here the frequency will change from 10,000 to 11,000 that is what is called frequency shift keyed information let us say 10,000 to 11,000 so that should be getting reflected here right as a step change but because of the dynamics of this and the low pass filter etc we will have this okay changing edge is gain bandwidth product okay the rate is going to be determined by the gain bandwidth product as a first order system that is what is shown here. So this is the step change at the input and the output is going to change exponentially from the initial value the same value output change is going to be the same as the input change right so that is what is seen here so that 1 volt change is getting reflected here. Now I am applying such 1 volt pulses at the input of the VCO which is converting it to FSK at the input of the FLL so then at the input of the VCO in the feedback path I am getting this exponential change reproduced and then it is going to reach the 1 volt value so this is depicting the first order system behavior. Now we come to second order system this has already been discussed in the last class where G1 is a second order with first corner frequency at omega 1 second corner frequency at omega 2 omega 1 less than omega 2 transfer function of the system with this G2 as constant independent frequency we had derived this earlier in the last class and we see that G1 by 1 plus G1 G2 can be 1 by G2 divided by this this is a gain stage with feedback amplifier gain equal to 1 by G2 this is the frequency independent gain of the feedback system that is its dynamic is related by the denominator which is now governed as 1 plus S by some factor plus S squared by some factor that is normalized this is the normalized things are written S squared by natural frequency of the system what is this natural frequency if this zeta is going to 0 that means the coefficient of S is 0 is a second order system a resonant system which resonates at this natural frequency with infinite amplitude because this goes to infinity at put S equal to j omega this becomes minus omega squared by omega n squared omega equal to omega n this becomes 0. So that is what is called natural frequency of the system it will oscillate at the natural frequency omega n is the natural frequency and is equal to square root of omega 1 omega 2 into the DC loop K and zeta is called the damping factor in control system okay and zeta is by comparison now 1 over 2 into square root of loop gain square root of omega 2 by omega 1 plus omega 1 by omega 2 in this omega 2 by omega 1 is the dominating factor because this is greater than 1. So this is always less than 1 1 over 2 zeta is also defined as quality factor Q and that is equal to G10 G2 that is the DC loop gain divided by omega square root of omega 2 by omega 1 plus square root of omega 1 by omega 2. So now considering that second pole is far away from the first pole then we can neglect this and say that Q is equal to square root of loop gain divided by square root of omega 2 by omega 1. So we have seen the behavior time domain behavior of such systems okay for Q greater than half or zeta less than 1 is represent what is called under damped system okay a step into it will result in always arranging at the output coming to steady state after all this ringing stops. So Q greater than half or zeta less than 1 that is demonstrated here in the system a step is input is given ultimately it equals the step however this is ringing Q equal to 1 indicates that is going to be 1 peak which can be counted Q equal to 10 also has been drawn and you can count 10 such peak 1 2 3 4 5 6 7 8 9 10. So this is the way you can judge the behavior of the time behavior or what is called transient response of a second order system and the mathematics of it is given there this has already been done for you in networks. So these are the things that are shown this is for Q equal to half where there is no ringing and it is the fastest reaching the steady state okay here Q less than half takes unusually long time to reach the steady state the best is to make Q greater than half so as to have just one ring so this reaches very fast the steady state and just peaks once and comes to steady state also fast settling very fast. So such a systems therefore for transient response to subside very quickly must have always Q typically of the order of 1. So all these systems whether it is a frequency lock loop or a phase lock loop or an amplifier op amp okay or a feedback amplifiers using transistors must be designed if they have to have high speed operation possible with a Q of equal to 1 the response is characterized by good rate of rise with one small peak. Now second order system frequency response is what is going to be considered next the steady state behavior okay peak magnitude of G of max occurs at omega max and omega max can be shown to be okay equal to this is the transfer function and put S equal to G omega we get G of equal to G of naught divided by 1 minus omega squared by omega n squared whole squared magnitude plus omega squared by omega n squared by Q squared okay square of this 1 minus this whole square that is the real part squared plus the imaginary part omega by omega n Q square the phase is the imaginary part omega by omega n Q divided by 1 minus omega squared by omega n squared the real part than inverse of that in this now we can maximize this magnitude and that has been done and maximum occurs at a frequency omega x max equal to omega n natural frequency square root of 1 minus 1 by 2 Q square and the value of that maximum is G of max equal to G of 0 into Q directly proportional to Q for high Q circuits divided by square root 1 minus square root of 1 minus 1 by 4 Q squared so magnitude G of at omega n is G of max okay at omega n is going to be G of 0 into Q maximum rate of rise in the phase shift occurs this also we can do take the phase shift and maximize this and that is going to be delta phi by delta omega this an important thing that maximum occurs at a frequency omega equal to omega n and it is equal to 2 Q divided by omega n this is depicted okay in this graph this way it is going to be decreasing ultimately at 40 decibels per decade and it might peak for 2 equal to low values is going to decrease like this and the phase is going to change from 0 to 180 degrees say second order system for 2 equal to 10 for example okay is going to peak and then this phase change is going to be more rapid okay than lower queues okay this is directly proportional to Q it is 2 Q by omega n and this is going to be Q into right you can see here G of naught divided by 1 minus 1 by 4 Q squared and Q is very large this is very nearly occurring at the natural frequency and the peak is G of naught into Q this can be exploited in design of amplifiers where we can make it maximally flat that means this whole thing can be equated to the peaking can be equated to 1 so that occurs when Q is equal to 1 over root 2 that you can show it is a maximally flat amplifier designed okay and for good wide bandwidth okay designed. So Q equal to 1 over root 2 is for good transient response Q equal to 1 over root 2 is for good steady state response Q equal to 1 over 1 is the good transient response. So these are the 2 things that you have to remember Q equal to 1 is a system with good transient response you cannot have both simultaneously and Q equal to 1 over root 2 has good steady state response let us now consider FLL as a second order system so again in order to understand the steady state behaviour of this okay you have to again find out the loop gain okay and obtain the transfer function from here to here under this situation. So what is done is delta phi naught by delta phi I earlier was represented as a first order system now because of the low pass filter introduction that 1 over loop gain gets modified by this factor 1 plus S by omega LP. So 1 by 1 plus S over GB it was earlier as a first order system now this is a factor which is modifying it as a second order 1 plus S by omega LP. So it becomes 1 by 1 plus S by GB plus S square by GB omega LP. So what happens again it has a certain Q and the natural frequency which is square root of GB into omega LP and Q is equal to square root of GB by omega LP. So if you want to make the Q of this FLL equal to 1 for it to have a good transient response you have to have GB equal to omega LP okay. And then the natural frequency extended up to GB itself okay. Example consider now for designs of these systems consider a system with G10 equal to 10 power 5 as the DC gain G2 equal to 1 so that means full feedback is given so it is a voltage follower or a current follower that you are designing okay where output follows the input and the first order frequency omega 2 which is greater than omega 1 is located at 3 into 10 to power 6 radians per second let us this is the device that is given to you determine omega 1 where should you locate omega 1 to obtain ideal transient response to a step input that means omega 1 should be located at fairly low frequency in order to separate out omega 2 from omega 1 by an extent equal to the loop gain this is the simple design principle the DC loop gain okay so be equal to ratio of omega 2 by omega 1 very simple. So omega 2 by omega 1 gets fixed as equal to loop gain DC loop gain for making Q equal to 1 X naught by XI is equal to 1 that is the situation discussed omega 1 therefore is equal to omega 2 divided by the DC loop gain so 3 into 10 to power 6 divided by okay 10 to power 5 which is the DC loop gain G10 into G2 which is equal to 30 radians per second the value of the gain bandwidth product in this example therefore is this corner frequency omega 1 into DC loop gain 10 to power 5 okay because G2 is 1 so 3 into 10 to power 6 is the gain bandwidth product of this system that you have designed just now so the design is very simple right make the Q equal to 1 the DC loop gain has to be equal to the ratio of the second pole to first pole example 2 consider a system with G10 equal to 10 to power 5 and G2 has now been made equal to 1 over 100 that means you are intending to design an amplifier with the gain equal to 100 because 1 over G2 is the gain of the amplifier with feedback so it is amplified with gain equal to 100 what should we do in the design omega 2 should be greater than omega 1 and omega 2 is given as 3 into 10 to power 6 we are taking the same example as before X naught by XI on the other hand is 100 if the step response of the feedback system is to remain ideal Q of the system should be 1 omega 1 now is going to be equal to omega 2 divided by the DC loop gain so that DC loop gain is reduced by a factor of 100 for this amplifier now 3 into 10 to power 6 divided by 10 to power 3 the gain bandwidth product of in this case is GB equal to G10 okay into omega 1 which is 3 into 10 to power 8 so it is 100 times more than the earlier amplifier okay feedback amplifier so that means for a higher gain you can effort to have higher gain bandwidth product because the DC loop gain is going to be reduced by the same factor as the gain approximation of a second order system second order feedback amplifiers can be approximated to your first order system when Q is much less than half so then these systems always are going to be exponentially reaching the steady state they are low speed systems okay so in such situation the approximation simply means that the omega 2 is far away from omega 1 so in the useful range it can be considered as only governed by omega 1 the first corner frequency so for Q much less than 1 therefore GF 0 is 1 plus G omega by omega NQ right so that means it can be considered as the first order system okay that is what has been stated there will be no magnitude error because you know that becomes very small that we have already discussed okay because square of this is negligible compared to 1 so however there is bound to be a phase error in these systems phase error which is tan inverse omega by omega NQ which can be approximated by omega by GBG2 for frequencies within the bandwidth of the system so let us consider another system consider the op amp with F1 equal to 10 to power 6 hertz F2 is equal to 4 into 10 to power 6 G0 equal to 10 to power 6 this is typically the type of op amps IC op amps okay specification that is available in order to design a feedback amplifier with unity gain and Q equal to 1 and first corner frequency must be shifted from 10 to power 6 okay this is the where it is occurring down to you can say F1 dash which is 4 hertz why because we have the loop gain equal to 10 to power 6 now okay and the first second corner frequency is 4 into 10 to power 6 that therefore results in F1 dash as 4 into 10 to power 6 divided by 10 to power 6 okay so in order to make Q equal to 1 so it has to be shifted to 4 hertz that means 4 hertz is the first corner frequency and the gain loop gain is 10 to power 6 so 4 into 10 to power 6 okay is the gain bandwidth product that is all that you can have for this as gain bandwidth product in spite of the fact that first corner frequency itself occurs at 10 to power 6 without this kind of arrangement of shifting the first pole down now gain bandwidth product of unity gain feedback amplifier is 4 into 10 to power 6 if the same compensated op amp which as F1 dash at 4 hertz is used now okay for higher gains F2 is equal to 4 into 10 to power 6 G naught is equal to 10 to power 6 higher gain means feedback factor is going to be for example for a gain of 100 is going to be 1 over 100 for gain of 100 it should be 1 over 100 so the DC loop gain gets reduced by a factor of 100 4 into 10 to power 6 by 100 4 into 10 to power 4 the Q of this feedback amplifier therefore is 0.1 much reduced so it is a low Q system that means it does not rise very fast it is a sluggish system so a compensated op amp okay for unity gain is useless okay as high speed op amp for higher gains so this is demonstrated that means for every gain okay you actually design the amplifier to have a Q of 1 okay by shifting the corner frequency that means internally compensated op amps are not very efficient because normally they are compensated for the highest DC loop gain that means full feedback and for lower DC loop gain for that means higher gains they are pretty useless and therefore it is better to use externally compensated op amps where you can decide the compensating ahh capacitor such that you can shift T first pole down to such a frequency that Q of the system remains equal to 1 as Q is much less than 1 the transient response is sluggish in order to increase the Q to 1 F dash need to be shifted from 4 hertz to 400 hertz so that the gain bandwidth product is the highest possible for so 4 into 10 to power 6 instead of 4 into 10 to power 4 second order system with a 0 now we have considered normally first order and second order when I say first order and second order right what I mean is within the range of frequencies where the loop gain remains greater than 1 the order is what counts beyond that the amplifier is pretty useless as an amplifier so we consider the amplifier in the useful range of frequencies to have a behavior of frequency response which is decided by first order or second order now if there is a 0 also coming into picture then the 0 always causes the gain increase so at the ultimately at high frequencies right the ahh effect of gain decrease is compensated for by the effect of gain increase and it reduces to a first order at high frequencies. So let us therefore consider what the effect of 0 is the introduction of the 0 will make the loop gain have GF equal to GF 0 divided 1 plus 1 over loop gain now becomes this with only this modification factor due to the 0 coming here so that means this 0 will influence the numerator also now apart from influencing the denominator however in the denominator the coefficient of us which is mainly the Q determining factor is independent of feedback okay earlier GF 0 was figuring in okay this ahh feedback factor okay and G1 0 also was figuring in G1 0 has to be made very large in order to desensitize GF 0 okay with respect to active device parameter. So GF 0 has to be desensitized with respect to G1 0 means G1 0 is made large that means Q of the system is very high okay Q is directly proportional to square root of G1 0. So this arrangement of introducing a 0 in the original transfer function causes it to be independent of G1 0 that means you can fix the Q of the system based on the location of the 0 okay that is not possible with ahh the earlier system which has all poles. So this flexibility results in ahh fixing the Q independent of desensitization effect. So natural frequency remains the same as before okay and further the disadvantage of the system is the rate at which gain falls off outside the band okay of the signal that is reduced because of the numerator S coefficient. So when noise white noise effects the system the noise gets increased because of this 0. So that is the disadvantage of the feedback amplifier with a 0 okay. However Q of the system now if you compare the coefficients can be made independent of G1 0 in this manner. So by locating the 0 suitably you can make the Q equal to 1 that is omega natural is made equal to omega Z in order to have Q equal to 1 because these factors become negligible. So this is the major advantage of introducing the 0 however the disadvantage also is there in terms of noise increase. So this is the design that has been made the natural frequency of the system is made equal to the ahh 0 frequency omega Z in case this is negligible in order to make Q equal to 1. So this can give ahh Q equal to 1 without shifting the corner first corner frequency that would have reduced the effective bandwidth of the system considerably. So this is a technique of designing white band amplifiers okay ahh by optimizing Q equal to 1 okay by locating the 0 at the desired position of the natural frequency of the system. In conclusion we have now discussed lot of dynamics about feedback systems in general whether it is for a phase lock loop or frequency lock loop or an op-amp in feedback the dynamics is decided by the same factor what is it. The dynamics of the system makes sure that it is frequency independent over a wider bandwidth why how much the loop gain plus 1 is the factor by which the bandwidth improves. The gain bandwidth product is a measure of the quality of the feedback system and it is dynamics that means whether it is transient response or it is steady state response both are affected by the gain bandwidth product of the feedback system not the open loop gain not the open loop bandwidth but it is the gain into bandwidth DC loop gain into bandwidth okay which is of concern as a quality of feedback and the speed of the system okay. So please remember that gain bandwidth product is an important property associated with any feedback system it is not necessary that it should be only in amplifiers that this is a measure of the quality of feedback and this technique of shifting the first corner frequency okay such that it becomes optimized feedback system with Q equal to 1 okay responding very fast to input changes or for that matter any change anywhere in the input okay. So this is called frequency compensation is an important term that is associated with feedback designs so frequency compensation what about higher order systems higher order systems are always brought back to second order in order to optimize the performance of the feedback system for transient or steady state it is just the value of Q equal to 1 or 1 over root 2 okay these are the factors which make the feedback very useful in practice that is higher order systems if they are not used without this kind of frequency compensation are liable to become unstable that means they may oscillate okay at the natural frequency of the system cause the additional poles cause additional phase lag okay causing them to turn the negative feedback into positive feedback at those frequencies result in instabilities therefore we will not even discuss the higher order systems in feedback in most of the present day applications the feedback look gain magnitude is maintained okay less than 1 when the effect of higher order poles come into picture that means within the range of usefulness of the feedback system that is the gain bandwidth product right it is having loop gain okay much greater than 1 and therefore it is a useful feedback system right beyond that right we do not mind it increasing the order of the system it does not okay effect the stability of the system. So we have also consider the effect of saturation in breaking a feedback loop that means if any system before starting is in saturation right the signal has to bring it to active region or an offset voltage or input can bring it to come out of saturation into the active region that is the starting circuit similarly capture simply means that the feedback system has to be brought by maybe suitable signals applied at various points to remain in active region as it starts functioning okay locking is after capture right locking simply means that output follows the input output at the feedback point follows the input and that is when the loop gain is much greater than 1 the error goes towards 0 okay or it becomes a nullator at the input and therefore right the moment this is broken by either saturation one of the blocks going to saturation okay it is going out of lock again it has to brought to be brought back into locking that means you have to come into the active region of every block and make the loop gain much greater than 1 at which it captures okay and then it can keep itself locked okay over a wide range called the lock range. So this is the conclusion of this in the next class we will be discussing the same feedback in circuits using network theory approach right we have an amplifier block now and feedback block both of which may have some properties like infinite input impedance finite output impedance finite transfer parameter from input to output finite feedback parameter from output to input. So how to design a good feedback amplifier so that we can reach the ideal amplifiers which we require.