 Today in our 26 lecture on active filters by Q enhancement we are going to cover the rest of the techniques of Q enhancement. We had already in 25th lecture covered how starting with low pass passive filter we can convert it into an active filter by embedding the low pass filter along with inverting amplifier of gain K forming a loop. So in such situation of forming the loop please remember that we had made use of the low pass filter of second order passive and put it along with an amplifier of gain – K and put this loop like this and here we had the facility of summing it with input so that the transfer function is going to be that of the low pass filter modified by this gain K stage with full feedback loop modifying the omega naught of the overall system with feedback. It is going to have a new natural frequency omega naught which is the natural frequency of the low pass which typically is for R1 equal to R2 C1 equal to C2 equal to C omega P is going to be 1 by RC and QP is going to be 1 over 3 so that is for the case of R1 equal to R2 C1 equal to C2 equal to C R1 equal to R2 equal to R. So then we saw that the natural frequency is going to be dependent upon 1 over RC of the low pass and then square root of 1 plus K so you can fix the natural frequency of your filter based on 1 over RC and square root of 1 plus K and square root of 1 plus K is determined by what Q you want to have so even though QP is less than 1 1 over 3 in this case it can be enhanced by root of 1 plus K so the gain bandwidth product of this amplifier has considerable influence on the performance of this filter the Q of the filter is going to be Q active is going to be Q passive into square root of 1 plus K that is the expected value but it is going to change from that to it is going to be 2 K times F naught by GB K into Q P into square root of 1 plus K which is the actual Q for which it has been designed. So this is the sensitivity of this achieved Q to gain bandwidth product it is very sensitive if this factor is close to 1 we will see that there is a possibility of the entire circuit having Q A becoming equal to infinity when this factor becomes close to 1 that means F naught has to be restricted to a value which is GB divided by Q divided by 2 K that means F naught should be less than GB into Q A for which you want to design and that divided by 2 K sorry GB divided by Q A into 2 K that means that is the sort of restriction on this kind of filters which make use of this negative feedback. Next then went over to second order high pass filter design where the basic filter topology was once again similar to the low pass here we had the high pass filter which is passive which was nothing but CR CR embedded in a similar situation like that in the case of a low pass with feedback and this is the summing stage. So then the natural frequency now got shifted to a frequency lower than that of the passive filter omega P and again is equal to 1 by RC and Q A is fixed by Q P which is 1 by 3 for this special case of equal resistors and equal capacitors. So again what happens now is Q of the filter after design becomes dependent upon the gain bandwidth product here also but it is now going to be 1 that is Q P into square root of 1 plus K that is the design value for which you are selecting the K that divided by again F naught divided by GB into Q A into let us say 2 K. So the same factor F naught Q A into 2 K divided by GB comes into picture but it is going to decrease the Q earlier okay in the case of low pass it was increase in the Q and we have shown that it is going to decrease the Q in this case okay and by the same factor this we had demonstrated experimentally and all the experiments have been shown where F naught has been increased okay for Q of 5 and we have seen the sensitivity of the actual Q on the gain bandwidth product. So again in order to have our design Q this factor F naught Q A to K by GB should be much less than 1 this is the criteria for low sensitivity to the active device parameter as well as passive devices stem cells are concerned K etc come as ratio of resistors are capacitors depending upon the type of amplifier you are designing. So the K sensitivity to Q here is equal to half because of the square root when K is very large it is very nearly equal to half okay. So it is not so sensitive to the ratio K as compared to GB when F naught is high Q A is high let us consider the second order active notch to be designed starting with a passive Q notch here the notch filter has M equal to P equal to alpha N equal to 0 let us see the network the network the passive network therefore is going to be similarly embedded so this is the notch passive notch once again we have this embedded in a negative feedback situation with something taking place VI then the transfer function is going to be okay we will see why this alpha comes okay this alpha okay is coming because of the passive network attenuation in forming a notch filter so this minus K is the gain stage S square by omega P square plus 1 means okay this M is equal to P equal to alpha N equal to 0 that is no coefficient of S here. So when you modify this the denominator polynomial now will change only the coefficient of S square and the constant so S square by omega P square will be now becoming 1 plus K alpha and this constant 1 is going to become 1 plus K alpha in the denominator and the coefficient of S is only due to the original denominator coefficient it remains unchanged natural frequency of this active filter because this constant 1 plus K alpha and this 1 plus K alpha make the omega naught of this system which is root C by A okay to be the same as omega P and the QP of this system because this divides the whole thing when normalized okay so it becomes a new QP QA is equal to QP into 1 plus K alpha. So instead of increasing by root of 1 plus K now it is directly proportional to K alpha. So this is the advantage that means by using for a same Q you can design the whole thing using low much lower gain stage okay that is a great advantage in the earlier case for high Q circuits K is directly proportional to QA square here it is proportional to QA that is a major difference that means the error due to finite gain band product the phase error due to the finite gain band product will be correspondingly vastly reduced for this structure that means it will be less sensitive to the gain band product than the earlier structures that we have discussed both high pass starting and low pass starting network. So this is the second order passive noise so we take VI and attenuate it by RB by RB plus RA that is the alpha and this network as a transfer function of SCR when R2 equal to R, R1 equal to R, C1 equal to C, C2 equal to C this will be having a transfer function which is SCR squared plus 3 SCR plus 1. So if you subtract this from that so alpha VI okay minus SCR VI divided by SCR squared into 3 SCR plus 1 okay becomes this that is it just simply becomes S squared by omega P squared plus 1 into alpha divided by S squared omega P squared S by omega P to P plus 1 this is the transfer function of this low pass output here V naught is taken here when VI is given that for that we have to satisfy this condition RB by RA plus RB should be exactly equal to QP because at the frequency natural frequency the attenuation of this will be alpha into QP so that should be actually the sort of alpha should be equal to QP so it is behaving exactly like a notch filter that we desire. So this is the thing then alpha is equal to QP this particular thing S coefficient in enumerator gets cancelled so another condition with negative feedback okay so this is the negative feedback this the DC negative feedback is there so there is no DC path for this so there is a proper negative DC feedback for this circuit if you sum up here we get QA equal to QP into 1 plus alpha key QP equal to 1 by 3 and alpha is equal to 1 by 3 is therefore QA is 1 by 3 into 1 plus K by 3 so for any desired QA you can decide the value of K. So these are the letters therefore design for a notch filter at 50 hertz this is very one of the most important filters to be included in biomedical application to get rid of the power line interference 50 hertz QA better be high so that we get rid of for the notch what happens is we are getting rid of the 50 hertz exactly by making the Q area that is the 50 hertz right. So all the other signals come through without any harm so this is one or alpha in this case right with the passive thing with active thing it can be made equal to one. So QA is equal to 100 alpha is equal to 1 by 3 and evaluating K for Q of 100 you get K as 897 right because this factor of 3 makes this 300 okay that is therefore is nothing but that into 3 again is 900 minus 3 897 is the value of K R1 equal to R2 equal to 100 K chosen C1 and C2 become equal to 100 by pi nanofarads is 32 nanofarads approximately. So this makes use of an instrumentation amplifier which is available with TI INA126 or anything you can use okay and select R dash so as to make the gain K the required value in this case the gain has to be again made equal to 2K because we are using passive attenuator with 2 RS and 2 RS here this RS can be made small compared to RA so that effect of RS does not come into picture in the overall look. So this has been designed for QA equal to 100 RS has been made 0.5K and these resistances are 100K okay so alpha is made equal to 1 by 3 so if R1 is made very small this extra factor that comes modifying the transfer function is going to be close to 1 so we get minus K QP S square by omega P square plus 1 by this for this the advantage of this is obvious right this is what is simulated so what happens here is that the here K has been varied from 100 corresponding to K equal to 100 corresponds to Q of okay 1 by 3 plus 1 plus 100 by 3. So 11 also that is for this particular structure and K has been increased okay from 100 to 500 to 900 you can see how the Q makes it narrower and narrower. So actually the notch is nothing but 1 minus band pass okay so you can see that is higher the Q narrower the width is as far as the region of rejection is concerned so we want it to be very narrow so that it does not remove the useful signal content in the biomedical signal which may be within this range itself. So we want to get rid of only the power line frequency one factor of the becomes important you have to exactly locate this power line frequency it has to be stable and that value okay should be exactly located at the power line frequency that is another problem that will deal with tuning of filters later. Now what is the effect of the gain bandwidth product on such circuits this can be easily investigated we had seen earlier that when this factor got improved here the constant factor got improved by K alpha in this case right and K got reduced by K replaced by K into 1 minus 2 K into S by GB or 2 K plus 1 into S by GB that is the factor actually because buffer stage also it was there we had made it 2 K plus 2 okay the same factor comes on this side also here also 1 plus K into 1 minus 2 K plus 2 S by GB comes into picture as a replacement of K because of the phase error of due to the gain stage that is 2 K used in the network. So what happens this causes enhancement of Q okay it is reducing the coefficient of S this increases the coefficient of S in the case of high pass we had noted by the same amount this causes decrease in the coefficient of S they are exactly same and therefore they cancel one another the effect this effect gets cancelled with this effect okay as far as the coefficient of S is concerned okay. So this enhances Q this causes decrease of Q by the same amount so in compensation okay the Q becomes insensitive to the gain bandwidth product that is an important advantage of the circuit that uses notch filter for its Q enhancement basic passive notch filter for Q enhancement. So now let us consider how Q enhancement takes place using variety of basic filters in the loop like low pass high pass or notch has been discussed but are we going to always this design only that specific filter after Q enhancing no as far as denominator polynomial is concerned that is getting modified by putting a loop with a basic low pass or a basic high pass or a basic notch passive filter but the actual transfer function that we require for the second order that could be band pass okay not low pass or high pass so how do we alter this or notch so how do we alter the numerator polynomial okay retaining the denominator polynomial that we acquired okay that is the desired denominator polynomial that is an important feature of filter design that means the denominator polynomial should remain in general as this with omega naught and QP to actual okay whatever we want as far as the numerator polynomial is concerned for low pass it is some constant okay some constant K and for high pass it is that into some omega naught square by for notch it is this and for a band pass what is it going to be is going to be just S by omega naught. So this is for band pass so only the numerator polynomial has to change that is the characteristic of any network once you fix the dominant sort of that characteristic equation of the network you can change the numerator polynomial depending upon where you feed the input. So let us look at this this is the basic band pass structure using the same passive network R1 C1 R2 C2 now please understand that whether the resistance is here and capacitance is here whether the capacitance is here and the resistance here okay that means whether it is low pass high pass okay R band pass right this network characteristic does not change that means the denominator polynomial remains C okay for any transfer function from anywhere to anywhere okay in this network it will have that is from any point as input to any point as output okay you will have the transfer function as okay always something the numerator divided by S squared by omega P squared plus S by omega P QP plus 1 if it is an active network the same thing is valid if it has modified using low pass that is what we have done this is low pass and minus K okay then this comes from here output to this this you ground because it is an input voltage source so ground that so as far the feedback loop is concerned it is low pass minus K again low pass so it is the same one that has been discussed in the previous lecture right which is enhancing the constant okay and the constant instead of 1 will become 1 plus K and here that will be minus K okay as far as input is concerned instead of earlier we had fed this as 2R and 2R here and this has been grounded now you just connect the R back and input the source at this point where C was earlier grounded right so what happens now there is a SCR okay coming at the top so it from here to here in the numerator you have been able to modify the polynomial function so if you write down the polynomial for this right it will be exactly equal to minus K SCR that is a modification and the denominator remain same as before which we have discussed in the last class SCR squared that S by S squared by omega P squared S by omega P QP plus 1 plus K okay once again we have QA equal to QP into square root of 1 plus K so one root of 1 plus K by 3 and omega naught is equal to okay this into omega P square root of 1 plus K into omega P okay so this is equal to the Q okay so that is what we are going to now design that means we have converted our low pass Q enhancement topology into a band pass just by changing the input okay so let us look at that it has been designed like that okay band pass filter with QA equal to 5 and center frequency of 50 hertz QA equal to 5 square root of 1 plus K by 3 is equal to 5 so square root of 1 plus K is 15 K is equal to 224 225 minus 1 F naught is equal to 50 hertz 15 this is square root of 1 plus K okay by 2 pi RC is F naught R equal to 100 K C comes out as 0.477 micro farad gain at the center frequency therefore is nothing but this is KS by omega naught okay this is going to be divided by 1 plus K therefore let us see that so it is going to be K by 3 right so that is the gain at the center frequency at resonance okay so K QA by root of 1 plus K okay QA is nothing but QP into root of 1 plus K right so that is about 75 here okay so we get this experimentally it has got enhanced from 75 to 79.032 okay it is 75 because square root of 1 plus K is approximately 15 and QA is 5 15 into 5 75 but it has been boosted to 79 this is the effect of the gain bandwidth product this enhancement to 79 from the expected value of 75 now when it gain bandwidth product so QA divided by 1 minus K QA omega naught by GB is the enhancement okay here it is 2 instead of 2K it is K because we have not used that addition using 2K and attenuation by half we are directly using again stage K so the phase error is going to be just K plus 1 into omega by GB. So this that is why the shift has gone from 75 to 79 you can just roughly estimate the this factor K is in this case 224 QA is 5 and F naught is for which it has been designed is pass filter at 50 hertz 50 divided by 1000 GB is 10 to power 6 okay this is the factor okay which means 10 to power 6 this is going to be 10 to power minus 3 25 okay into so it will be about 0.0 or something right 25 into 250.05 also so that is the magnitude of this so to that extent it has been shifted as you can readily see. Now Q enhancement using positive feedback let us look at this V naught by VI now instead of minus K which was put in the loop okay you have now changed the polarity to plus K which means actually this particular N with minus K is negative okay which means overall product okay that is K into N okay in the negative feedback situation right is made negative. So we have NS by DS into K coming into picture in positive feedback so M is 0, N is 0, N is 1 so that is the situation. So N is equal to 1 okay M is equal to 0, P is equal to 0 we are starting with this basic what is that passive network so what is that passive network simply this is one such network RC it could be this it could be high pass first and low pass next it does not make any difference for the transfer function it remains the same as before this is exactly equivalent to this okay or there is a third network which is what is used in what is called wind bridge again it uses two resistors and two capacitors okay so you can just put actually this way low pass okay this is high pass this is low pass okay so this also is having the same characteristic so see the number of elements remain the same but the topology keeps changing right however the transfer function remains the same so whether it is this or this or this right the transfer function is SCR by 1 plus 3 SCR plus SCR square which is what we want to start with so using any one of these three networks okay we get the same transfer function that is desired star okay and now this one put in the feedback will make it NS by NS plus actually in this case it is NS by NS minus KDS this is what was shown NS KNS divided by DS minus KNS this is what it becomes and the resultant effect is the numerator remains the same as before KS by omega naught does the same polynomial NS but denominator has S square by omega naught square plus S by omega naught QP into 1 minus K into QP this is the effect of positive feedback plus 1 so this is what we get okay and omega naught happens to be remaining same as omega P in this case omega naught remains same as omega P remains unaltered equal to 1 by RC okay for any one of these networks okay and Q is nothing but original QP okay which is 1 by 3 divided by 1 minus K by 3 this is the Q active. So now if K by 3 is less than 1 as a K by 3 is 0.1 what happens the Q becomes 1 by 3 into 1 by 1 minus as a K by 3 is 0.9 let us make it so 1 by 1 minus 0.1 so it is 10 times 1 by 3 that means 3.33 that is the enhancement effect so K by 3 should be pretty close to 1 in order to make QA even just higher than 1 right so that is a technique of Q enhancement that may be preferred we will see later only for not very high Q values because for very high Q K by 3 has to be very close to 1 like 0.999 kind of thing okay so this sensitivity to K is what prevents this technique of Q enhancement being used for high Q circuits so that is Q sensitivity to K this is nothing but ratio of passive resistances used in the amplifier. Now actual network the amplifier K is realized by using R2 and R1 here negative feedback this is the non-inverting amplifier that we had already seen having a gain of 1 plus R2 over R1 that is the gain okay non-inverting amplifier has been used this is the band pass network as I told you we could use CR okay RC or R and C in series R and C in shunt okay anyone of those networks to get the same effect okay so then the transfer function is this actual circuit right let us try to realize a active Q of active RC filter Q of 5 and then this leads to we know okay that is 1 by 3 divided by 1 minus K by 3 is equal to 5 that is what we want okay this leads to a K of 2.8 see it is pretty close to 3 that is how we are able to get this value of 5 so C1 equal to Z equal to Z R1 equal to RT equal to R is presumed here okay omega P is equal to 1 over RC that is made now equal to 2 pi into 40 hertz let us say because we want the thing to be at 40 hertz band pass R equal to 100 K gives you C equal to 339.7 nanofarads so what happens now we have to have again resistive add here 2 R1 and 2 R1 from making it effectively R1 thevenin equivalent therefore the gain has to go to 2 K so we needed again of 2.8 so 1 plus R2 over R1 is made equal to okay 2.8 into 2 that is 5.6 RB by RA comes out as 4.6 so RA equal to 1 K you have RB equal to 4.6 K okay this RB and RA has to be altered here this is the RA this is RB so 1 plus RB by RA then gain becomes so R2 okay equal to R1 equal to R equal to 100 K C2 equal to C1 equal to 39.7 nanofarads so with that it has been simulated and you can see it is occurring at 40 hertz and it is exactly equal to 13.987 what is that this is the gain at the center frequency for this okay gain at the center frequency is going to be you can look at it this resonates at omega naught with this so this goes to 0 S squared by omega naught squared gets cancelled with 1 okay S equal to J omega omega equal to omega naught we are investigating so this becomes equal to 1 okay this also becomes equal to 1 so it is nothing but gain at the center frequency is nothing but K QP by 1 minus K QP this is the gain at the center frequency at F naught okay V naught over VI at F naught so that is equal to K into Q actual that you have designed it for K is 2.8 Q actual is 5 we have designed it for so this is 14 okay theoretically it comes out as 14 and you can note that it is going to be 13.983 so that is well with the theory there is no effect of gain bandwidth product seen on Q okay so because this is the gain at the center frequency directly proportional to Q that and the K so we see that there is no effect of gain bandwidth product on the Q at all in this positive feedback situation but it is very sensitive to K variation that is demonstrated okay this is the phase shift and phase shift becomes exactly 0 at the center frequency of whatever frequency 40 hertz that we have designed it for right this is 40 hertz 100, 990, 80, 70, 60, 50, 40 so this is exactly behaving as a pan pass filter now QA is QP by 1 minus K into QP 1 by 3 by 1 minus K by 3 now sensitivity of QA to K is what is of interest to us so what is sensitivity definition again this we are done earlier sensitivity of QA to K is nothing but delta QA by delta K differentiate this expression with respect to K and multiply by K divide by QA so this is what is defined as sensitivity okay so if you do that you will get this as QP K by 1 minus QP K this kind of thing we have evaluated for sensitivity to gain bandwidth product also here we are doing it for K which is the passive parameter that decides the Q okay passive device parameter right so that is ratio of resistances so QA into K that means it is directly proportional to the actual Q so K in this particular case is the amplifier gain 2.8 QA is 5 so it is about 15 itself right that means if 1% change occurs in K 15% change occurs in the Q that is something that you have to bear in mind as far as this kind of circuits are concerned this is illustrated by changing K okay that changing K from the original value of 4.6 that we have evaluated for Q of 5 has been done by changing it to 4.65 and then 4.7 okay that means only varying by about just 1% right so you can see that it has varied considerably the Q has increased considerably because the gain at the center frequency is directly proportional to the Q into K so it just goes up like anything goes up okay so from 4.6 to 4.65 to 4.7 variation is huge as far as Q is concerned so this particular technique of Q enhancement is to be cautiously used okay when poor tolerance passive components are available to us for design so let us just investigate the sensitivity of this positive feedback situation to gain bandwidth product the same method is adopted as before K is to be replaced by K into this one 1 minus the amplifier that we have used in the our actual practical network is 2K gain so 2K plus 1 into S by GB is the phase error so that phase error in the earlier case of low pass high pass and not was resulting directly in okay change of coefficient of S even though in the case of not it just got compensated okay because of increase and decrease by the same amount okay due to GB. So now in this particular case this 1 minus K 2K plus 1 by S by GB will be occurring within the coefficient of S which itself is small if for IQ okay so that with this error coming within this is going to make the error not in S coefficient it will be in S squared because this S gets multiplied by this S and ultimately let us write down the whole thing so S squared by omega naught squared plus S by omega naught QP into 1 minus QP into K into this factor GB plus 1 so what happens this becomes S squared by omega naught squared S by omega naught QP into 1 minus QP K as before it remains as before okay whereas we have now additional term as far as S squared is concerned S squared by omega naught okay QP QP get cancelled so that into K into 2K plus 1 okay into S by GB that means it becomes S cubed divided by GB this is the additional term that comes because of the finite gain banded product this S cubed by GB can be approximated as a coefficient of S squared changing you can put it as 1 plus okay so SS that is okay so SS cubed okay K omega naught by GB which is nothing but okay you can just say that S okay sorry S squared okay so that into K into 2K plus 1 by GB into omega naught that is the factor by which S squared by omega naught squared is going to change this is S squared not S cubed so it is going to change the coefficient of S squared by only this amount the coefficient of S squared changing by this amount it has no Q in it unlike the previous case right so if K into 2K plus 1 omega naught by GB okay is the only factor coming it is going to change the coefficient of S squared by omega naught squared that means the natural frequency is going to get shifted down okay by some amount but it is not changing the coefficient of S okay that means Q changes only marginal and that Q is going to slightly increase okay but there is no danger of okay it going into instability mainly because K itself is of the order of let us say 1 over QP QP being in this case 1 over 3 K is nearly equal to 3 so this factor is very small okay whereas in the previous cases K is of the order of QP in 1K Q actual in N1 case and QA squared in the other case right so we can actually neglect the sort of effect of the gain banded product on the performance of these filters okay finally we are going to show here that these are the observations that we have made based on all the demonstration that we have had all higher order filters require higher Q at higher frequencies as second order building blocks that I have demonstrated that building a high order filter which is maximally flat or elliptic or tibishev involves cascading okay let us say higher and higher Q's okay at higher and higher frequencies so this is the technique of design of any filter so we require higher Q filters at higher frequencies filters with high Q should not be designed using positive feedback that is the first conclusion that we just made okay because they have very poor passive parameter sensitivity positive feedback can be used for Q enhancement for designing low Q filters safely because they advantage of this is that they use low gain stages and therefore the effect of delay is not going to be enhancing the Q too much positive feedback permits independent adjustment of Q and F naught because the omega P remains the same as omega naught in positive feedback you can independently adjust Q and F naught whereas changing Q is by changing K and the same K is also determining F naught sensitivity of Q to passive parameter variation in filters using positive feedback is as high as Q itself is an important thing as far as the passive parameter is concerned okay the sensitivity of Q is as high as Q itself for passive parameter variations filters with high Q should be designed using only negative feedback this is what we have learnt okay because then the passive parameter Q sensitivity is always going to be less than 1 okay sensitivity of Q to passive parameter variation filters using negative feedback is always less than 1 because Q is either proportional to root of K which makes it have a sensitivity to K of half or proportional to K which makes it have a sensitivity to K of one so this is the typical nature of passive devices fixing the Q so in conclusion what we can say is negative feedback structures have good passive parameter sensitivities and they are somewhat sensitive to the gain measured product of the device so selection of the device is important in negative feedback okay for design of filters okay selecting the proper gain measured product for the device and as far as positive feedback circuit to enhance Q is concerned it has to be restricted to low Q designs for high frequencies it can be used the effect of gain measured product is not so dominant thank you very much.