 So, today the 29th lecture on state space filter will be covering frequency compensation of filters and tuning. Effect of gain bandwidth product on state space filter performance was the topic of last lecture. Queue enhancement occurs due to gain bandwidth product. This is an important thing. It makes the queue of the designed filter become sensitive to the gain bandwidth product variation and that is because of the phase lag error in the loop caused by the gain bandwidth product. So, we have here the actual designed queue of the filter queue A, queue active getting chain to a new value of queue because of the gain bandwidth product as queue A divided by 1 minus this delta phi minus delta phi is the lag error in the feedback loop. The feedback loop may in this case comprise of summing two integrators and a summing amplifier which is called the resonator block that is the primary feedback loop in the state space filter. A queue forming loop has another feedback coming from the band pass output here through another op amp it gets connected to this. So, this is not this is a secondary loop and the queue is high very little of feedback is coming from this secondary loop. Primary loop is just this. So, this particular loop has two integrators causing accumulative phase error here as well as here of S by GB and S by GB phase lag. This we had seen in the discussion of feedback amplifiers and integrators. So, its magnitude error is negligible. How about a magnitude error occurs because of the integrator an extent of omega naught by GB. So, whereas the summing amplifier has no magnitude error but only phase error and if it is just an inverting amplifier it would have been just 2S by GB lag minus 2S by GB whereas because of the additional summing action due to this resistance R it will be 3S by GB cumulative phase error here in the loop is going to be 5S by GB this is what we are going to show okay in the next lecture. Basic building blocks of active filters are second order negative feedback systems. So, second order active filter consists of amplifier and two integrators which have frequency dependence. So, basic building blocks themselves are actually amplifier or the summing amplifier with two integrators and each amplifier and integrator is formed by op amps in negative feedback as they themselves are most probably a first order or second order systems okay. So, if that is the case what is the effect of the increased order this order is theoretically a second order system that we are designing and with the help of three op amps each of which can increase it by one order or two orders right. That means it will be essentially either from second order going to chain to fifth order or may even chain to say 5 plus 3 eighth order. So, what is the effect of this increased order on the loop that is going to be discussed in terms of cumulative phase error in the loop. So, a non-inverting amplifier okay is going to be just this feedback this is R this is R by K. So, then the gain of this is 1 plus K ideal value okay V not over VI. However because of the op amp whose gain can be approximated by GB by S for example if it has just one dominant pole otherwise it will be GB by S okay with the second pole also coming into picture. So this itself is going to make this so called resistive feedback section become a second order and how to exploit the second order effect in making its Q equal to 1 over root 2 for maximally flat response of this amplifier maximally flat magnitude or making it suitable for switching application okay by making the Q equal to 1 so that there is just one peak was discussed earlier in frequency compensation of amplifiers. Today we are going to use these blocks okay some of these blocks as basic building blocks of a filter which is also a negative feedback system. So what happens because of the loop gain so ideal value changes from 1 plus K to 1 plus K divided by 1 plus 1 over loop gain always which is 1 plus K into S by GB okay. So this is the loop gain of this is GB by S into R by K divided by R plus R by K which is going to give us this as the error deviation from the ideal value that is coming as a phase error. Now if it is an inverting amplifier input is going to be just failure so the feedback remains the same so whether it is inverting or non inverting okay it does not make any difference as far as the phase error is concerned. Only thing is ideal gain now is minus K this divided by this minus K divided by 1 plus 1 plus K into S by GB as the loop gain remains the same as before. So in the case of this is the loop gain GB by S into 1 by 1 plus K so phase error in the amplifier is 1 plus K S equal to j omega omega equal to omega naught if you put actually tan inverse okay this which is for small phase angles is by itself. So this is the phase tan inverse minus tan inverse okay 1 plus K omega by GB at omega equal to omega naught it is this for small phase angle this is roughly equal to minus 1 plus K into omega naught K. So we are evaluating it at the highest frequency usable in the bandwidth so that is omega naught. So integrator that is the normalizing frequency is omega naught minus omega naught by S so K is replaced by omega naught by S in this case so what happens this K is replaced by omega naught by S so it has a magnitude error this is the magnitude error this was not there in the resistive feedback session this magnitude error is independent of frequency it is omega naught is 1 over CR so it is 1 over CR GB it is fixed however this can be neglected compared to 1 in most of the cases it only changes omega naught to omega naught dash. So it contributes to a phase error of S by GB so effectively it is minus omega naught by S which is integration 90 degree phase shaped deviating by this amount as phase error. So phase error in the integrator is minus omega naught by GB so the cumulative phase error of the loop that is formed by this integrator integrator inverter is going to be 1 by 1 plus S by GB square which means it is going to add as 2 S by GB due to this 2 S by GB okay so this becomes 1 plus 2 S by GB approximately and this is 1 plus 3 S by GB due to the summing amplifier which is having this kind of combination. So this is R this is R this is R so what happens here because of this R and R this is R by 2 so K is going to be 2 so we have 2 plus 1 3 S by GB as the contribution of phase error due to this so 2 S plus 3 S is 5 S by GB so lag so minus 5 omega naught by GB is the phase error and this is what is going to cause as problem the Q therefore of the filter built using such uncompensated integrators and summing amplifier will result in Q actual divided by 1 minus 5 omega naught Q actual by GB. So this was the deviation okay as the frequency omega naught 1 over RC is change to higher and higher values this error keeps on increasing and it becomes close to 1 it is going to become infinity and the whole system is going to be unstable. So it is going to oscillate at its natural frequency omega naught at that point of time so this is what we had seen demonstrated in the earlier lectures. Now this has to be compensated how do we compensate so since the summing amplifier for example uses only a single stage it is contributing to a maximum phase error of minus 3 omega naught by GB and how to compensate for this phase error main reason for phase error is because of the loop gain being equal to GB by S into this attenuation factor okay which is 1 over 3 okay coming in the loop. So it is because it is a first order system so it is necessary to convert this into a second order system so that the phase error can be compensated. So we had seen this happening okay even in amplifier systems that it is better to use instead of a first order system in feedback a second order system in feedback where you can optimize the performance of the so called second order feedback system. So in control some terminology these are called PID controllers right in negative feedback. Similarly here therefore instead of using just one op amp with the dominant pole you can have op amp with the dominant pole assisted by a close okay second pole or you can make the second pole come into existence by using another op amp with the dominant pole compensation in negative feedback put in the loop in cascade. So this GB by S is going to be now replaced by GB by S of this okay and this is a buffer stage for example I have designed here so this resistance comes into picture only for the feedback as well the ideal operation is concerned this is nothing but then a later so this resistance is of no consequence okay as well the input is concerned now current flows through it and this resistance also has no effect as far as the input is concerned so this input gets transferred to this here input and this remains same as V not equal to VI so this is a buffer stage ideally however because of the feedback arrangement here right so this is going to have a face error introduced how much face error is introduced here okay so this is a buffer stage so it is one divided by one plus okay now the feedback is okay again 3 S divided by GB GB by S of this okay in feedback loop this is grounded so we have again 1 over 3 feedback so this can be adjusted to anything you want right so let us say you have this as R by N then this becomes okay N plus 1 from 3 it changes to N plus 1 just like K plus 1 N plus 1 so you can adjust this depending upon what we want let us see why we are introducing this kind of thing so we have GB by S as the loop gain modified by this so again 1 over loop gain so this summing amplifier just take any one of these inputs is ideally speaking minus 1 okay either this one or that one okay feeding through this or feeding to that divided by 1 plus okay 1 over loop gain of this this is nothing but 1 over this which is S by GB into 1 plus 3 S by GB into its own feedback which is going to cause this to come as K plus 1 here which is 3 so effectively the this particular thing from here to here the transfer function V naught to less a VI is going to be this so that is single stage summing amplifier this is the case of buffer stage gain that we had written earlier okay 1 by 1 plus okay and if you use a single stage summing amplifier the error would have been minus 1 by 1 plus 3 S by GB now if you use the composite stage it is becoming V naught to VI minus 1 plus 3 S by GB okay let us see where you are taking the output please look at this you are taking the output if you take the output here okay it is just this okay and if you take the output at V naught 1 by VI it is going to be multiplied by this from here to here it is 1 by 1 plus 3 S by GB from here to here transfer function is 1 plus 3 S by GB multiplied so this whole thing gets multiplied by 1 plus 3 S by GB this is V naught 1 so that is what is happening here you get the thing as okay V naught 1 by VI as minus 1 plus 3 S by GB 1 plus 3 S by GB plus 9 S square by GB square now phase error of this output V naught 1 by VI is very nearly 0 it becomes nearly equal to minus 1 now you can get any number here as V naught 1 by VI if you only change this for example this whole thing instead of 3 you make it N okay so it becomes N plus 1 so this multiplying factor becomes 1 by 1 plus N plus 1 S by GB so this factor here becomes N plus 1 so and if you take the output at V naught 1 okay at that time so let us write down what happens at that time so this will become this transfer function will become that as N 1 plus N plus 1 S by GB divided by 1 plus 3 S by GB plus okay N plus 1 into 3 S square by GB square so this way by making N for example equal to let us say you want to get a lead error of 1 plus 3 S by GB so you can make it become equal to let us say 6 that means N equal to 5 that corresponds to a lead phase error of 1 plus 3 S by GB okay for this inverting stage so you can therefore use it for compensating for additional phase cost in the loop by the integrators so this is a technique of making it work with perfect compensation for the phase lag error that is introduced okay that this particular stage which is a combination of buffer with a normal op amp acting as a summing amplifier okay this can give zero phase error and positive phase or lead error to compensate for the lag error that occurs in the previous stages. Now same thing can be done by using a composite integrator which is formed by using GB by S cascaded with a buffer for example so this will be 1 by 1 plus S by GB 1 by 1 plus S by GB its own loop gain of GB by S causing this to change to this so effective loop gain of this composite structure is GB by S okay into 1 by 1 plus S by GB so when this is put in an overall feedback loop what do you get you get the ideal value which is say 1 by SCR okay minus 1 by SCR from here to here assuming that this is an ideal of amp then divided by 1 plus 1 plus 1 by SCR into 1 over loop gain that 1 over loop gain is nothing but S by GB into 1 plus S by GB so 1 over SCR being equal to omega naught we can rewrite this as minus omega naught by S divided by approximately 1 plus now this factor 1 into S by GB plus S square by GB that is all GB square you can ignore the other factor because it will be SS gets cancelled omega naught by GB into 1 plus S by GB so omega naught plus omega naught by GB into 1 plus S by GB is ignored compared with these factors so this is approximately the transfer function from here to here if you take the output here now therefore V naught 1 by VI therefore is going to be multiplied by this that is minus omega naught by S into 1 plus S by GB divided by this which is exactly compensating for the phase here so that is approximately equal to minus without any phase error so this is compensated integrator frequency compensated integrated without any phase error so we can use integrators without phase errors by connecting these similar integrators together and assuming amplifier without any phase error forming a feedback system there which does not have any phase error which means Q enhancement is stopped of course you will see that instead of using three op amps it is using double the number of these op amps okay but improved performance in fact instead of compensating for the double integrator look like that one can have integrator with phase error cascaded to another integrator with lead of the same value as the lag error of the first integrator that forms what is called ackaberg MOSFET circuit which actually uses only three op amps and gets the Q enhancement suppressed okay so this is what we are going to show okay so this is something that we have understood now this is the integrator with perfect compensation because of the numerator having the lead error same as that of the lag error in the denominator so how to obtain this ackaberg MOSFET circuit let us consider an integrator with inverter which is part of the summing amplifier that is coming into picture in our loop that we had already seen this is the summing amplifier that was added so that resistance R also is part of the whole thing so how to convert this into again a non-inverting integrator so you put this in an overall feedback loop with this becoming positive and negative instead of negative and positive and the inverter put in the feedback loop now so this is a negative feedback so this is an ideal integration now VI V0 okay which we have already derived and this is the summing amplifier okay which has a transfer function now of minus 1 by 1 plus 3S by GB you can see because it is GB by yes here in feedback loop okay 1 by 3 so it is having minus 1 by 1 plus 3S by GB as the transfer function that multiplied by GB by yes is going to modify this feedback performance as again what we have already written that similar to this except that okay we have this hmm so it is going to be modifying this let us write down the transfer function so it is minus V0 by VI from here to here V0 to is going to be minus omega naught by S that divided by the denominator is going to be 1 plus 1 over loop gain so the loop gain is going to be 1 plus that is going to be always there by S 1 plus K here this loop gain is going to make it come as GB by yes into 1 by okay 1 plus 3S by GB this 3 can be again change to something like N plus 1 okay if you need okay so this brings about denominator which is equal to minus omega naught by S divided by 1 plus once again we have the same thing 1 plus omega naught by S similar to what we wrote earlier S by GB into 1 plus okay let us put it as N plus 1 S by GB so instead of 3 let us replace it by N plus 1 so you can have you know any number of these resistances coming here so they will keep increasing the error okay why we are doing this is becoming obvious to you now this results in similar stuff that we had written earlier if you take the output here now it will be just getting multiplied by this factor of 1 plus so minus omega naught by S that is V naught 1 by VI into 1 plus okay N plus 1 S by GB divided by 1 plus okay S by GB plus N plus 1 S square by GB square okay so this is what we get here as the phase error so we get this as just is it becomes a now at this point it becomes a non-inverting integrated so this actually is not right this is going to be 3 S square by GB square so N plus 1 is 3 so this is the way we have to modify these integrators in order to use them for frequency compensation is what we have done here integrator with phase error okay because it is using a single stage here we have used composite stage for the other integrator okay so that the phase error here for this integrator if you take the output here for forming the loop is going to be just one S by GB lead there is one S by GB lag compensating perfectly the overall loop phase error thereby preventing the peaking from increasing above H naught into Q or increasing the Q and hunvent is suppressed this is what is called akabat MOS per circuit this is demonstrated very clearly by using uncompensated op amp that means the Q enhancement has occurred what in the simulation what I have done is I have used 2 ideal integrators followed by a summing amplifier which is an op amp based one with 741 as the op amp or summing amplifier so it gives you Q enhancement of because it is going to be 3 omega naught by GB so Q is going to get enhanced to this Q originally used to 5 so that gets enhanced as frequency is progressively increase you can see this similarly the thing is also proportional this error is also going to increase the transmission at the notch because directly proportional to the same thing Q 3 omega naught Q A by GB as far as the notch transmission is concerned from 0 is going to be keeping on increasing as omega naught is increased now with this particular phase error getting compensated that means I am now using the output here where I will adjust the total phase error cumulative phase error that is why this R has been additionally put here so that okay you get here this as 2S by GB this as 3S by GB so it only gives you a phase error of S by GB lead which is compensating for the S by GB lag error with that kind of thing okay I have now used the summing amplifier with zero phase error because I have used 2 integrators which are ideal so that arrangement as mentioned earlier where I have adjusted so that the numerator phase error is same as the denominator phase error for the summing amplifier then I get no Q enhancement at all as the frequency is change over the same range by the control voltage of this voltage control filter so you can see this effect even the notch output remains close to zero okay which is theoretically expected one now switched capacitor filters any filter using LC or RC has its pole frequency directly proportional to pole frequency is proportional to 1 over RC it is inversely proportional to R into C in case of LC it is 1 over root LC so the sensitivity of omega naught to R or C is about minus 1 here this is half here to these components it is okay but tolerance of components definitely has a great influence on this fixing omega naught okay precisely depends upon the tolerance of R into C or L into C tolerance of common as great influence on the accuracy with which it is fixed resistors and capacitors have poor tolerance and large temperature coefficients in integrated circuits whereas ratio of resistors and capacitors are very good tolerance an order of magnitude better than the absolute values so we are not able to design precision filters if you make use of the RC's in monolithic IC's so what to do switched capacitor gives a solution to this what is it let us consider the integrator the basic building block of filters VI is driving a current of VI by R1 into the integrator which is charging the capacitor C1 okay and that is how V naught is equal to 1 over C1 integral IC DT IC is VI by R1 so we have the time constant 1 minus 1 over C1 R1 deciding the integration VI by DT okay so this parameter is of great importance for precision what is done in switched capacitor is that resistance R1 which is troubling us is replaced by a switched capacitor this capacitor receives a charge of C into VI that is the charge received okay in the first clock phase it is connected to the input to receive this charge and that capacitor charges that changes or switches over to the virtual ground in the next phase this is the first phase of the clock the switch connects to input the capacitor to input then the capacitor is connected to the virtual ground in the second phase of the clock within a time period T so what happens the entire charge of this Q is transferred to C1 that is the purpose of the exercise the entire charge that has been accumulated in C is changed to C1 okay instantaneously so the charge transferred per unit time is current that means this is doing the job of an equivalent resistance okay which was earlier put as VI by R equivalent okay so R1 so switched capacitor realizes R equivalent equal to T by C this is the basic principle of switched capacitor network the switching frequency is chosen to be so high okay it should be a much greater than twice F max of the signal that you are processing you can consider therefore that the signal is remaining static okay without any alteration with respect to time during the process of entire charge transfer so these are the two phases of the clock so non-overlapping clock 5 1 during which VI is connected to C 5 2 in which this charge is transferred to the other capacitor so the R equivalent is T by C or omega naught is 1 over R1 C1 is C by C1 into 1 over T C by C1 into the clock frequency that is an important thing so omega naught now becomes ratio dependent ratio of capacitors can be fairly accurately fixed in integrated circuits absolute value of the capacitor may be as bad as 20% 30% tolerance whereas ratio tolerance is of the order of 1 to 2% so there is the trick okay to adopt these RC filters to switch the capacitor versions which can be easily giving precision in integrated circuits for filters so this has the program in facility that once you use clock you can change the clock thereby change the omega naught or tune the omega naught to whatever frequency you want programmable omega naught because omega naught is the normalizing frequency of the filter so the characteristic of the filter remains unaltered Q is what determines the characteristic of the filter and the frequency omega naught can be conveniently varied without Q getting altered okay in the case of our by quad ratio of capacitors have tolerance on order of magnitude better than okay the absolute values temperature coefficient of capacitance is very close to zero so however switches introduce switching noise in the entire system this requires additional band limiting prefilters and smoothening post filters the supply voltage scale getting scaled on the switches themselves are pretty leaky these days and therefore for low voltage application these switcher capacitors are really not dissolution so by part base switcher capacitor filter is MF 10 this is what is manufactured by TI cost of this is only $2 for 1 kilo units center frequency can be varied from 2 hertz to 20 hertz clock frequency FC it can be varied from 10 hertz to 1 mega hertz so F naught to FC ratio is 1 is to 50 you can also therefore make the approximation without any qualms gain bandwidth product of the up to 2.5 mega hertz this is important so we have already said that F naught into 2 product is a important criteria in the selection of filters okay key parameters of slew rate also is an important that has to be considered 7 volts per microsecond what does it do the sine wave is going to get it started around the zero crossover because for a sine wave the slope is maximum at zero crossover so it is slope is equal to omega into A so omega into A is the peak of expected output it should be less than slew rate for no distortion so please bear in mind this depending upon what the peak value of expected output is the maximum frequency is also fixed based on this slew rate limitation so this is normally not taken into consideration in most of the design which results in the distortion okay appearing in filters apart from the fundamental limitation due to the gain bandwidth product which causes Q enhancement which needs frequency compensation technique to be adopted okay to work close to the gain bandwidth product work up to the gain bandwidth product then what happens is the slew rate limitation need to be taken care of so one should be aware of the limitation in amplitude of the output of a filter based on this tuning of filters need for tuning now apart from switching capacitor right technique there is no other technique of making these filters become precise in active RC other than physically tuning the filters so how do we tune this filters with the advantage of DSPs and microcontrollers becoming useful now in electronic systems today most of the analog circuits okay become precise only by letting them get tuned so whether it is amplifier design or filter design okay tuning of these for precise performance parameters to be obtained is a common technique this is digitally assisted analog circuits are the things of future and today need for tuning key parameters characterizing filters are Q which is a ratio H naught is a ratio no problem F naught is the problem okay normalizing frequency is dependent upon absolute values of R and C okay so if you want a precision filter you have to have precision R senses that is not possible in integrated circuits okay so what do we do tuning of filters is what is to be done that means we have to have an arrangement where a reference can be used for tuning the filter precisely to whatever frequency you want and then this filter can be used okay thereafter for actual application digitally programmable analog reconfigurable front end and back end filters or amplifiers are the ones that are currently in both voltage control filter we know that integrator replaced by integrator okay with multiplier making it voltage control right so what happens in this case is if this is the control voltage this is VI earlier this VI or V naught was driving a current of V naught by R now because of the multiplier introduced it is V naught VC by let us say VR which is 10 for precision multipliers standard value of 10 mass or IC multipliers it can be just 1 volt okay so if you use a multiplier like this then the current in this is V naught VC by 10 R that means the R can be replaced by 10 R by VC that is a omega naught which was 1 over RC now becomes omega naught equal to VC by 10 RC so it is directly proportional to VC control voltage now such a VCF is put inside let us say it may be the double integrator loop so this is the control voltage terminal which is controlling the time constant of the two integrator simultaneously like we had shown it earlier so if you use such a filter and use a summing amplifier to sum up HP this is what is called HP high pass filter this is band pass this is low pass these three outputs are available so you can selectively take some of HP and some of BP and some of LP and add then the zero of the entire second order system can be located anywhere on the explain so this is a general voltage control filter not only the omega naught is directly proportional to VC the alpha 1 alpha 2 alpha 3 selection will make it become available as any second order system okay the queue and omega naught can be fixed okay independently alpha 1 equal to H naught alpha 2 equal to 0 alpha 3 equal to 0 it is high pass alpha 1 equal to 0 alpha 2 equal to H minus H naught alpha 3 equal to 0 it is band pass alpha 1 equal to 0 alpha 2 equal to 0 alpha 3 equal to H naught is low pass these are the three standard outputs available apart from that alpha 1 equal to H naught alpha 2 equal to 0 alpha 3 equal to H naught is also available in the case if it is not available you can have the band stop and make it get located at any point as the 0 okay so by having H naught this different from H naught alpha 1 is equal to H naught alpha 2 is equal to minus H naught by Q alpha 3 is equal to H naught it is what is called an all pass an all pass filter actually passes all the frequencies with the same magnitude that is magnitude is independent of frequency only the phase changes right first order all pass for example is 1 minus S by omega naught by 1 plus S by omega naught second order all pass is 1 minus S by omega naught Q plus S squared by omega naught square by 1 plus S by omega naught Q plus S squared by omega naught square. So these coefficient of these all polynomials of S keep changing in the numerator or in terms of poles and zeros okay they are mirror images. So that is the standard design procedure which can be done by this basic building block. Now voltage control filters are also useful in music synthesizers okay voltage controlled oscillators voltage control filters and voltage controlled amplifiers from the core of most of the music synthesizers. Now manual tuning of second order filter so we have to tune this exactly to a specific frequency omega naught that omega naught let us say is a reference frequency VP sign omega reference T is given as input to such a filter first and we have the alternative of okay using the magnitude information or the phase information we would prefer to use the phase information obviously for example if it is a sort of band pass filter how will you tune it exactly to desirable center frequency one things that this is the technique of maximizing this but maxima and minima by definition okay there is no variation of magnitude okay with respect to frequency delta T by delta omega is 0 for maxima or minima so they should not be used mathematically as sensitive measure of whatever you want to tune to. So what changes maximum is the phase for a band pass or high pass or low pass we see that phase at omega naught okay variation delta phi by delta omega we have shown this earlier is directly proportional to Q. So this we are shown earlier in the second order system delta phi by delta omega naught okay delta omega at omega equal to omega naught is minus to Q by omega naught. So this we have demonstrated for the second order high Q system also that as Q varies the phase changes rapidly around omega naught that information is used so that this becomes a phase detector now. So multiplier analog multiplier as a phase detector this is getting VP sign omega T as the input and this is VP dash sin omega T plus phi. So what happens to this phase detector so V average is the multiplier output so we have here VP sin omega T this is VP dash sin omega reference T plus phi so output of this is VP dash by 10 okay sin omega reference T into sin omega reference T plus phi which is cos phi minus cos 2 omega reference T plus phi. So this gets eliminated by the slow pass filter so the cut off frequency of this is so chosen okay so it is let us say much lower than omega twice omega reference so it removes that and only the DC gets selected and DC average right is nothing but VP dash by 20 this becomes 20 okay cos phi that is what we have written here so V average which is VP VP dash by 20 cos phi and that is the characteristic that is drawn this is how phase varies okay average okay varies with phase. So this green color one is nothing but the sensitivity of the phase detector which is delta V average by delta phi so that is assigned this is going to be therefore delta V average by delta phi which is the sensitivity factor of the phase detector this VP VP dash by 20 sin phi so minus so that is how you get this and it is maximum at phi equal to pi by 2. So that information is used only at this point it can be used as a phase detector very efficiently because the sensitivity is maximum at this point sensitivity is 0 that is 0 phase shift okay and at pi also it is 0 so one tends to use the low pass and high pass output which have a phase shift at omega not equal to pi by 2 rather than a band pass which has a phase shift of 0 or 180 so that in sensitivity of V average the changes in phase shift is maximum at y equal to pi by 2 that average is 0 so we keep on changing the control voltage for a fixed omega reference here keep on changing the way control voltage until this average goes to 0 if this is low pass or high pass low pass or high pass that is how the thing has been built now high pass filter with omega reference F reference at 1 kilo hertz R is equal to 1 kilo ohm C is equal to 0.1 micro farad in the VCF F naught becomes 1.592 VC by 10 into 10 to power 3 hertz the low pass filter has 100K and 1 micro farad to get rid of the 2 omega reference when V average is 0 VC is found to be exactly C.6.3 so VC was varied parametrically here in the simulation software and one could see that this is the low pass filter output for this value of VC which is precisely 6.3 volts it is going to be perfectly tuned to the frequency 1 kilo hertz so this is the band pass output just shown that for 1 volt input with a Q of 5 H naught of 1 it remained at nearly 5 volts for all these roughly so it is insensitive so the peak is insensitive it could have been tuned to this this this or this whereas in phase detector it is precisely getting tuned to this point so this is the advantage of using phase detector which is the best method of tuning and it is demonstrating that magnitude as almost remained close to the peak okay for all these control voltages which are varying around small DC of micro volts right. So multiplying DAC use as a multiplier so you can use instead of analog multiplier DAC as a multiplier so V naught is in situation V reference into the N bit analog output okay based on the switch position okay BI can take 1 or 0 okay so 2 to power – I is equal to 0 to N – 1 in N bit converter so V reference is the variable so VX okay this is VY so this DAC 12 bit DAC 7821 cost about 3.15 dollars okay can be used in place of the analog multiplier and then it becomes a digitally controllable okay tuning of the filter. So in conclusion we have seen that tuning is an important part of the analog systems okay which facilitate digital tuning or DSP control or micro controller based control of analog systems okay we have demonstrated the efficacy of phase detector for tuning all these filters we have also shown how frequency compensation can be letting us use the filter closer to the gain bandwidth product than what was possible earlier because F naught into Q should be earlier without compensation should be made much less than the gain bandwidth product.