 Today we are on to lecture 30 connected with automatic tuning of filters and review of filter design. So this is the last lecture on filters review. We had in the last lecture talked about frequency compensation. Now that frequency compensation for the double integrator loop can be done in several ways. One is the non-ideal effect of the op-amp that is finite gain vented product effecting the integrators and the summing amplifier can be compensated for by changing the summing amplifier to a modified structure so that the delay cost by the integrators and the summing amplifier can be compensated for by the modification of the summing amplifier. Let us see how it is done. We have the summing amplifier which is normally made up of a single op-amp with two resistors one for the double integrator loop or resonator block and other for the queue forming loop. So that this is the summation. You might as you improve increase the order you can have more summing okay resulting in modification of the summing amplifier which means actually there can be depending upon the order R by M okay such effective resistance okay R R R M such resistance resulting in effective R by M. Now how to compensate for the delay error caused by this. If it is just a first order system with this feedback okay directly from here then the loop gain is GB by S okay and then this resistance R by M plus R by M divided by R by M plus R is the feedback factor along with the GB by S forms the loop gain. So that results in the sort of amplifier having a phase error if it is a sort of ideal summing for this loop it is minus 1 and for this loop also again minus 1 getting added adding these two voltages it will be 1 plus okay 1 plus M into S by GB. So resulting in a lag okay additional lag error of 1 plus M G Omega by GB okay causing angle phase error corresponding to Delta Phi which is equal to minus 1 plus M evaluated it Omega naught the extreme frequency end of the band of usefulness. So that is the lag error caused by the op amp gain banded product. So now modifying that using that op amp GB by S and another as a buffer stage which is going to be 1 by 1 plus 1 over loop gain of that which is GB by S of this okay and 1 by 3 here. So if it is R by M that will be resulting in 1 by 1 plus N plus 1 S by GB. So this is the general thing. So when you combine this together to form a feedback pair then we have the general summing change into minus 1 divided by 1 plus 1 over loop gain which is GB by S into this okay. So 1 plus M which corresponds to this feedback okay into GB by S so it will be S by GB into 1 plus 1 plus N S by GB. So it becomes a second order system okay with a transfer function of this type okay from here to here if you take the output here now this will be multiplied by 1 plus okay from here to here it is 1 by 1 plus N plus 1 S by GB. So here it will be N plus 1 S by GB. So both the numerator and the denominator have the error cropping up which is dependent on GB. So here it is 1 plus N S by GB here it is 1 plus M S by GB the other one is the second order factor. So if you make N equal to M in this case we have made it equal to 2 so it is perfectly compensated for phase okay N equal to M. N greater than M you have a lead error. So whatever lag is there due to the integrators can be compensated by appropriate league error lead error in this combination. So this is a nice technique for compensating for any complex feedback loop involving feedback amplifiers okay. Then we went over to switched capacitor filters and these switched capacitor filters okay wherein the resistance of the integrator is replaced by okay a capacitor okay. So in the following manner connected to the input for some time in the first phase and connected to the virtual ground transferring the chart to C1 resulted in an R equivalent which was T time of the clock time period of the clock okay divided by C. So this way the RC time constant of the integrator okay 0 over 2 okay T by C into C1 you can ratio of capacitors and the clock frequency 1 over T. So that way it became programmable however switching is one of the thing that introduces noise in the system again which requires further filtering by analog filters. So we are now suggesting how analog filters themselves can be tuned fairly accurately by a tuning manual tuning of the second order that is this was also mentioned earlier. So we have the voltage control filter that has been designed by transforming the integrator into multiplier in combination with the integrator so it became voltage controllable. So the time constant became VC by 10 RC that is the normalizing frequency of the second order filter. So now how to tune this the exercise is that we have BP sign omega reference T coming to the input of this and you are permitted to change VC so as to make this filter get exactly tuned to the omega reference that means omega naught has to become equal to omega reference by changing VC that can be tested out using a face detector because we had shown that if you take either the low pass or high pass output okay you have 90 degree phase shift occurring at omega equal to omega naught okay if incoming frequency omega is omega reference then at exactly omega naught equal to omega reference the phase shift between low pass output or high pass output and input is going to be pi by 2. So this phase shift of pi by 2 can be tested by the multiplier which is going to give at this point V average which is VP sign omega reference T and this will be VP dash sign omega reference T plus 5 that means this will be VP dash by 20 cos 5. So 5 equal to 5 by 2 this average is going to 0 that is a measure of tuning that was what was covered in the last lecture also and we had seen how in simulation we could accurately adjust this voltage to go to 0 thereby making sure that omega naught is exactly equal to omega reference. So this mode of thing is going to be automatized now this K VCF now is nothing but sensitivity of the VCF what is it that is the change in phase at this point okay delta 5 by delta VC is defined as sensitivity of voltage control filter how the face changes here of this filter as VC changes okay. So this is nothing but delta 5 by delta omega naught into delta omega naught by delta VC omega naught depends upon VC okay in this following manner it may be dependent upon in any functional manner we do not really care it need not be necessarily linear. So at around that operating point VCQ we can find out okay how delta omega naught varies with delta that is what is called K VCF okay. So that depends upon how phase changes with frequency okay omega naught how phase changes with frequency omega naught please remember that omega reference the incoming frequency here is fixed okay. So we have to now find out how phase changes with omega naught on omega naught changes with respect to VC in the following manner. So delta omega naught by delta VC is 1 by 10 RC it is linear otherwise you have to take the particular mathematical expression to derive the value of the slope around the omega naught. And KPD is nothing but it is delta V average by delta 5 it is input is phase change and it is output is the average voltage. So delta V average by delta 5 is called KPD sensitivity factor of the phase detector okay. So these parameters are important okay in situation when the whole thing is put in a loop in automatic frequency control. So let us see what that is that is nothing but what is called as phase lock loop let us see what it is. So here the filter now is designed okay such that the multiplier makes it voltage control. We have point 8R directly going to the integrator and an R going through the multiplier and coming to the input of the integrator. So effective resistance of the integrator is now that is what is determined by this point 8R shunted by what is determined by the control voltage okay. So this equivalent resistance here is this VI which was directly coming here okay is now modified by this factor of VC by 10. So this multiplier is VX VY by 10 it is reference voltage is 10 volts. So we have this resistance changing as this resistance is change okay in terms of VC the following manner. So the current is now going to change from earlier VI by R to VI this is VI okay or V naught okay VC by 10. So the current is instead of V naught by R becomes V naught VC by 10R. R is replaced here by 10R by VC. So effective resistance of the integrator is 10R by VC in parallel with point 8R. So VC for this is permitted to change from plus 10 volts to minus 10 volts that is the range within which the multiplier functions right. So when it is plus 10 volts it is point 8R in parallel with R when it is minus 10 volts it is point 8R in parallel with minus 10R both are resulting in effective positive resistance. So one case is point 8R parallel R that is one limit that is the highest that is lowest value of resistance the highest value of resistance is point 8R parallel okay minus R when VC takes minus 10 volts. So point 8R parallel R is roughly equal to R by 2 okay and point 8R parallel minus R is point 8R into R okay divided by point 2R. So it is changing by about 4 times R right. So the range of R is going to be changing when VC changes from minus 10 to plus 10 volts R is going to change from okay 4R to almost point 8R parallel R. So that is it it is about okay point 4R it is about a 10 fold variation of resistance occurs because of this arrangement. So let us remember this for calculating what is called the lock range of this PLL it is going to be locked now to 90 degrees what how does it happen this multiplier phase detector this is the phase detector the output is now okay being fed to an integrator which is a low pass filter okay this R dash with Miller capacitor of C into a naught is what is low pass filtering actually this is an ideal integrator now so this is nothing but a PID control okay where this voltage okay which is the V reference in this case it is 0. So the average is compared to V reference which fixes of the phase V reference here is 0 because we want this voltage to go to 0 average right. So when the control works as negative feedback satisfactorily that becomes negative feedback only when we take the output from low pass not from high pass it becomes high what is a positive feedback if you take the output from high pass. So the sign change is important here is the low pass filter output that is compared with input fed to the phase detector which is formed by the analog player and then the integrator and then this control voltage is applied here now this becomes if it is negative feedback it is negative feedback it gives us the automatic control. V for this is taken as 0.1 micro pad R is 1 key Q is 5 H naught is 1 so Omega reference changes from direct control by VC to slightly indirect control because of this parallel combination of resistances right. So this kind of thing has been purposely done in order to make sure that wherever it is staying as far as the control voltage of the multiplier come integrator output is concerned as far as this output is concerned it is capable of going only to the saturation value of this op amp which hopefully is same as that of the multiplier input limit which is plus minus 10 volts. So in which case this is the one that sets the limit of lock range for the entire setup okay lock range is the range of frequencies of these input over which this locking of phase to 90 degree is going to be working out satisfactorily with loop gain much greater than 1 so that the error is always close to 0. So this kind of system level discussion of this phase follower has already been done by us when we discussed most of the analog systems so it is nothing but a phase follower which is also nothing but a voltage follower. This V reference can change the phase at which it is getting locked in this case it is locked to pi by 2. So this is actually summarizing the phase lock loop here we have the voltage controlled filter here this is our input okay which is VP sign omega reference T and this is the control voltage coming from the integrator output okay this is the phase detector. So this is connected to reference equal to 0 so that this voltage goes to 0 when the control starts functioning okay. So KVCF is the sensitivity of conversion of this phase difference to sort of this input okay to phase difference okay the input is actually constant here and it is VC that is changing so this VC change will change the omega naught of this okay such that the low pass filter output phase becomes exactly equal to pi by 2 in order to make this voltage become equal to 0. This is the basic principle of the phase lock loop. If you connect VP VP dash cos phi equal to a specific value V reference right the cos phi gets adjusted to that okay. Now this is now having time wave input as far as this input is concerned that frequency has been adjusted to be 1 kHz input and 1 volt magnitude. So let us see what happens this is the time of connecting the circuit in simulation. So once is this is the band pass output okay and this is the notch output if it is getting tuned to the incoming frequency and it is a sine wave that frequency must be eliminated in the notch and it should be becoming equal to H naught into Q at the band pass output. So you can quickly see how the control voltage is now changing from its constant value whatever it is. Why at what value of constant it is there we can see it must be the natural voltage at which the control loop is not functioning or not getting any input okay omega reference. At that point of time the integrator is not having any input okay either from EI or from the output of the low pass in which case integrator output is solely determined by the offset of the op amp which in this case may be about 1 to 2 millivolts and that model has put it at that value and this goes to saturation because it is having high DC gain it just goes to saturation of plus 10 volts. So from 10 volts it is starting and getting control automatically and settling down at the value at which okay the input of the integrator goes to 0. One end of the integrator V plus is connect that is plus input is connected to ground. So the minus input also goes to 0 when this reaches VC reaches the value of control voltage required to make omega naught of the filter come exactly equal to omega reference and at that point the band pass output goes to the maximum of 5 volts you can see roughly equal to 5 volts 4.83. So there is a small error due to the offset okay which is present. Now this same thing F reference is changed what is now done is this reference is changed now from 1 kilo hertz to 2 kilo hertz you can just see that 1 to 2 kilo hertz amplitude remains same as 1 volt. Now again the control voltage automatically changes of that of the integrator output automatically changes this is the transient coming from plus 10 volts going towards the steady state value so as to make the band pass output reaches maximum and notch output reaches minimum 0 right. So the notch output gradually goes to 0 and band pass output goes to its maximum right. So this is what is seen very readily when we actually do the experimentation. Now it is the so this is changed over to square wave input it does not matter what the input is as long as it is periodic it has certain fundamental and its harmonics this whole scheme keeps working it just gets tuned to the fundamental frequency or the what is that the harmonics of the input waveform okay depending upon how close the initial state was initial state of the filter was okay. So square wave input you can see the square wave input okay let us say VP is the square wave okay. So if the square wave obviously is made up of a sine wave fundamental component which is greater in magnitude than the square wave. So if the square wave magnitude is VP we know that the fundamental of the square wave is 4 by 5 it means greater than VP all the harmonics will be having magnitude reaching a negative peak okay at this at this point okay where this is peaking. So it is to compensate for that it has only odd harmonics so you can see that is how the square wave is formed the first harmonic okay. So you can see it will be 4 by pi times 1 volt VP is 1 volt 4 into H naught into Q at the band pass output and at the notch it will be the square wave minus the fundamental which will result in the square wave looking as just this okay. So if it is exactly tuned these two heights will be exactly same. So you can see this is what is appearing at the notch that means this is the result of all the rest of the harmonics of the periodic square wave okay. So this gives you the harmonic content so the RMS value of this will straight away give you okay the harmonic content okay and this is the fundamental. So if any periodic waveform is applied it actually locks on to the fundamental okay and in the notch you can actually measure the harmonics content of that. So this is a useful thing as a distortion analyzer wherein you give the input waveform which is a pure sine wave okay and to anything that amplifier or data converter which is required to have its distortion measured okay. So you give that and output of that unit you give it to the self tuned filter automatically you get the fundamental at the output of the band pass and the rest of the harmonic content or distortion in the notch output. So it can be used as distortion analyzer or spectrum analyzer. So the usefulness of this is in distortion analyzer and this spectrum analyzer spectrum analyzer. So when also signal is deeply buried in noise okay and the signal is very narrow band signal this is the best way to extract the signal okay using very high Q okay. So the signal to noise ratio improves drastically in that kind of self tuned filter output. So square wave input that was earlier 1 kHz and the control voltage has remained this okay it is having the double the waveform which is not really DC it is having some unfiltered 2 omega component you can see in this. So these are very interesting simulations that you must do in order to experience the ultimate in negative feedback in analog okay. So this is 2 kHz you can see the DC what is that the average has shifted now close to 0 whereas that is what you will get okay the VC equal to 0 means the multiplier series resistance has no effect now it is solely determined by 1 kilo ohm okay or 0.8 kilo ohm. So it is going to be 1.56 divided by 0.8 which is close to 2 kHz. So theoretically it fits in very well with the experimentation. So this is what is called the phase locked loop this is the 2 phase locked loop it is the frequency can be anything it is all the time locked to 90 degrees in phase okay by this loop. So the lock range of the system okay this is normally decided by the for example in this case the control voltage is applied to the multipliers here. So the multipliers can only work up to plus minus 10 volts and also most of the time the control voltage of this may be only going to up to the saturation level of these op amps. So whichever is the one that is limiting the loop gain to become okay very low or equal to 0 is the one that decides the lock range of this closed loop system. So whichever comes first in this particular case we have seen to it that none of the other ranges are limiting except the saturation state of the amplifier or the multiplier input state that limits the lock range. Capture range now something about the capture range I would like to tell you now almost all these control systems there is this capture range. So when it is not having an input fed that is what we talked about the output is also not there. So we do not know what the output is going to be as for the multiplier is concerned just going to pick up some noise and this is going to be at some state or if that also is 0 here we have an offset which is non-ideality okay this offset will simply take this because the loop is not closed because input is not there. Input has to be there so that these get the reference inputs okay to make the whole loop work or make the loop gain much greater than 1. So bringing this whole system to a situation where all these blocks are in the active region such that the loop gain is much greater than 1 where this error goes towards 0 in this so called integral control system is what is called what is that starting the circuit and capturing the loop okay. So once the loop is captured automatically the error goes to 0 that we have seen okay and the phase is kept locked and the system has high loop gain almost at all points okay. So because at pi by 2 it has the best sensitivity possible so it remains pi by 2 as the frequency gets changed. So lock range is that range where loop gain is kept much greater than 1 all the time capture range is 1 where the loop at starting whatever point it is there either plus VC or minus VC is the starting point at that starting point depending upon the input the low pass filter output does it exist so that this into this because the sensitivity of this in this case depends upon VP, VP dash by 20 cos 5 so it depends upon the magnitude VP dash VP is already there let us see then it depends upon VP dash when both VP and VP dash are not there then the sensitivity of this multiplier itself is 0 so the loop is not closed okay. So basic thing while starting is as soon as you apply VP sin omega t this input is there it does this input output exist in order to facilitate okay this output to come and change the VC okay in the proper direction that is the capture range okay negative feedback loop means this VC should change at that time in such a direction as to bring this error close to 0 okay. Now this is called self-training its application is in terms of what is that this getting locked okay to face of pi by 2 and it can be used as a discharge analyzer or the what is that spectrum analyzer apart from that how to use active RC filters for precision application in integrated circuit where resistances and capacitances have poor tolerance so you cannot have precision design of filters done at all. So then we have to take request to tuning so you can actually resort to one method wherein this is the filter to be used second order filter you make it voltage controllable and tune it as before okay as a PLL so this VC is derived from the output of the integrator as before it gets tuned automatically to the incoming frequency the incoming frequency is very accurate okay. So this is what actually tunes it it may be the clock okay which is very accurately fixed by the crystal so this accuracy is unquestionable right. Now once this is fixed this control voltage is known that is sensed and fed to the ADC and stocked in the memory and outputted by the DAC and used until okay this comes to the what is the calibration cycle again where the loop disk again is closed at which time you cannot use it. So whenever calibration cycle is coming into picture this calibrates itself accurately and then keeps that control voltage that is sensed at that time here until your measurement is over using the filter at that point of time input is changed to whatever signal that you want to use it for filtering okay. So this is one method another method is you use this self tune filter as master and make it get tuned this omega naught okay is omega reference VCM by let us say 10 RMCM okay by this loop then use the same control voltage for identical structure as slaves. So this structure is exactly similar to the master here here any number. So apply the same control voltage. So VCS by 10 RSCS of this structure is omega slave. Now since VCS is same as VCM we have to substitute for this VCS this VCM then we get omega reference by omega S as ratio of resistors okay and therefore we are able to tune this exactly this slave filter this slave filter as omega slave is equal to omega master which is nothing but omega reference okay into RS CS okay divided by RMCM okay. So omega S by omega reference is sorry RMCM by RS CS ratio of resistors and ratio of capacitors. So omega S therefore is equal to RM CM into omega reference by RS CS okay. So this is similar to the switched capacitor thing but even better because there is no switching at all involved and it is continuous time you can keep this connected as a higher order filter by connecting this output to this input and so on any number of slaves. This becomes a programmable filter the moment you change omega reference the normalizing frequency of all the slaves change simultaneously so that if you have built let us say nth order but it remains nth order except it is bandwidth keeps changing just like in the switched capacitor. The programmability feature which is there in switched capacitor comes to this also and ratio of components is maintained at better accuracy than the absolute values. So let us now consider another topic design of fourth order band pass or band stop is the aim without going into any high level mathematics. Let us try to design a fourth order filter just by knowing something about the second order band pass prototype. Center frequency is 5.3 kilohertz maximally flat magnitude whatever this required. Second order state space filter will have R equal to 30 kilo ohm C equal to 1 nanofarad we have purposely used universal active filter block 42 okay and the model of this is incorporated in T naught EI so one can easily simulate this and test exactly. So for a Q of 10 H naught of 1 the second order filter has been built so it is going to be exactly 5.3 kilohertz center frequency and H naught in the Q so the it is going to be about 20 decibels okay with H naught in the Q is 10 okay. So now cascading to second order filter what happens this is the main principle to be understood by cascading to second order filters with a certain Q the band width of the filter becomes narrower right. So wherever it was getting halved now it is going to get one fourth right quarter of the earlier thing that means it becomes narrower. So cascading or synchronously tuned amplifiers cascaded cause reduction of band width that is the basic principle whereas if you want to broadband cascading stagger tuned amplifiers which are tuned differently from okay each of it is staggered from the other okay. So now from a single tuned circuit right second order we are actually cascading instead of two stages with the same center frequency once with stagger from this by amount of bandwidth itself right. So this distance should correspond to the bandwidth okay of the original thing so if that happens then it becomes maximally flat. So this mathematics of this is demonstrated here by 1 by 1 plus X square X minus alpha X plus alpha are the two points at which the center frequency is located. So by alpha it has been changed. So by doing the mathematics you can show that it becomes maximally flat okay when X is equal to that is alpha equal to 1. So that means the bandwidth is the separation from zero okay of these two filters then it becomes maximally flat. We have just shown you the demonstration for alpha equal to 1.1 that means now if you just exceed the optimum then there is this small ripple that comes it is equivalent to what is called to be same filter maximally flat one is called Butterworth okay. So alpha equal to 1 is Butterworth we have increased the alpha to 1.1 this is actually simulated we have at the first filter at let us say 5.3 so okay cascaded to another stagger tuned one the distance is about 10% this Q is equal to 10 is what is chosen okay. So the that is 10% distance we have changed one of those the two resistances from original value of 30K to 33. So the frequency gets shifted down okay and now you can see that it has become maximally flat over that area this is the region where it has the decrease and increase here I make it maximally flat okay over the widest possible region. So if you just increase this distance further away from this optimum value so we have increased it further and shown you the fact of the ripple in the passband that is the Chebyshev filter okay. So it is not very is difficult to transit from Butterworth to Chebyshev by merely manipulating the staggering of this band pass filters okay. So that is what is demonstrated here this is the other filter which was originally at 5.3 now it is staggered more than 10% now notch filters so we have the same filter being used as notch 5.3K with a staggered notch cascaded to it both of which are second order notch filters so they are initially we tried it with 5.3 and 5.3 at this same point notch is occurring it is called synchronously tuned. So then it is the notch depth like the band pass output okay became square of the original H naught Q square okay like that here the notch this is lower than 1 so it becomes still lower when you multiply with okay the same quantity again so we stagger now why should we stagger because we are not sure of the fact accuracy of notch so it may be anywhere within this range so this is a notch and we have coupled it with another staggered notch now so okay that resulted in a composite characteristic which is having double notch okay in its characteristic okay or there is ripple in this top band now. So by it is only change by 5% if you change it more what will happen is transmission okay is going to increase so that is what we do not want this is almost less than 30 decibels okay of transmission so if they are coming close the transmission is better. So review of filter structures right BataWat and TBCU filters are basically all pole filters useful when white noise dominates over signal rates of attenuation at the pass band edge or slope that is the thing these are mainly all pole filters useful particularly when white noise dominates the signal okay or we are now comparing with white noise and coloured noise. Coloured noise means narrow band noise white noise means it is uniformly spread all over the band stop band it is there in the pass band but we cannot do anything about it dominant coloured noise is not effectively removed by this filters inverse TBCU and elliptic filters poles and zeros presence of zeros helps in eliminating narrow band dominant noise components in the stop band attenuation in the stop band is decided by N-1M number of poles okay because N is the number of poles and M is the number of zeros it is only N-M poles okay ultimately deciding the stop band attenuation. When white noise is dominant signal to noise ratio improvement is not as much as that of all pole filters because of this so depending upon the kind of noise that is available we decide about the filter input output relationship of a second order filter with a zero this is the way it is this zero okay is bringing about a zero in the stop band this denominator is bringing about the peak in the pass band. So let us understand this for a second order filter okay this is the square of the magnitude function input output relationship of all pole filters okay is just this for the second order so it is peaking in the pass band. So starting from one it is going to a peak the zero is characterized by outside the band let us see so outside the band is indicated X equal to one is within the band okay up to and this will correspond to X is equal to square root of two that means 1.41 times T what is the pass band H or normalizing frequency so that is where the zero is located okay so root two times let us say omega naught or omega P X is equal to omega by omega P so here it is getting a peak at a frequency which is omega P into square root of one minus one by two Q square this we had derived earlier and the peak value is QP directly proportional to QP one minus one by four QP square okay this is one that is the peak if it is H naught this H naught into QP so that is the plot that has been given here this is the peaking point you can see the whatever we have written earlier okay that is the frequency at which peak occurs. So the peak for IQ circuit is pretty close to the normalizing frequency this is how it is got so we have it here now for QP equal to two and alpha equal to 0.5 this is what is got that is getting compared with okay you can see that it peaks and goes to zero and what is that that is nothing but the elliptic filter that we have earlier called that with ripple in the pass band and ripple in the stop band but it is permitting lot of white noise to come through here it has been optimized QP equal to one is made okay for it to become maximally flat instead of peaking so for this value of alpha equal to 0.5 you get QP as one by making the numerator polynomial have the same coefficient as the denominator polynomial. Edition of another first order okay can get rid of the white noise okay it goes to zero here right and you can optimize it in the pass band by making QP now equal to one over root two so this is the design beta equal to 0.5 you can get alpha equal to 0.5 QP can become equal to okay one over root two that is the design so these are the responses right in fact so summing up all these things this is how the present day filter design is going to be carried out okay have both digital and analog system many of the present day systems are portable hence battery operated or subsystems have to be designed using the digital device technologies for single chip solutions so we have to have low voltage okay that is being used for the analog subsystems as well three volts so leaky switches and switching noise in switched capacity filter is not a viable option L replacement method requires too many of these components okay and there is no flexibility Q enhanced method can lead to one with single device active device but less reliable and then the multi active device structure like by quad it is also less flexible H naught Q and omega naught can be independently fixed in the by quad and that is the ideal building block for current filter design whether they are switched capacitor or state space or master slave filters so this sums up the thing the op amp IC UAF 42 is costlier than the filter built using quad op amps so currently but if you make it popular by demand then the cost of that also will come down thank you very much for listening to these lectures we have a follow up lecture on filth oscillators which is nothing but a filter with Q equal to infinity second order oscillator nothing but the same filter topology that we had already shown okay earlier that is the harmonic oscillator so it starts with again the simulation of second order differential equation with dv by dt coefficient being absent