 We start with the filters as we said last time, the word is active. The active filter essentially stand for the word, there is an active device during filtering. So some property of an active device has been used, otherwise we shall see soon that even a normal RC or LRC filters can do as much a good job as otherwise but why active we will see later. Another filter requirements as I shown you last day was we have four kinds of filter we may use. One is something which passes almost everything except up to a frequency, we call it low pass. Something which passes all of it from starting from some frequency below which it is nothing is passed, so it is called high pass. You are allowed to pass between two frequencies the signal but otherwise block everywhere or do not pass anything there, here or here and the finally you are passing everywhere except some band of frequencies. So these band reject, band pass, high pass and a low pass. These are the four filters which we will be using in any of the circuits which we will use in your whole career. We are right now as the word filter says does not say whether it should be analog filter or it can be a digital filter. You can actually have either kind of them. Normally even if I am doing a analog sorry digital filter, we actually convert partly to digital in a half way and what we call switch capacitor filters, these are essentially digital filters but actually copy out most of the analog functions. So the basic filtering still could be said to be an analog filter. Of course there are many digital interesting filter, your signals and system theory and communication system theory will deal with many more of them later in your career or later in your classes. What is the typical low pass filter shown to you, maybe I will censor it or first. This is typically ideal filter, this is what I am expecting ideally but when I actually pass through any active filtering system, what I see is till certain frequency called omega p, see certainly the transfer function does not show a constant value but repels up that is keeps on changing and till this point which is essentially a pole, at this point the gain starts falling or the transfer function starts falling its value because that is a pole we said. And ideally it should have reached to 0 at this point but it did not, it actually crossed that point and before it actually settles to 0 again repels, again repels. So the way we defined is, this is of course called the ripple which is passing through and we may assume it is a universal one frequency ripple in real life that also is not to ripple itself will have different components but right now assume single frequency ripples. Ideally what did we want, we want flat response here and sharp fall there, this is ideally a filter low pass filter was asking for. What we are essentially getting a fixed value and plus ripple up to a frequency where the pole starts that band we call pass band that means where the transfer function is such that everything is passing beyond this depends on the number of poles, this may be 20 dB down, 40 dB down or 60 dB down whatever it is the value start going down and when it reaches the so called ideal filter value we essentially say that we have reached, there is a band between the pass band and before everything becomes 0 let us say this band is called transition band that means from higher to low the frequency range is called transition band. Ideally what was, do we really need transition band, ideally we want transition band should be 0 okay that is what ideally we are looking but what we are seeing in real life there is a transition band, there is a fall time or fall, there is up to which it will keep falling. Beyond this we would have expected something to become constant but it did not but it also ripples and maybe finally it settles to the value which you are expecting. The band in which nothing is passed as for the low pass requirement we call it stop band. So for any low pass filter we are interested in what is the ripple magnitude, ripple frequency, what is the expected pass band you are getting, what is the transition band you are getting and what is the stop band you are getting and of course as I say ripple frequency itself will be varying but assume right now it is single frequency ripple occurs. So this is a definition for any filter this, so ideally for a low pass I have expect a max be 0 okay that means this should have come here. I want transition band frequency omega this low minus omega p should be as 0 as possible and again this a max be 0. Ideally this is what was expected and if that occurs we say we have a ideal low pass filter. If you look at the other band pass requirement it is similar a typical good band pass filter would have expected to have this transition between this frequency and this frequency the transfer function should have value which is constant and everything is passing okay. Below this frequency we do not want anything to pass so we say lower stop band beyond this frequency nothing should pass we say it is upper stop band. So there is a lower side stop band and upper stop band in between there is a pass band. However in real life this is not starting from here it actually again ripples and some kind of a Gaussian function one gets or rather signum function it gets and then that means beyond this value stop band I mean pass band is this but there is a ripple over it here then it goes down so there is a pole there is a somewhere 0 starting here there is a pole going down and one can say it again ripples before it actually reaches to 0 okay. So essentially in our design what are we expected to do that where it starts where it ends and what is this ripple magnitude is going to be exactly that is what we must do and what should we do minimize them as much as possible same way this is some transition band there is a transition band I also want transition band to be as close to 0 as is possible. We have already said that a pass this band pass filter is essentially a combination of low pass and high pass if I say this is my FL and if I say this is my FH please remember I repeat listen that is why I gave the name separately this is my FL this is my FH that means first there is some high pass starts and then there is a low pass starts so in between both are passing and so done the band pass. If I separate the upper frequency if I make this as FH and if I make that as FL that means high pass sorry low pass followed by after certain frequency high pass there is a frequency band in which nothing is passing so we said say band reject so basic design of a filter is essentially this and this is just a setting of the frequencies to suit your band pass or band reject requirement so in design of course in this course we should have done all of them time is not enough so we will only do one good low pass and one good high pass and if we get that then we say okay this is the circuit which can be then manipulated to create a band pass as well as a band reject filter. Now having showed you this is my requirement and this is what ideally and non ideally I am going to get I want to come closer to ideal but before we go to act networks active networks I just want to see a network itself okay normal network itself we are done this earlier as well if you have a RC network okay 1R and 1C okay if I take a transfer function of this V0 by VNS which is 1 upon CS upon 1 upon CS plus R which is 1 plus RCS or it can be written as 1 upon 1 by RC if I define a frequency cutoff frequency is 1 upon RC this is nothing but 1 upon S by omega 0 what does that mean there is a pole in negative half minus omega 0 and if I plot the board S plot therefore at this the gain will start falling so a simple RC network actually acts like a which filter low pass filter now one can think little interestingly if you see now ideally I wanted this sharp fall is that correct I must all my design why I am doing active filters all my in any passive filter this is going to occur is that clear so what I say can I do some mischief okay or close to this I have come closer to ideal and the active part should help me in getting these as close to the ideal value as is possible if I do that then I will actually create a filter which is close to ideal low pass so why active part is necessary because I want to shift the slope down if I do so something else may happen I will say whether I can tolerate that or not I will check that but at least I want fall to be as sharp as possible let us say it is 120 dB down at this point is that clear so it will swap very shortly is that clear so essentially I am trying to see I have done this pole 0 theory and I am now started looking here is some way there should be multiple poles coming there if they are they occur at very close phase or same point then my fault can be but if I look at the other filter equivalence of this which is interesting okay so basically a active low pass filter ideally could be a RC circuit as I just now did followed by a voltage follower is that correct follows by a voltage follower so whatever is the voltages here will be transferred here because a voltage follower so this is the easiest way of making a low pass filter we are not seen as transfer function so far so we do not know whether it is going closer to ideal but this should do as a low pass filter with an active device what is the purpose of all over here if any load occurs at the capacitor it will load that out this buffer will separate your input response from the output loads is that clear that is why I put a buffer in which is my voltage follower by doing this I am still achieving one upon RC kind of requirements and I am getting low pass with of course 20 dB down gain falling I can do similar thing for and as I say high pass is just the opposite of low pass and I say okay replace R by C and C by R wherever there so I put a C here or here again derive this transfer function which I get S by Omega 0 upon S by Omega 0 plus 1 where 1 upon RC is Omega 0 now you see how many things are happening here how many poles are there how many zeros are there there is a 0 at S is equal to 0 itself and there is a pole at S is equal to minus Omega 0 is that clear so there is a 0 and there is a pole so till Omega 0 which will dominate the 0 will start dominating is that correct so initially what could occur you can see from here 20 dB 0 means 20 dB per decade rise start rising okay at the pole what is the fall should be minus 20 dB but what is it rising already with 0 plus 20 dB so at this point onwards 20 minus 20 0 dB gain becomes or output function becomes constant is that high pass requirement at frequency above 1 upon RC I want transfer function to have unity value or no attenuation just passes out inputs is that correct so how do I create in this case you have a CR here pass way follower and this will act like a high pass filter this is 0 this is 0 and pole together constant okay so obviously now one thing you must have got from there if I have a low pass I would avoid zeros is that actually 0 should be at infinite okay if I am looking for high pass I will bring 0 whenever I want flatness to start is that clear so that means I have now a trick that if I have a S in the numerator I am creating 0 S plus something also but S is in the numerator and S plus something in the denominator if that kind of function I create then what I will create high pass if only one upon S plus something I create I will create a low pass is that point here this is the trick of filtering that okay create properly zeros and poles such that you actually achieve low pass and high pass values the only thing trick is how much to get whether this value should be 1 or it should have gain function if I want to gain function what should I do I should do actually are here then it will also give me some gain out of it like a normal amplifier I may amplify it also okay now this tricks are always played in many of the filter designs but using this what is the main problem we saw that that 20 dB is the all that we could get in you cannot have fast rise or fast falls so this though this is a filter no pass high pass can be always realized by this try yourself one and you will see as it will cut off somewhere and it will show you but the response will not be closer to ideal but why should you always have ideal yeah the reason is okay so there is something you must remember many circuits do not mind for example that it should actually be cut off completely as far as anyway it is like a noise going on a different letter anyway I do not care very much what happens but if I do care it single end system I will worry about it so depending on the what is going it going to dry or where from the next stage will be your choice may not be you should not see that I should have a ripple or should not hurt now way I have figured out and that is who will work today are not necessarily I will complete I have figured out basically if I see a function as I am getting I know any conic section take any conic section straight line parabola hyperbola circle ellipse or for that matter any curve for that matter can always be represented by a very simple functions what is that function is called polynomials if I write a 0 plus a 1x plus a 2x square up to n polynomials I assure you that any curve can be fitted okay it may require thousand terms to actually get closer to the curve you are looking for it is a fit function technique so I figured out that if I can generate some functions which also gives to be like those 0s and poles which I am looking then I can have a this equivalent of a I repeat what I said I create a polynomial or I create a function whose numerator and denominator looks like 0s and poles but their values can be tailored then I can say I am right now creating a filter for that function is that okay because that response will be something closer to filter is that correct this is the technique which most designers are most circuits in the analog we use that we actually create some kind of a known functions whose values mathematically we can derive and then try to fit it on those on a circuit and we say how close we are getting these two filters which we shall see little later one is called Butterworth filter the other is called Shabashio filters why I am showing you because in Butterworth filter the ripple can be minimized what is the advantage of a Butterworth filter that the ripples are very small and therefore it is called maximally flat maximally flat so all Butterworth filters are so designed that their ripples are very close to small value of 0 kind it is a flat here but the price I pay because of the function I am using to create the ripple 0 or ripple small there will be a transition band is that correct there will be a transition if I want to reduce transition band then I will say the number of poles which are required higher and higher that number of remember each pole will require let us say one circuit of this kind equivalently saying I just now showed you each pole requires one such circuit if there are n poles I will require n such circuits so I figure out that if I want I use a Butterworth filter then I can maximize the flatness but to reduce this transition band I may require larger number of poles is that correct on the contrary Shabashio filters are more this they say we cannot reduce very much the ripple okay but for a smaller number of poles I can reduce the transition band is the two difference clear in the maximally flat filters Butterworth filters ripple is smaller small very small at the cost if I want to reduce the transition band I will require larger number of poles or larger number of sections as I call in the case of Shabashio I say I cannot reduce this very much I tolerate ripple okay but for a smaller number of poles I will give better transition or at least same transition I would have got in Butterworth okay with a smaller number what is the advantage of smaller number of sections of smaller poles a circuit is smaller numbers what is the cost cost reduces if your number of sections goes down the price goes down but price going down at what cost the ripple is that correct so if you want a very good filter then you pay price okay higher price is that correct so maximally flat filters will require larger sections but will care much less ripples is that correct this is something all designers must keep in mind all IC designers are always told that anything I use because we have a silicon area known area let us say few millimeters by few millimeters say 1 centimeter at best I may have both side so many million transition I am putting if I do not put additional hardware I can put something else there is that clear that means the price of silicon is very very high area silicon what is any equivalently we say like in Mumbai we say it is a real estate silicon is a real estate any small micron by micron you throw you are losing money okay so whole trick and integrate circuit design sir how to minimize area or increase density is that correct so if you are doing the same function with larger circuit oh yeah it will do better but you paid price high price for that okay so one must understand some chips we sell very heavily high at high price if someone like young man says I want exactly this filter this should do this this man says no no I do not have only this much money so the trick of design is what is the customer wants okay so in all our engineering this part is never explained till maybe graduate of course dual degree will learn in 4th year we are never telling that all this our theory etc is as good as the money in your pocket that is it okay so this has to be understood why we learn all the tricks because we do not know what customer at the end will give me a response you say it may make or not okay so I have now suggest you here is a circuit active filter which is a generalized two pole filter how much how many what do you mean by two poles how many capacitor RC should be two RCs this is generalized why I say generalized what does that mean generalized means all four filter should be clear can be created out of this is that great generalized I never say it is low pass high pass band pass I from this circuit by proper choice of the Y1's and Y2's and Y3's and Y4 I should be able to create low pass high pass band pass band is that clear to you what is the advantage of such generalized network then because this is one time created any user during design can just manipulate this and say okay here is a low pass so he does not what is the other constant design do you know what is the cost of design is the time taken by so many engineers are called man years or man hours is the cost larger the circuit redesigned by you every blog then it will be costly every time so if I have ready met block which can be manipulated to create different things I have saved a lot of money that is essentially called semi custom something is prepared and used again and again so this is something equivalent of a semi custom that means a generalized block can be configured as either low pass high pass or any other is that clear to you why I shown to you this is something which all designers should know analysis why this is not very great this is same as what low pass high pass we did but here is some analysis I will show you I have an input V in to one of the conduct it can be admittance or conductance why is G plus JB why is G is the conductance B is admittance so why is equal as we said Z is equal to R plus JX same way it is the opposite of that so I say okay here is a Y1 component Y2 component Y3 component feedback this is two feedback this is our follower which I am not disturbing please remember this follower I have kept as it is there is another feedback which I have brought here to output let us say Kishchoff law at node VA what are the current what is the current entering V in minus VA into Y1 is that correct is the current passing through Y network theory simple trivial. Yabadokonse branch janeh currents K from VA to VB and VA to V0 is that there are two more branch current so branch current at VA must be 0 total current so this current must be this plus this so I write VA minus VB into Y2 VA minus V0 into Y3 they must be equal Kishchoff law at node VA three current summed up okay same way I summed at VB okay I said node VB VB minus VA a current either a current or what is current here no current enters open that property I used so I say VB minus VA into Y2 equal to VB by Y4 VB into Y4 okay then the third equation I see it since this is a follower this is V0 but that should be same as VB okay by what is the that is active part did you get what is the active part I used opamp properties have been used in actually writing the networks is that correct these equation I could not have written if I have not used opamp properties what properties I used same no current enters same potentials both properties I use and I use this is a follower okay so I use this the property of opamp and then could write such equations if I solve this you solve yourself though I have solved but you solve yourself at then you can do analysis I did just look at the final transfer function you write and verify V0S by VNS which is my either you write TS or you can also write HS whichever way you are writing either T is the transfer function okay normally I have been using H as the transfer function but I do not know from wherever earlier I had taken there was a T so I used so maybe your colleges so I get Y1 Y2 upon Y1 Y2 this kind of function please note down this function rest you do yourself Y1 Y2 upon Y1 Y2 plus Y1 plus Y2 plus Y3 into Y this is the transfer function I have got for the circuit I have given there is that clear simple so Kishav Locos all career or a expressions up V0 by VNS I will now explain a specific example which filter do you want me to do we will see little later but let us do this example now in this I create the first Y1 Y2 I put it as R1 R2 and this Y3 I4 I put as two capacitors in our case G is 1 upon R and Y is CS okay be rather not but why these are why only this is capacitance so CS is the admittance of that and conductance of this is 1 upon R1 and 1 upon R2 this is the expression I just got for the transfer function G1 G2 upon G1 plus G2 plus SC4 G1 G2 SC3 so now you look at it at S is equal to 0 what is the value of HS yeah view all of your expression and please remember all that I had done is I have substituted here the actual Gs and Cs okay so I got G1 G2 upon G1 G2 plus SC4 G1 plus G2 plus SC3 is HS okay at S is equal to 0 what is the value of transfer function S0 hey S0 hey a term Chalagaya G1 G2 by G1 G2 so one what does that what does that means at S is equal to 0 there is already a transfer function has a value one means it could be normalized to some other 0 so it has a value is that correct so which filter I am looking for now initially there is a value which filter I am looking low pass I have a value of the function of the output gain available at this value is that clear now when S tends to infinity so there is a S is equal to j omega when omega tends to infinity one can see the transfer function will go to 0 is that correct that means finally it is going to 0 that I see it is 1 and finally it is going to 0 so your concert characteristic different low pass different this is what we say initially value and then it is going towards 0 so obviously I am looking for a low pass so can you now get this that jibe by just substituting here the proper values of this I can create a low pass filter how many poles you are seeing here two poles so at least fall will be better than one single pole system is that correct it will be better than the single poles. So, I can create a different filter of my choice of different cutoff why different because R and C is in my value of me at me so a generalized system so why it is not so good because it is now once made at best you have only those two poles to manipulate additional is that clear you cannot actually get to this person requirement so this tolerance of yours is how much will be depending on that I said how much money you keep in your pocket that is the way it is it come in a salt curke dikhay hain I may a but I know key a maximally flat let us see whether it is so I wrote the transfer function as it is is it clear what I just substitute G is 1 upon R and vice yes and I get this function if I manipulate it better and I give a definition that Tau 3 is RC 3 and let us assume R are equal my assumption is R are equal but I never said C's are equal R's are equal but see why I do not want to make C is equal because then the time there will be same poles so I okay there are two different poles one is related to RC 3 the other related to RC 4 so this is the transfer function I want please note down HS is 1 upon 2 RC 4 S plus C 4 C 3 R square SS square Kine poles I am a part SS square term I am a part dope holes if I define this I can write a j omega magnitude is the magnitude Rokey the expression I am in a thodasa directly like I you can also think how I have written okay I say a core method of those section if you give me a corner SS square get a car now it is a partial fraction career because separate separate image career 1 upon S minus a into 1 upon S minus B transfer function key theory can gain again because it is equal to a1 into a2 is that correct so a transfer function when I a second function when I say the input the product is that point clear I repeat a transfer action h1 is using one pole a transfer function h2 using second pole is that correct I can create two separate functions output of the first is given to the input so the transfer function will get multiplied so 1 upon S minus a into 1 upon into S minus B automatically appear so is that trick clear to you once you have two poles you actually can create single pole sections and then keep output of that should be given to the next input keep doing so as many sections you will put those many poles you can create is that correct if you put seven sections that is why a word I use section each one pole will create one section I have number of sections so I can create as many poles I want and the trick is they get always multiplied by H1 H2 H3 H4 this is the way we actually implement higher pole functions anyway is come in a value Nikola or maximally flat requirement take that d by this function with the omega should be 0 you see it is the maximum minima karna kele of the differential 0 correct magnitude in a flat renna change so no change of course less than omega 0 for it is co value solve here okay I can get a condition Tau 3 is Tau T4 and C3 is to C4 a condition meet Karna in a so maximally flat situation liars is that point clear I differentiated that below that cutoff and I say at this point if I want no ripple then what should happen if I meet these conditions then I would say I am going to have a maximally flat situation is that correct over ripple kajo party O we can remove by actually choice of proper tauts and seeds is that clear or rather seeds C3 should be twice of C4 in our case if we substitute back in a normal function the cutoff of this transfer function value will occur at 3 dB point what do we 3 dB point means normally in Bode plot what is the 3 dB point is the corner we start right there 20 dB but in real life that is not the point where it start where 3 dB below it starts falling it is a continuous curve it is not Bode what did he suggest that instead of that continuous curve you actually make two points and then you say it is your cutoff point so you can actually see if I substitute here this is 1 upon root 2 dB at omega equal to omega t4 1 upon t4 that is the corner frequency of the Bode is that correct so if I get the pole you can see this is the solution many up on root t4 1 upon root 2 Rc4 why we said in real life actual curve will be something like this sorry a 3 dB isn't it is key over car so Rc4 okay so one can see that I can create a low pass and how much is now fall will start from because I made that condition tau 1 is to tau 4 right here the gain will starts falling by how many dB is 2 poles and I am after a point or dope all of them in a shift key under what condition under this condition the two poles are matching is that clear and the maximally flat situation attained so I got maximally flawed falling down by 40 dB so at least 20 dB ke bhajaya 40 to karbiya meh thoda tha stop transition band ko sharper kiya kum kiya meh and also what did I do flat kiya is meh kitne section lagay doh section aur kuch bada nahi to kya karna padega ek aur section lagau if you aap isko aasai hi chate hain to number of poles should keep on increasing that means number of sections will keep on increasing that is the problem with larger amount of money you spend so you have Butterworth filter have a powerful okay for this kaap na problem hain ki it require larger section to make it sharper okay now if I say I do not use that condition of maximally flat then I must say can I get sharpness with a lower number of poles and then I will call they are Sheveshaw filters we will come back to Butterworth in more details but this theory has to be understood what we are really trying in design of a is that point here please remember only two filters will design low pass hi okay last time if you say bola tha what is a all pass filter which we are not discussed maybe here we can quickly show you what is it up on before we go to the next kaahoga hi isko bolthe hain all pass all pass ka kya meaning hai ye to edhi transfer function bana h ka magnitude nikala h j omega by h 0 samjo is ka value nikala that the constant hai everything is passing to filter do kuch kya nahi usin a flat come a sakta hai iska transfer function kareeb kareeb aisa hona chahiya equivalently hain this is not the exact one what is it trying to tell you this transfer function 0 aur pole ek hi jaga par hain 1 upon RC par hi hain 0 hai a pole hai uska kya meaning hai 20 dB 8 jatah hai to usra minus 20 dB shuru kar deta hai wahi par that every frequency you get flat is that correct. However even if you see this function is ka transfer function to thik hai aisa hai magnitude but is ka kya change hoga iska phase okay phase can vary depending on RC values phase can vary from 0 to 180 degree. So whenever the transfer function filters you create and you want a particular phase at the output between in and out then you connect the last stage of your filter can be all pass filter with proper phase requirements is that clear to you what is the advantage of this is ka gain to one hi ho gya nahi sab kuch pass hai okay but is ka phase adjust kya ja sakta hai kya kya minus plus jo value is ka hai the phase tan inverse nikalingi do is ka phase varying RC par is that correct so the net phase at the output compare to your input need not be just the product of their magnitude h1 h2 h3 h4 but also I can adjust my final phase with the input is that correct many requirement the output should meet should be either 180 degree off or less than 180 for the stability case. So we actually create the last stage as all pass filters okay though it does not change your whatever h1 h2 you brought everything it will pass anyway but it will give you a phase bit whatever phase difference between the final stage of filter to the actual stage where you are connecting so please remember the all pass filter as such word seems to be funny why are if everything is passing what is being filtered nothing is actually phase is also please remember any complex system will have something e to the power j5 magnitude into e to the power j5 so it is the magnitude may be one unity everywhere but the phase is not is that clear and that phase variations can be attained by proper choice of RCs is that clear this essentially is the trick of phase controls we do something called phase control oscillators okay we do use some of some such property there is that correct phase control oscillators okay all the filters which we are going to talk are essentially in real life this can be a polynomial of n by n or m by n it can be s3 s4 sn same way SS b0 s something more up to sn but this may say many square terms he will be luckily so cabling is you know bullies can I am by quadratic the numerator be quadratic function a or denominator be quadratic function so in filter we actually try to see whether both Butterworth or Shebesha functions are by quadratic in nature is that correct we are looking for by quadratic functions what is that now I am talking about it is called implementation of it a real life I will KSA implement so we say okay let us look functions if I have a by quadratic function like this a2 s2 a1 s by 0 upon s2 d1 s0 this is also given in Cedars Smith and me I can have I can be part of this function I can create into low pass by pass band pass high pass whatever it is so look at the numerator may subcurch s terms any zeroes only constant value hey something called K if that functions are K upon SS square omega 0 by Q s SS this plus omega 0 square have any time you heard of the word Q Q cut quality factor omega L by R 1 upon omega RC they are also quality factor now this quality factor. If I apply a step input to any amplifier or any circuit the output essentially does not respond instantaneously is that point clear what did I say if I give okay maybe that figure which I have I can still use if I apply a step input something like this this is my input so the output does not respond immediately it starts rising and let us say trans function as a value one it should have actually reached here it did not so what it did what is this word we call ringing okay ringing we start ringing of course settles okay ideally catch I had a message as a name ring over Jada is got your damping factor is okay it is called Jolly say Niche like a scope will take key mall key rather it is got definition yeah yeah 1 upon 2 Q K which is the damping factor or in some books it will be Zeta it will be called Zeta so Zeta is 1 upon 2 Q so what does that essentially means that ideally I want something to happen like this okay depending on higher value so you can see if I put a very high Zeta 1 ke karib hai so ye asa ho jai it means it may reach at isentotically only to the actual value isentotic come in him kaya infinite pejake one whenever it is in a Zeta come look who are the body Jada ring kar kaya so I have to adjust my Zeta so that very close to that value I will get flatter response is that clear to damping factor has something to do with the quality factor by a function which is this okay so if the transfer function say has a nature of K upon SS square Omega 0 by Omega 0 is the pole Omega cutoff frequency into S sorry upper plus Omega 0 is it quadratic SS square plus S plus something so it is a quadratic term numerator does not have any quadratic term constant as a function low pass the low poles and low pass the however if the numerator has a term which contains SS square it may give up at multiple 0 at 0 multiple to 0 that 0 okay. What do you do? Faster rise kia or bad may SS square SS square term to a new circle cut karengi to a flat hokia Milan co high pass in band pass instead of SS square you now want after another cutoff to occur to ask your S look okay so you can do this analysis but if you have a function which is K dot remember the denominator is same everywhere and in the band reject you should I cut off at some other point so SS square plus Omega 0 square with the as a function hey so it will be band so if I can create by these two names I give Butterworth and Shabesha functions closer to this then I can say I am creating low pass I pass band pass band I say even this function should give me low pass why did I say so it has a 0 okay so initial value can be pumped up okay but then after the pole it actually starts falling down in a way so it is key value we will be just as I the S into S plus Z hey so high pass the manner is exactly so the generally this we may not use but this is also possible in designs we always use these four functions are the easiest to make okay and these are the four functions will realize in either Butterworth form or in Shabesha form what are the advantage of that to Shabesha gives me lesser number of poles circuit to create sharper fall at the cost I will get better word will not give me repel or very little repel but we will take larger number of poles to get the same as if Shabesha of a visual let us say five section but it may take seven sections or eight sections that clear so the at the cost of money real cost I may create Butterworth even for higher poles but ideally I say I can choose either Shabesha or Butterworth to my specifications if this is my quadratic form let us take the low pass one it is H0 upon some constant clear H0 omega 0 say divide kardiya or us come in a H0 kardiya so it is H0 S square upon omega 0 square 1 upon QS upon omega 0 plus 1 yeah transfer function heck how many poles it may take you have to go do poles of H0 create the gain chalega lahir chaiyana H0 kilo gain lagga one or the two ne padta kucho gain factor chaiyai to the amplifier banaya this circuit has an amplifier here is that correct R3 R4 a amplifier a gain function dia me is that point clear Abhi Nikala Kishav law solver KSK transfer function Nikala into gain is made only thing is it into is ka gain be here the J transfer function or what transfer function same aate haikyaan dek sakti it follows the similar pattern then what do we say by quadratic form is type KS may fit bethega the filter karni is that clear there is say you know even buffer will do the same as well as this buffer may I am here to gain then on each other normal amplifier be R2 R1 a graph of it will be organic even open does the same thing what buffer does the only difference between buffer and an open means that we do not want to use any additional components because I do not need gain there if I need gain I will put resistance so essentially if R is shorted here I am going to get a unity gain that is what I am saying even in open that are 0 is still lower only thing is now it will be a function of these value there it with the intrinsic value is that the difference clear now R3 R4 will also affect that R0 value is that clear so R0 will not be same as what we naturally device was giving it will get modified by these feedback factors is that correct so there is slight difference but the nature is same R0 will go down and RI will increase will remain same independent of that is that clear to you only thing we are there we are not interested in actually getting gain out of it we only want resistance transfer natural resistances available so put a buffer in all unity gain follower okay so it is coming a transfer function banana the code abhijay say many banana just directly more say with that I fit said except R1 R2 C1 C2 as a square plus S time bigger bracket C2 plus C2 into R1 R2 plus R1 C1 into 1 minus AB0 bracket close plus one let us say R1 and R2 are same and that is R and let us say C1 and C2 is also same which is C okay then I get this function is R2 C2 S2 RC3 minus AB0 S plus 1 what is this why I am doing this please look at my transfer function for low power SS square Omega 0 by Q into S plus Omega 0 square if I create Q is equal to this and Omega 0 equal to this then I am actually doing bike what is that clear if I choose my Q which is under root R1 R2 C1 C2 divided by this R1 C1 mind this this and I choose my Omega 0 is 1 upon R1 R2 C1 C2 under root of that then I am essentially converting this function into SS square plus Omega 0 by QS plus Omega 0 square which is my bike where transfer function for low pass is that clear abhijay low pass catcher function banana I converted it equivalently in this form this we already said by theory now if it you can see the expression between the two is identical so they are identical only thing the values correspondingly should be like this if these values are there these two functions are identical is that correct that means with these values this function will represent a low pass filter is that clear you see this bike bike ride function which is this SS square Omega this we have already proved by theory this is a bike ride though poles and each I know I equivalent in a sir now please remember what did I do I have to implement this function so I said okay here is a circuit with me I tried this circuit by my earlier theory also but I know I put this circuit I evaluated this transfer function is that okay then I say if I compare with my bike ride function I get these values and if I get these values that this function and that bike ride function is identical is that correct since that was a low pass filter this also should do me a low pass action is that clear with cut off frequency now much 1 upon RC R square C square means 1 upon RC and Q is RC divided by this much is that clear so if I get this the cut off frequency is 1 upon RC Q is 1-3 upon AV 0 and for this can you think why I put this condition Q is AV 0 should be less than 3 what will happen if AV 0 is greater than 3 what will happen look at this function. So minor as poles will go into the right half plane okay so what happened what stability got keep ways margin I am sure 180 save. This come up all I am sure left half may run a right half may jara condition is that now condition clear this low pass filter will keep acting as a low pass as long as AV 0 is less than 3 AV 0 is decided about what is AV 0 value there 1-R2 by R1 or which is a so R3-R3 by R2 R3 by R2 ratio should never be greater than 3 is that correct it should be never be greater than 3 if that value is never greater than 3 then stability will achieve this function will implement a low pass filter is that correct at this cut off what is the fall it will start at this cut off. Is that correct. What I am trying is. What I am trying is then I am not sure whether face stability is guaranteed so what make a filter is that clear I want ideally this so I say okay I want to come as close to the ideal value so poles is that clear so a definition say I am going to be a filter is by quiet function okay. Okay. Is that correct this will become a high pass okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay okay. I already explain you how 0 is a point is say output if you have 2 paths out of phase at a given frequency they have same value then the 0 is created output goes to ? is that correct that condition always can be created by proper choices of frequency as well as the components you use. This is another which is integrated circuit filter, this is an IC filter, this is all component used. What is the difference between operational amplifier and operational trans conductance amplifier? So if I have an amplifier whose output current is controlled by input signal then I have an operational trans conductance amplifier, typically I will not show you the buffer stage which I have not shown you, I will not show you the output, it should be removed, so amplifier is modified a little bit, defam plus gain stage, two gain stages are applied and such is called OTA, operational trans conductance amplifier, forget about what is inside this, OTA, you have its symbol, what is the difference you see, so you have a different signal of V2-V1 or V1-V2 and there is an output current. And there is a bias current called I bias, you see, it is output current I out, so its gain is called Gm, that is transistor Gm or normal Gm of defam, Gm1 plus Gm2 is actually Gm effective, multiplied by K, K is a multiplying factor, so I can improve the trans conductance by this K factor, okay, size double triple, so K is essentially a size factor, okay, a amplifier may a transistor Gm, 2 beta dash W boil into I bias, so this is the function, what is V out, current, please remember, is any current can go here, no current can enter open, so whatever is coming here must go through the capacitance, is that point clear, no current can enter open, the current can only pass through capacitor as it cannot get into this input open, okay, so I out into 1 upon CS is my V out, so I out by V in is capital Gm, which is what the trans conductance amplifier give and assume right now K is 1, okay, so Gm is Gm, however, I out by trans conductance, capital Gm times V in minus V out, so V in minus V out is the difference potential, so Gm times Vd is the actual current which is coming out, correct the terms, so V out by V in is 1 upon CS upon 1 upon Gm, Gm by C per a pole, is that correct, Gm is conductance, so 1 upon Gm is resistance, so 1 upon RCS, low pass filter function, 1 upon RCS, pole is that 1 upon RC, but R is 1 upon Gm, so Gm upon C, this is the pole, is that correct, let us say I want a low pass which has a very high cut off frequency, so what does that mean, that is a filter has a bandwidth of 10 mega hertz or 100 mega hertz, C should be small, but C can be created in a silicon circuit, whatever smallest 0.01 perform, one femto farad, tens of femto farad I may create, but its only proportionally Gm should be reduced, isn't it, sorry Gm should be increased, Gm has to be increased what will happen here, size has to be increased, so if there is such a big transistor and there is such a big capacitor, then bandwidth can be increased, but in real life you will not be allowed to increase W by L 4, 8, 12, so this cannot be increased, this cannot be reduced, this means that bandwidth is always limited, is that correct, why Gmc filters are not very very popular, so they are used, where they will be used, less than a mega hertz cut off you want, 100s of kilo hertz, 500 kilo hertz, so this simple filter can be used, what will happen to you, Gm will have to be increased, alternatively where can we increase Gm, I bias say, but if I bias is increased then power dissipate, okay, so we have to now worry in a chip, how much power I will be given to dissipate, what is the smallest value of capacitor I can create during whatever processing I have and therefore what is the maximum size I will be allowed to because the area constraint, so maximum frequencies can never be greater than a mega hertz, is that correct, why mega hertz, that is maths, but in reality such OTA filters can be used only for limited frequency range up to a mega hertz maximum frequency range, preferably 500 kilo hertz, okay, this is the maximum frequency you will get, okay. This C which you can see, why cannot be small, in this we have just said that there are parasitics, they are all coming in parallel, so you can actually make it small, but the rest of the capacitance you will kill it, one of C1 plus C2, C1, C2, C3, so it is difficult to reduce C, net C, difficult to increase GM and there we say GM by C is limited up to a mega hertz or preferably 500 kilo hertz, most of the low frequency continuous filters are OTA filters, most of the low frequency up to 100 kilo hertz, guarantee it works fantastic, so GMC filter, what is the advantage, low power relatively, very few components, so very small chip, very cheap chip, okay and can filter it up to few kilo hertz, 100 kilo hertz, okay. These filters are extensively used in almost every hardware, where audio signal frequencies are in nature, whatever systems you are looking, mostly in audio, then these filters are ideal. This is also a bike ride, because it also uses the same low pass function, I will tell you a figure, 2 OTAs and 2 capacitors can be manipulated by different signals at V1, V2, V3, V3, 0, 1, 0, 1 and you can create all 4 filters out of that, we will come back to it, like next time I will start on this, just wanted to show you, abhi jo ek dikhaya maine issi ko use krte hua, issi theory ko use krte hua, hum ne ek universal filter banaya hain, jo ki chaaro filter design kr sata hai, but none of them will be good, iss ke honne ke ba, then we will show ke a chahi chahiye toh fir kya ka, toh fir butter berth aur shebe chahi tarah najar nalinga aur achha filter banaya hain, okay, yeh circuit mein aapko fir se batonga, just to show you that same block can be utilized to make universal.