 Today we are starting on 24th lecture on active filters. Let us therefore consider what we had done so far we had in the earlier lecture discussed about passive filters RC and RL combination of low pass filters first order filters these are called and then we had seen the second order filter RC RL second order filters these result in cascading of just first order filters which have Q less than half always so they are just having the poles always on the negative real axis of the explain so Q is always less than half that is the characteristic of a passive RC or RL second order filter therefore we if you want to realize higher Q filters it is only possible with the help of passive RLC filter can give you Q of any value low or high low pass RLC Butterworth design filter design Tebyshev filter design inverse Tebyshev filter design all these and elliptic filter design all these can be carried out with the help of only passive RLC blocks we had also discussed about frequency transformation that is an important topic as a means of converting the respective filters like high pass filter or band pass filter or band stop filter to the low pass prototype and carrying out defilter design using frequency transformation and frequency transformation involves conversion of S normalized frequency to 1 over S and that is low pass to high pass transformation we had also then low pass to band pass transformation then low pass to band stop so all these transformations therefore help us in doing the filter design all in the domain of low pass okay and shifting it to the center frequency from 0 to the resonant frequency in the case of band pass okay resonant frequency in the case of band stop but the transformation is S to 1 over S plus 1 over S. So these involve shifting inductor with series resonance at the frequency to be shifted capacitor with parallel resonance okay at the frequency to be shifted as far as low pass to band pass is concerned and the other way about okay in the case of low pass to band stop. And we are today discussing how to convert this passive filters to active filters active RC filters we would call them active RC filters because inductor is an element which is very bulky as far as low frequency signal processing of most of our base band signal is concerned. The inductor becomes too bulky compared to other components therefore is it possible to reduce the bulk that is the primary reason active RC filters started emerging as a means of making the components compatible with the days technology in electronics. So limitations of passive filter is mainly that passive RC filter on the other hand can be only used for Q less than half and therefore there is a need for converting the Q of a second or the passive RC filter from Q less than half to Q any value positive. So active element along with RC can achieve this by one technique called inductor simulation that means every inductor in our RLC network is going to be replaced by a simulated inductor that is the technique. So all of the previous designs of passive RLC filter can be directly converted to this kind of active RC filters with inductor simulation. The problem of large size inductor therefore can be resolved this way that L is going to be L by R this is a time constant in dimension is same as RC time constant that means L dimensionally can be obtained by using a capacitor and two resistors. Theoretically this is the thing dimensionally one like one Henry inductor can be simulated theoretically by using one Farad capacitor and one ohm resistor. So this is what the dimensional equivalence tells us. So let us therefore see whether it is possible to do this okay by inductor simulation technique. Now Q enhancement by feedback okay this is the other technique that we are going to talk about this is mainly going to cover most of the saline and key filters that are available in the early ah presentation of active RC filters. Q of passive second order RC filter can be enhanced by using feedback and amplification. Then the same ah time analog computers were using for second ah nth order differential equation simulation integrators and summing amplifiers for simulation of these differential equations. Simulation of differential equation is mathematically equivalent to designing filters this is what we are going to see to ah later a simulator of a second order differential equation is popularly known as by quad circuit. So we will see the third method which is one of the most popular methods of today's IC design of filters analog filters. The traditional approach to filter design through Q enhancement and inductor simulation are increasingly replaced today by the by quad method because of commercial availability of the universally active filter blocks UA of 42 and UF 10 UF 10 is a switched capacitor by quad. We will see all these ah basic filter ICs also later in the course. Active filters by inductor simulation is the topic of today's lecture. All filters used in base band applications particularly in telephony require large valued inductances resulting in large sizes if you have to use RLC filters. That means if it is RLC filter that is used in bed band that is voice ah telephone ah sort of speech filters then inductor will be 2 large these filters need to be designed as active filters simulating large inductances using active devices and RC components. Let us see how one can methodically understand the synthesis of these ah inductor simulators. There is a popular theorem ah which I would say normally one should present in a network code this called Miller's theorem. A voltage amplifier with gain G having an impedance Z connected between its input terminal and output terminal results in at the input as far as the voltage that is driving the amplifier is concerned that input will see an impedance VI minus G times VI by Z that is the input current. So II by VI is 1 over Z in the input impedance seen as a grounded impedance what is it equal to it is equal to this Z the impedance connected between input and output divided by 1 minus G the G is the transfer function of the voltage control voltage source. So this is what in essence the Miller's theorem is it appears as a different impedance okay if you start with Z it becomes Z by 1 minus G this is what the Miller's theorem says. So let us see how simulation of inductance in series with resistance can be done it is not that we require an ideal inductor to be simulated we may sometimes ah use inductor with a certain Q okay that means a series resistance or a parallel resistance. So when the series resistance is 0 or when the parallel resistance is infinity the inductance is said to have high Q Q equal to infinity the Q of an inductor Q of an inductor is equal to the comparison of the inductive ah reactance with the series resistance how big this inductor is in its impedance compared to the series resistance or okay it is going to be the parallel resistance divided by the inductor. So the that is defined as the Q of an inductor parallel. So actually R parallel can be shown to be Q square roughly into R series this is the transformation for high Q. So this is something that you might have studied in networks course please remember this is an important relationship to be understood in terms of Q of a coil. So let us therefore try to simulate ah inductor L in series with the resistance R1 let us say the same R1 is used for simulating it as Z equal to R1 then impedance in is R1 by 1 minus G that you want it to be equal to R1 plus SL L is again C into R1 into R2 because CR square that is the dimensional equivalence that we had got. So we can put this as C into R1 into R2 then we get G equal to R1 get cancelled and G becomes equal to SC R2 by 1 plus SC R2 which represents a first order high pass filter please understand this this represents a first order high pass filter that is what you are going to put here the first order high pass filter SC R2 by 1 plus SC R2 okay is the G then in order to convert it into a voltage control voltage source we put a buffer stage at the input and a buffer stage at the output you get a voltage control voltage source with this kind of transfer function and you put the Z which is equal to R1 between the input and the output and you get an equivalent which is just this that is what we have just now shown. So R1 in series with SC R1 R2 is got by using this circuit so this can be converted now into a special case now if you just put a short circuit between this and this that means you get rid of this buffer that results in this circuit as a special case of the previous circuit instead of using a buffer then what happens just this R1 is going to load this low pass filter that is all so that results in the equivalent at this corresponding to DDC resistance of R1 plus R2 to ground and the inductor that is simulated which is SC R1 R2 that remains the same the DC resistance instead of R1 now becomes R1 plus R2 you can prove this also by taking this circuit and evaluating the current through this R1 and show that this VI by current through R1 is nothing but equivalent to R1 plus R2 in series with SC R1 R2 this is one of the famous circuits okay and this is never seen as an inductor simulator circuit however this is seen slightly differently you consider that you have a resistance from this point to ground of R1 plus R2 and you want this resistance not to shunt the input so what you do you connect this point the junction between R1 and R2 to the output of this buffer stage thereby shorting this for all high frequency utility that means capacitor is acting as a short okay at any frequency of use of VI so if the capacitance is acting as a short R1 comes between this input and this output since G is equal to 1 okay R1 appears as an infinite resistance that means there is no current that is going to be drawn by R1 okay that means this combination of circuit does not load this input this is called bootstrapping this connection of this point to this output is called bootstrapping so as one of the applications of this kind of inductor you can treat it as an inductor getting simulated in series with R1 plus R2 and at the minimum frequency of interest this is an open circuit compared to R1 plus R2 this is how you can design a bootstrapping circuit it is primarily needed to provide a DC path for the bipolar transistor bias current to flow DC current to flow to ground however it should not shunt the input okay so this bootstrapping helps it to simulate an inductor in series with R1 plus R2 this R2 is split into A and R2B for biasing it at a certain high voltage so that there is a certain current passing through this so this circuit is very commonly used in single supply op amp structures okay where you want to provide from the input a DC path to ground which should not shunt the input. Now let us continue with our discussion on inductor simulation we just said let that non ideal inductor be an inductor in series with R1 why should it be in series it can be a non ideal inductor with inductor in shunt with R1 so R1 by 1 – G should appear as R1 okay parallel SL okay so it is nothing but okay 1 by 1 over R1 plus 1 over SC1 R2 R1 so again R1 and R1 get cancelled and G comes out from this as – 1 by SCR2 it is very important what does it mean it requires an integrator you know this already as an integrator block you convert the voltage to current here by putting an R2 to the inverting terminal ground this is virtual ground so current in this is Vi by R2 that flows through the capacitor and develops a voltage which is plus minus so you get a voltage – Vi by SCR2 this we have discussed earlier in integrator you put the buffer state there it prevents the loading of input by R2 so we have a voltage control voltage source with this kind of transfer function now you put an R1 between its input and output you get the inductor simulated with parallel resistance R1 that means you see that an ideal inductor is sufficient to simulate an inductor ideal integrator now if you remove the buffer state just like we did before that means you connect this to this this you remove then R2 will send that combination so we have R1 parallel R2 with the buffer removed with the same inductor C which is CR1 R2 so this is another circuit using single op amp just like the bootstrapping circuit which can simulate a non-ideal inductor but resistance will be in parallel with the inductor it is a very simple circuit okay using an integrator as the block to simulate a non-ideal inductor. Let us now move over through simulation of ideal inductor how is it going to be possible so we again do the same thing R1 by 1 – G the Z in is going to be equated to ideal inductor SL which is SCR1 R2 so R1 R1 get cancelled G becomes equal to 1 – unity gain and an integration inverting integration – 1 by SCR2. So that is what we have to simulate we have the need for the integrator so the integrator block remains the same however we want to add the input voltage VI to it if you put just a buffer stage this VI comes here this is grounded and therefore it gives you just the integration – 1 by SCR2 that should be an input component directly from the input voltage appearing at the output so how can you do that we have to have a current in the integrator which is VI by R2 so we will lift this output to 2 VI so that this is till VI these 2 are equal resistances so this is R and R this will be 2 VI this is VI VI by R flows here and the same current flows through this and develops a voltage to VI this VI can be connected here so this VI becomes now available to you at the output and addition to this drop which is through the current same current VI by R2 still flows through there and develops a voltage which is – 1 by SCR2 into VI so the voltage here is this VI – VI by SCR2 which is what is wanted by us so through this inductor now the current flows which is VI – VI so VI VI get cancelled plus VI by SCR2 divided by R1 so it is VI by SCR2 by R1 which is an inductive current ideal inductor so ideal inductor gets simulated this circuit is known as a circuit that simulates an inductor connecting a capacitor at a port is called a guireta guireta means rotation okay guiration means rotation so the the phase shift okay between voltage and current changes by 180 degrees in order to make it get converted from capacitive current to inductive current that is what is meant by guires so we have the ideal inductor that gets simulated this way let us therefore design for example a band pass filter a second order band pass filter with center frequency equal to 5 kilohertz and a band width of 1 kilohertz okay this kind of design we had already seen how it can be done with R C and L in shunt so we have seen that this can simulate a second order filter so what will be the transfer function this is R this is C and L we can see that the transfer function is 1 over R divided by 1 over R total admittance SC plus 1 over SL total admittance connected here and admittance linking input to output at the emitter which is nothing but multiplying by SL SL by R okay S squared LC plus SL by R plus 1 that is the band pass it resonates at this frequency this is nothing but written as S by omega naught 2 S squared by omega naught squared plus S by omega naught Q plus 1 so Q is equal to R by root L by C which is set to be equal to 5 in this 5 kilohertz with 1 kilohertz band width will mean 5 by 1 which is 5 that is called quality factor of the band pass filter R that is equal to R by root L by C this we had seen in the passive filters class so that is equal to 5 and omega naught which is 1 over root LC that is equal to 2 pi into 5 kilohertz so from this we can evaluate the for C equal to 0.1 micro field R it comes out as 1590 L for resonance is 9.87 milli henries this is what we have to simulate using the generator circuit that is what has been done so L equal to C R 1 R 2 is 9.87 milli henries R 1 equal to R 2 you have we have taken and that comes out as okay 314 ohms okay putting that value of the induct component 1590.1 micro farad 314 ohms and 0.1 micro farad one gets this simulation result it is it is seen that the center frequency is exactly equal to 5 kilohertz 5.008 it is giving and band width is coming out to be 1 kilohertz this is the state of f adds as far as the simulated inductor is concerned we have ahh please note that output is taken instead of here at this point this difference is it has a gain of 2 apart from what we had written here that means the if you take the output at the ahh op amp first op amp output it will be twice this input so this output is going to be actually twice S by omega naught q divided by 1 plus S by omega naught q plus S squared by omega naught squared that means when it resonates at omega naught the output will be equal to input here but output will in this output will be twice this input so we get again of 2 that is what is seen in the simulation also this particular point it is going to be 2.2 it is shown as 2.133 this is the effect of the gain band width product of the op amp that we will see later that means it is different from what we have designed it for because the q has got enhanced because of the gain bandwidth product the transient response of this even for large signal you can see that it is going to be a band pass filter so whatever impulse that is applied here okay that is going to cause ringing and as pointed out earlier in our passive ahh filter design lectures the q is equivalent to count the number of peaks go from this peak to 1 tenth of that value so we have gone from this to 1 tenth of its value and the number of such peaks that you get equal to q so we get 1 2 3 4 5 so thales with whatever design we have carried out now we have increased that series resistance here from 1592 ahh twice its value okay so 3180 so we get double the q that means q is going to be 10 so q of 10 3180 is the resistance used in series so it has been double that means q gets doubled 1 2 3 4 5 6 7 8 9 10 going from VP to 110 VP now increasing q by negative resistance we can enhance the q of such a circuit further by negative resistance simulation this indicating a basic principle of simulating whatever you want negative resistance we have simulated inductor the q is limited by this series resistance let us now enhance the q by simulating a negative resistance across the inductor this is VI and this is 2 VI so the moment you put a resistance or shunt what happens again same middle theorem is applicable at the input so we have Z in shunted by another Z in dash let us say Z in of the inductor is shunted by Z in dash which is now another resistance connected to the input amplifier input and output R shunt divided by 1 – 2 which is – R shunt that any resistance that is connected to the gain of 2 amplifier between S input and output APS as a negative resistance that means the inductor is shunted by R shunt which is negative so that can compensate for this positive resistance and result in an ideal tank circuit what is the ideal tank circuit is nothing but an oscillator so this will again use in synthesis of oscillators later but now we are actually using it for enhancing the q of an inductor that means the shunting effect should be such that defective parallel resistance which is R shunted by – R shunt okay which is nothing but let us see R into – R shunt divided by R – R shunt which is actually the Z in dash which is R shunt divided by R shunt – R as long as R shunt is okay greater than R it is still positive and therefore it is finite q okay so it can be used as a filter with enhanced q now I am demonstrating it by making RP equal to 3140 3180 is the resistance that we had put earlier in series this one so that can be compensated for by putting a negative resistance which is just slightly lower than this R that means it is going to be effectively negative that means a transient response of that will be exponentially increasing without even any input therefore the output will be at frequency F naught okay which is 1 over 2 pi root LC which is in this case 1 over 2 pi RC so it will F naught 2 pi RC becomes an oscillator with unlimited amplitude that is why as far as this op amp is concerned it is 741 op amp biased at okay plus – 15 volts and therefore it goes all the way up to about 12 to 13 volts and gets chopped off distortion sets in okay so that means it is exponentially increasing this is the transient response if you just further decrease the value of negative resistance then it goes into increase amplitude of oscillation and I mean we are increasing the frequency by changing see 2.01 micro farad micro farad and with a gain banded product of 741 being at 1 megahertz it now goes into limitation by this lurid it tries to oscillate but the rate of rise demanded is too much okay for it to oscillate at this frequency which is F naught equal to 1 over 2 pi RC and therefore it goes into what is called oscillation limited by both in amplitude and frequency limited by the maximum rate at which output of the op amp can rise that is 1 volt per micro second and not by saturation this just I want you to make a note of that you cannot therefore design any filter that you desire okay with any value of Q okay that is very high it might go into instability because of the effect of we will see the gain banded product of finite gain banded product of the op amp and this lurid so this is the topic that we have to discuss now effect of active device parameter that is the most fundamental limitation of active filters when simulated inductance is influenced by the parameters DC gain of the op amp it is transistor is the DC gain of the transistor amplifier and the gain banded product GB of the op amp again if it is a transistor it is the FT of the transistor MOSFET or bipolar that comes into picture that will limit the performance of the inductor that gets simulated using these active devices so it means it is sensitive to the active device parameters like finite DC gain and finite gain banded product out of this I want to show you that as far as op amp is concerned it is the finite gain banded product which is ultimately the deciding factor regarding how high a frequency this component will work as a simulated inductor garator circuit uses non-inherting amplifier of gain 2 here as we have seen and an integrator okay which is going to give you a transfer function of – 1 by SCR okay so from here it is going to be 1 plus 1 over SCR okay with half the voltage coming into picture or VI coming into picture so G is equal to 1 – omega naught by as is the ideal transfer function of this garator block from here to here now what happens let us analyze G changes that effect of change in G because of the gain of 2 amplifier in the front end and cascaded to an integrator next is to change that we have seen already that as far as the first stage is concerned it is going to be 1 by 1 plus okay 1 over loop gain so that loop gain is a by 2 as far as the buffer stage is concerned the next one is the integrator that is whatever integration it does – omega naught by as divided by 1 plus again 1 over loop gain which is this we had already done in discussing discussion on amplifiers so 1 over loop gain now becomes for the integrator 1 plus omega naught by as divided by 8 so this can be approximated as 1 – omega naught by as into 1 – 2 by a plus 1 by a which is 3 by a – omega naught by SA that is the first order effect on the that is why it is approximately equal to this the full effect is 1 into this 1 – 3 by a – omega naught by SA – omega naught by as gets modified as 1 – 3 by a – omega naught by SA so if you now observe the effect of this on the gain this is equal to 1 – omega naught by as – omega naught by SA plus 3 omega naught by SA plus omega naught by S square E full effect now taking the real part and the part that is controlled by S separately the one changes to 1 – omega naught by omega naught square okay so this is the part omega S equal to j omega is what is put we are investigating the whole thing finally at the resonant frequency so we are interested in the effect of the gain band with product and gain on G so effect of G is seen here as okay S equal to j omega naught is what is substituted so S square becomes – omega naught square so this brings in 1 by a naught – 1 by a naught here and omega naught by as into 1 plus 1 over a naught – 3 over a naught so this is the effect on the inductor that is simulated whatever is the coefficient of S effects the inductor value and this real part effects the resistive part that is getting shunted to the inductor that is simulated so you can now do R by 1 – G okay G being replaced by this then you get this as R by 1 by a naught when a naught is infinity this goes to 0 that means this is a resistance shunting the inductor and magnitude of the resistance shunting the inductor is a naught into R and the inductor okay which is 1 by SL okay L is equal to R by omega naught originally omega naught is 1 over CR so that is why L was equal to CR square that gets modified as 1 – 2 by a naught if the inductor value okay just slightly increases because of finite gain okay or the frequency of resonance shifts down that is of not much concern what is of concern is that the Q of the inductor becomes limited by the finite DC gain because of this and Q is going to be equal to a naught so instead of infinity Q is going to be equal to a naught L is going to be R by omega naught dash that is what happens this is the modification due to finite gain inductance is shunted by a negative by a not a negative resistance a positive resistance R into a naught okay the the inductor value is going to be slightly enhanced because of the finite gain finite gain okay now let us look at the effect of finite gain bandwidth product effect of finite gain is that inductor is shunted by R a naught that means Q is changing to a naught Q of the system is and inductor value is slightly increased now we see the effect of let us see gain bandwidth product so instead of a equal to a naught you now substitute a equal to G B by S in the same expression for G so when you substitute a equal to G B by S we see that original 1 minus omega naught by S changes to minus S by G B okay that is so frequency dependent okay capacitor actually the inductor gets shunted by a capacitor and the gain increases to 1 plus 2 omega naught by G B when you put a equal to G B by S and S equal to J omega naught this is what happens okay this becomes minus omega naught squared okay so this becomes minus 1 this is G B by S so 1 minus G B by S that becomes and this is going to be 2 omega naught by S A which is G B by S which is 2 omega naught by G B so it is going to be that if you now substitute R by 1 minus G as the whole thing R by 1 minus G and with G as this value you will see that this will be R divided by okay 1 gets cancelled this will be omega naught by S okay by R okay plus S by G B R okay minus 2 omega naught by so let me write that here equals 1 by omega naught by S R that is the inductor okay it is slightly changed because of the effect of this S by G B R capacity effect then minus 2 omega naught divided by G B into R that is the negative resistance okay this is so apart from the positive resistance you have a negative resistance which is R G B by 2 omega naught instant with the inductance there was a positive resistance of R A naught okay and that effect is due to finite gain okay so we had the positive resistance then now that is going to be further changed okay so that is 1 by R A naught apart from that we have a negative resistance due to the finite gain bandwidth product so that is the equivalent circuit that we see ultimately for the whole effect of A naught as well as finite gain bandwidth product of the inductor that is simulated let us therefore look at the whole thing so band pass filter with similarity and inductance C equal to 0.1 micro farad R equal to 1 kilo ohm V naught V I equal to 1 okay so we have seen that this the gain changes actually what happens the gain also changes because of the effect of Q changing so what is the gain change it is the same as that is happening okay due to Q change so let us look at the gain change so it is 2 divided by 1 minus we will see due to the Q actual changing to omega naught divided by negative resistance which is corresponding to this apart from R okay into A naught due to finite what is that DC gain of the op amp we have a negative resistance which is RGB by omega naught something that so the actual Q changes to okay this value and changing the gain at the centre by this amount to Q omega naught by GB this is what happens so let us consolidate the whole thing earlier the Q was decided by the DC gain so it was R 1 over R into A naught okay that was the Q now because of the negative resistance Q changes to minus GB R by 2 omega naught this is the negative resistance this is the positive resistance okay effect. So the Q was actually controlled by R right so that is that is coming in shunt so that R actually earlier the Q was this okay Q actual was this that it is now changing to this that is therefore defined the Q determined by R A naught as Q then you can see that it is going to be Q due to finite gain banded product is equal to 1 over 1 over Q minus 2 omega naught by GB so we have this becoming equal to Q by 1 minus 2 omega naught divided by GB. So if you say that you have an effective Q determined by this this is the capacity this is the inductor CR square because of the negative resistance coming across this which is GB R divided by 2 omega naught the effect of Q is Q enhancement takes place due to finite gain banded product that is what changes the gain also by the amount small amount 1.035 at the input that times 2 is the actual gain at the output of the op amp. So we have purposely increased the Q to 100 and the negative resistance remains the same the gain changes now from 1 to 1.47 by calculation if you use the enhancement method and it is simulation is giving 1.516 so it is closely approximated okay to whatever we have actually theoretically evaluated as Q enhancement and we make the thing oscillate by actually increasing the frequency further from okay 1.59 to 15.9 by increasing both the capacitors decreasing both the capacitor by a factor of 10 frequency increases by a factor of 10 the circuit oscillates negative resistance emulated is 31.4 kilo ohm positive resistance is 100 kilo ohm is nothing but RGB by 2 omega naught after being equal to 15.9 kilo hertz GB being 1 mega hertz okay with the same value of resistance R you get a negative resistance simulated because of the finite gain banded product as 31.4 kilo ohm that is much less than 100 kilo ohm so it goes deep into slew rate mode of operation just the same effect which we had seen earlier okay so the slew rate limitation comes into picture limiting the operation frequency due to slew rate okay which is 1 volt per microsecond in the case of operational amplifier and for 1 the amplitude also gets limited by slew rate this is the output okay across the so called simulated inductor this is the output of the ohm op amp and you will see that it is violating at this input this is at one terminal this where did we observe all these effects we have observed this effect okay at the inputs two inputs actually if it is working as an op amp okay at all times this voltage should be roughly equal to this voltage whereas you see because of the slew rate limitation right this whole thing has got converted into yeah these two voltages are different the moment those two are different the op amp is never considered as a null later because its limitation has come into picture these two voltages are becoming different output is always rising at slew rate okay and therefore input voltage is not 0 okay only at this point it is acting as a null later again again it is trying to trying its best to rise at maximum rate possible for it and then again changing the sign here okay so is trying to follow the input point so this is expanded that to highlight the whole issue now sensitivity of the actual Q to the gain bandwidth product can be illustrated by that this Q is the one due to finite gain A naught finite DC gain and that gets modified as 1 minus 2 omega naught Q by GB because of the negative resistance that comes about due to the finite gain bandwidth product so sensitivity of Q to GB is roughly equal to if you differentiate this and see it is minus 2 omega naught Q by GB itself higher the Q higher is the sensitivity okay sensitivity of omega naught by GB is on the other hand not so much it does not depend upon the Q just depends upon the GB it is pretty small that if this Q is very high okay then the sensitivity of gain bandwidth product to Q also is very high so we cannot design high Q inductors okay using this kind of technique unless we select the gain bandwidth product to be very very high compared to F naught so this gives us the limitation due to the gain bandwidth product so this is summarized here Q enhancement takes place due to the finite gain bandwidth product Q actual is equal to Q due to the finite DC gain or the actual resistance that is put across the inductor divided by 1 minus 2 omega naught Q by GB omega naught actual is equal to omega naught by 1 minus omega naught by GB this omega naught active Q active sensitivity is too high for high Q structures high Q inductors so this is the limitation this limitation due to active parameter sensitivity comes about in simulated inductor unless we take care of proper selection of the gain bandwidth product of the op-amp and the slew rate so these two fundamental limitations have to be borne in mind in the design of inductor for high frequency application. Now coming to the generalized garret we replace all the resistors by impedances these are all resistors and this was capacitor and this was also resistor so replace these resistors by Z 1 Z 2 Z 3 Z 4 Z 5 then we can show that Z in equals we can do that very simply this gain from here to here is now 1 plus Z 2 by Z 1 times VI okay then this is VI itself so the current in this is VI Z 2 by Z 1 divided by Z 3 VI VI get cancer so that flows through this and this voltage is going to be VI this VI minus VI Z 2 Z 1 Z 3 into Z 4 so and the current in this therefore VI VI again get cancer is going to be VI Z 2 Z 4 by Z 1 Z 3 Z 5 this so Z in seen is Z 1 Z 3 Z 5 by Z 2 Z 4 and let us consider this circuit that we have already discussed okay with all the Z 1 Z 2 Z 3 and Z 5 replaced by R and Z 4 alone is 1 over SC that is one inductor simulator the next one we can have instead of that Z 2 as capacitor and all the others as resistor and it simulates the same inductor so using one topology we can get two varator circuits this way actually you can multiply the number of circuits different circuits that can act as garretors so in conclusion we say that the garretor simulation of the inductor this is the grounded inductor simulation is one of the ways of replacing RLC circuit where inductor is grounded by simulated inductors and getting the high Q required for the application of all types of filters okay we have illustrated it by using a band pass filter in the next class we will see the other technique leading to some of the saline and key filters which are well known that is negative feedback and positive feedback being utilized to enhance the Q.