 Today our 25th lecture will be continuing with active filters in the last lecture we had already discussed how passive RLC filters can be converted to active filters by inductor simulation techniques. Inductor simulation we had discussed non-ideal inductor with series resistance with parallel resistance without any resistance okay all these cases. And that circuit is known as guirater that can convert the capacitor terminated into an inductor that is a guiration means rotation by 180 degrees between voltage and current okay. To convert RLC filter to active filters this inductor can be therefore used active and passive parameter sensitivity is now come into picture although this active parameter is an important criteria for selection of the filter topology effect of finite gain and gain finite gain bandwidth product how this effects the inductor that is simulated using an active device is also discussed earlier. Now from that a criteria for design of a filter arose wherein the F naught of the filter into Q F naught is normalizing frequency should be much less than the gain bandwidth product of the active device. This is what we had shown in the last class today anyway we will continue after we indicate the Q of the inductor simulated is dependent upon the DC gain A naught of the op amp and GB of the op amp in this fashion. So A naught is the Q of the inductor if GB is infinity and if A naught and GB both are finite then that is the variation in Q it appears as a negative resistance of magnitude GB by 2 omega naught shunting the inductor whereas due to finite gain A naught it appears as a positive resistance which is A naught into R negative resistance is GB into R by 2 omega naught. Q of the filter simulated using such an inductor is going to be Q A whatever you have designed it for divided by 1 plus Q A by A naught minus 2 omega naught Q A by GB. So this is the limitation of the inductor simulation circuit. The quality factor now how to enhance it in another fashion that is what is normally adopted in what are called as salinity networks the Q enhancement takes place due to feedback negative or positive. The quality factor Q P of the second order passive RC filter since it is always less than 0.5 it is unacceptable for a general filter design. So salinity proposed use of negative or positive feedback to enhance the Q several topologies similar to salinity filters are possible these are going to be now synthesized systematically by us. Second order passive filter let us represent this by a polynomial numerator divided by denominator this is the way the passive filter is represented numerator by denominator say it is a second order filter okay then DS is primarily normalized S squared by omega naught squared plus S by omega naught Q this we have been adopting from the beginning is the way we have normalized the constant is made equal to 1 and then this is the normalizing frequency omega naught this is the Q of the filter. So if it is passive let us therefore term it as passive and Q P is less than 0.5 that is the characteristic of the filter and omega P is equal to 1 by RC. Now such a numerator polynomial by denominator polynomial is embedded in an inverting amplifier topology like this so that there is a loop feedback loop which is negative because of this inversion here and summation here it is a negative feedback loop. So V naught by VI now becomes converted okay this we had shown earlier equal to minus K into NS by DS divided by 1 plus K NS by DS which is minus K NS the numerator polynomial remains the same as before okay except for multiplication by minus K denominator polynomial gets modified as this DS plus KNS this is the technique that is adopted to change the characteristic polynomial of this network okay which is embedded in an amplifier like this. So what happens for a general second order passive RCRL filter if you write NS by DS as the denominator being the standard form that we had used earlier so numerator on the other hand is again normalized with the same normalizing frequency as denominator. So it is therefore M integer S squared by omega P squared plus N integer positive or negative S by omega P plus P positive or negative okay integer. So omega P is the natural frequency of the passive RC filter QP is the quality factor of the passive second order RC filter. So in such a situation V naught over VI gets transformed as numerator polynomial remains the same as before denominator gets modified as 1 plus MK 1 plus NK QP plus 1 plus PK. So what happens now omega naught natural frequency is nothing but this if you consider as A B and C root AC by B is the Q okay and root C by A is the natural frequency if you do that it is square root of 1 plus PK divided by 1 plus MK into omega P that is the modified natural frequency of the system feedback system Q also gets modified as QP divided by 1 plus NK QP square root of 1 plus PK square root of 1 plus MK this is an important transformation that we are looking for so this is what we are seeking this is the consequence of what we are doing right. So in case we want to improve the Q from the original QP which is less than 0.5 we have different methods possible PK can be positive MK can be positive okay MK into QP can be a negative quantity less than 1. So these are the possibilities of improving the quality factor. So PK being positive M equal to 0 N equal to 0 is first considered okay there is the low pass original passive filter then P0 MK positive M0 there is the high pass filter as the starting point in the passive block then P and M0 okay being NK being negative okay such that NK into QP is less than 1 is positive feedback okay situation. So these are the possibilities that are going to be discussed one after the other then when both P and M are present and N is 0 that is the notch filter situation okay that also is going to be discussed okay. Finally we are going to have P and M equal to 0 and NK QP a negative but less than 1 that is the positive feedback structure. So these four topologies for enhancing the Q will be discussed in the course of these lectures. Enhancement of QA can be therefore carried out by using any one of these combination that I have already mentioned about. So let us start with the low pass RC filter. So this transfer function we have written it so many times 1 by 1 plus SC1 R1 plus SC2 R1 plus R2 that the coefficient of S. S squared means 2 capacitors coming SC1 R1 SC2 R2 so that is the transfer function. Omega P is recognized as 1 by circle top R1 C1 R2 C2 and QP is this and R1 equal to R2 equal to R C1 equal to C2 equal to C omega P is 1 by RC QP is 1 by 3 this is a simple design procedure active low pass filter okay how to convert this. So this network with this value of R and C is embedded in this inverting amplifier with gain K the natural frequency of the active filter is now omega naught equal to square root of 1 plus K okay into omega P root C by A. So and QA is equal to QP divided by square root of 1 plus K. So this QP into square root of 1 plus K. So both omega naught and QA get enhanced by a factor of the original frequency and Q by a factor of 1 plus root of 1 plus K. By suitable selection of K we can design it for a specific Q and thereafter fix the time constant omega P which is 1 over RC based on what omega naught we desire for the low pass filter. The structure of the active low pass filter now is that we want to add VI in such a manner that this transfer function remains the same as what we have just derived okay. So for that we add VI along with the feedback voltage in this following manner by using an adder network which is passive here. So I make this R1 split into 2 R1 and 2 R1 so the convenience resistance is still R1 but it has a voltage which is getting added which is 2 V naught plus I mean 2 V naught okay with the convenience resistance of R naught 2 V naught into half which is V naught itself okay and VI into half okay that means instead of gain of K we should use in order to fulfill the same feedback we have to double the gain as 2K. So if that is done that is done actually by using a network like this now along with addition that we are just now mentioned. So a voltage control voltage source with gain 2K inverting amplifier gain of 2K is used in the following manner buffer followed by a inverting amplifier. So this achieves the transfer function that we have earlier depicted so let us design a second order Butterworth Lopez filter bandwidth equal to 40 hertz QA is 1 over root 2 for the second order Butterworth R1 equal to R2 equal to R C1 equal to C2 equal to C QP is equal to 1 by root 3 1 by 3 and therefore QA is equal to 1 by 3 into root of 1 plus K equal to 1 over root 2 which gives K value of 3.5 that is the K omega naught can be evaluated 2 pi into 40 is equal to root of 4.5 divided by R C. So R C comes out as 84 nanofarads for an R of 100 K that is the design it is over. So that can be simulated by putting those values in the circuit shown earlier in this circuit and that is related and you can see the cutoff frequency is very nearly what we have designed obviously that it is 40 hertz okay at the 0.7 point this very nearly equal to 1 actually it is K by 1 plus K less than 1 that is what you see K being equal to 3.5 K by 1 by 1 plus K is the numerator gain. So if K is high it goes to very nearly equal to 1 this is indicating that the Q is 1 over root 2 just suggesting a small hump there. Now let us enhance the Q Q is equal to 5 and F naught is same 40 hertz R is equal to 100 K C is equal to 0.6 micro farad in order to bring the F naught to 40 hertz C has to be enhanced to 0.6 micro farad K is 224 now because Q has been enhanced. So root K plus 1 by 3 is equal to 5 so root of K plus 1 is equal to 15 K plus 1 is 225 so K is 224 and knowing this value of root of K plus 1 root of K plus 1 that is 15 divided by RC 2 by RC is made equal to 40 hertz from which you get the value of C for an R of 100 K which is equal to 0.6 micro farad that is how you get the thing and we see that the frequency is 39.405 it is slightly reduced we will see the fact of the active device gain is to reduce the frequency of normalization from the 40 to 39.405 very nearly same. However Q has been enhanced considerably it has changed to very nearly 5.5 5.481. So this we have to reason out why it is getting enhanced from the design value of 5 so this is due to the gain band with product we will see this later qualitatively Q equal to 5 F naught is increased now F naught is increased from 40 hertz to 400 hertz with a C of 69 of errors. Now you can see that this is very nearly 391 hertz but the Q is nowhere near what we have designed it is 5 that we have designed it for but it is 36.555 in simulation. Now let us investigate why the Q is changing as the frequency is increased Q is increasing okay from the design value of 5 and that is the transient response actually Q of 5 means that should have been only 5 peaks and we see now very nearly 3740 peaks in the whole transient response of this for the same circuit okay. So what is it due to let us just therefore increase the frequency from 400 hertz to 600 hertz by using capacitance equal to 14 and of errors Q is still the same 5 what happens it does not need any input it is just going into exponentially increasing oscillation at the natural frequency of 600 hertz right. So it is only limited in amplitude by the supply voltage which is nearly 15 volts for this 741. So this is due to this oscillation or instability in the circuit which has been designed as a filter but it works as an oscillator okay oscillating at this natural frequency is due to the finite gain bandwidth product of the op amp. So Q increases from the specified value these are the effects seen in the experiment that we have carried out the natural frequency reduces slightly from the specified value at higher natural frequencies the transient responses are more oscillatory indicating Q enhancement effect beyond a certain natural frequency the system becomes unstable goes into oscillation at the natural frequency these are the observations that we have made by designing a high Q circuits at higher frequencies using the same topology and op amp. Now that is the effect of gain bandwidth product that we are going to investigate now amplifier using a buffer and an inverting amplifier of gain K has a transfer function this we already know if you are using a buffer the effect of the finite gain bandwidth product is 1 plus 1 over loop gain of that. So this is the loop gain GB by S is the loop gain so 1 over loop gain is 1 plus S by GB. Similarly for that of the amplifier inverting amplifier with gain of 2K the error in phase that is mainly this is the error in phase that is arising 1 plus 1 plus 2K into S by GB that is 1 over loop gain of that inverting amplifier combined together approximated when omega is much less than GB this is 1 minus 2 plus 2K into S by GB this is the cumulative phase error that is causing a total phase lag in the gain stage that we are using that is responsible for the Q enhancement in the loop okay always remember that the additional phase lag caused by the finite gain bandwidth product is the 1 that is causing us the trouble. So transfer function of the active low pass filter now changes wherever K is there replaced that K by K into 1 minus this phase lag. So 1 minus this phase lag okay comes in the numerator that is of not serious consequence however in the denominator the K changes from K to K into 1 minus 2 into 1 plus K into S by GB that gives an error coefficient getting subtracted from the already low coefficient of S okay and that results in modification of the denominator polynomial as S square by omega naught square which is the standard way of normalizing with the constant is made now equal to 1 that 1 plus K is going to be dividing throughout okay so K by 1 plus K here okay we get this modified as minus this factor due to the gain bandwidth product gets subtracted that means the coefficient of S gets reduced which is going to make the Q go here okay effectively therefore QA due to finite gain bandwidth product is QP into square root of 1 plus K which is what we normally design it for using the slow pass filter feedback however it gets enhanced by this factor and this factor can be rewritten as 2K into omega naught into QA by GB that has to be much less than 1 in order that it has no influence on the designed Q. So this is an important conclusion that we are arriving earlier we had seen that inductor simulator it is sufficient if you have omega naught QA by GB much less than 1 here since you are using an amplifier with gain equal to 2K the phase error has got multiplied by 2K okay omega naught by GB is the phase error that into Q is to be made much less than 1 so that means 2K for high Q because K root K plus 1 by what is that into QP is equal to Q actual right. So K plus 1 is equal to Q actual by QP QP is 1 by 3 actually in this case. So this K plus 1 is going to be this squared so that means K is prior proportional to the actual Q squared for high Q it is going to be very high. So the error is going to be considered this is what is to be borne in mind in selecting the op-amp okay therefore 741 is inferior to let us say LF356 or TL082 okay because there is a tenfold increase in the gain band with product in these two compared to 741. So if you are designing filters for high frequencies it is better to go for TL082 or LF356 okay QA is equal to 5 F naught equal to 40 hertz K is equal to 3.5 okay QA due to GB is going to be this 5 divided by 1 minus this factor is going to be such that is almost very nearly equal to point if you calculated for the concern the frequency F naught is equal to say 40 hertz GB is equal to 1 mega hertz that is for 741 and QA is equal to 5 that into 2 K is 224 right. So if you calculate that it comes out as 5.55 the actual experiment has shown as this to be you can see this to be 5.481 so that has got enhanced for the same Q and frequency okay it has got enhanced to 5.481 this is the simulated result and theoretical calculation has shown as that to be very nearly 5.55. And second example QA is further into 5 F naught equal to 400 hertz K equal to 224 GB is going to be this QP which is 5 by 1 minus 0.894 also so this quantity comes out as the earlier case it was 5 by 1 minus 0.0894 right so which is very nearly this right. So we have this coming out to be close to 48 let us see what it is in the simulation so in the simulation it is nearly equal to 37 okay. So all these approximations are valid only when this quantity please remember is very very small compared to 1 when this is coming close to 1 these approximations are not strictly valid okay. So what is the limitation due to GB on our power this thing for inductive simulation circuits it will be 2 F naught QA by GB should be much less than 1 whereas here it is 2 K into F naught QA by GB much less than 1 for the filter to be stable in case of filter using feedback. This is the conclusion that we can draw and now let us design a better worth low pass filter that is this 4th order 4th order better worth filter 4th order means it is can be designed by cascading 1 second order with another second order what is the logic behind this this first second order with low Q that is corresponding to 1 by 1.848 is 1 Q 1 this is Q 1 and is cascaded to another which has a high Q Q 2 is equal to 1 by 0.765 so this is the peaking that it shows with Q 2 equal to 1 by 0.765. So when you combine this when you multiply this you get this decrease is compensated for exactly by this increase so that it becomes exactly flat and of course beyond a certain point both of them are decreasing so this decreases rapidly. So that is the better worth okay of 4th order okay which is actually root of 1 plus X to the power 2 to power 2N that is 8 okay X is equal to omega by omega naught okay. So this is the better worth that has been designed and we are realizing this using the same network with 741 as the op amp so that is a finite gain bandwidth product which is enhancing the Q okay in this particular case okay the Q enhancement is seen clearly because it is a higher Q circuit. So we can see that the effect of gain bandwidth product is that this peak is raising more and this is not so much changed this change is only about 1% and this change from 1% from the theoretical value this change is on the other hand 12.4% from the theoretical value. So there is a peaking on this okay high pass filter design M equal to 1 now this can be actually concluded that whenever we are designing these filters suppose you are designing a higher order low pass filter function which is maximally flat let us say and it has a combination of low pass low Q filter with high Q filters cascaded then this one may be one such thing this does not get disturbed much but the higher Q things will be peaking more so effectively let us say a typical response will be looking like and something like this okay is what is going to appear due to because as you come closer and closer to the pass band edge you are having higher end higher frequency filters with higher Q design. So they are going to peak more due to gain bandwidth product if you use the same op amp for all the filter blocks of the second order then this is what is going to happen for a response which should have been strictly this okay. So this we can conclude qualitatively without doing any mathematics because of the effect seen here the Q is going to be enhanced frequency is going to move down slightly. Now let us go to high pass filter as a means of improving the or enhancing the Q. So M is equal to 1, N is equal to 0, P is equal to 0 earlier we had P equal to 1 and M and N equal to 0. So then V naught over VI in that embedded situation of inverting amplifier will give you minus K into S square by omega P square the original high pass nature of NS is retained but the denominator gets modified only this 1 plus K comes for S square by omega P square other coefficients remain the same as before. Then omega naught becomes root 3 by A so omega P now omega naught changes to omega P divided by square root of 1 plus K and Q obviously okay is equal to Q P into square root of 1 plus K because omega P Q P okay remains equal to the same here coefficient of S as 1 over omega P Q P. So we get this as Q A which is the same as the low pass case enhancement by a factor of root of 1 plus K. However the natural frequency decreases by a factor of 1 plus K compared to low pass where it increases by square root of 1 plus K. Now this is the network high pass passive filter network CRCR and active high pass filter conversion minus K with summing. Summing using capacitors we can do. So then we get minus K S square by omega naught square S square by omega naught square plus S by omega naught Q plus 1 omega P square is 1 over this omega naught is this Q is this and Q A so Q P into square root of 1 plus K required Q A is obtained by selecting K as before omega P is determined by this relationship that we had seen here. Topology of the active high pass filter now summing can be done by earlier we did using resistors split it into 2R and 2R. So here now we can use the same addition of input and feedback voltage using C1 by 2 and C1 by 2 capacitors so that the transfer parameters and feedback loop gain remains the same as before okay and V naught over VI remains same as before. If the lower cut off frequency selected as 0.4 hertz typically let us see assuming C1 C2 equal to C R1 R2 equal to R this is typically that of the biomedical designs of low frequencies Q P is equal to 1 by 3 as Q A is equal to 1 over root 2 we are designing for maximally flat response whatever filter of high pass okay K comes out same as 3.5 same as before omega naught now is 2 pi into 0.4 which is 1 by RC into root of 1 plus K so we get this for R equal to 100 K C comes out as 1.8 micro farads this is the simulated result it shows the what is the cut off frequency exactly at the 0.4 hertz this one so this is 0.4 hertz so the design is okay. Now you can convert this topology from 2 op amp to single op amp just connect this to this only thing is the R2 that earlier has been used is going to be a parallel combination of this with this that means we can call it R3 parallel R4 okay is made equal to R equal to R1 C1 equal to C2 equal to C then all the design equations and the Q's and these natural frequencies remain the same as before except that we are saving one op amp in the whole thing Q of the is enhanced by a factor of root of 1 plus K natural frequency for the low pass is multiplied by root of 1 plus K into OP omega P natural frequency of the high pass filter is omega naught equal to omega P divided by root of 1 plus K so it gets reduced and let us now investigate the effect of high pass filter with F naught equal to 400 hertz Q equal to 5 same 400 hertz for which we have designed our low pass we are designing the high pass now so that we can compare the effect of gain band product in these two low pass and high pass case that is the intention Q is also maintained the same as before equal to 5 K becomes the same as before because Q is equal to root of 1 plus K divided into QV which is 1 by 3 so we get the same square root of 1 plus K is equal to 15 15 squared is K plus 1 so K is equal to 224 F naught is equal to 400 hertz R equal to 100 K gives C equal to 265 pick of parents the circuit is ready and you can simulate that now see what has happened the effect of gain band product is directly seen here at this increase frequency of 400 hertz we had gone from 0.4 to 400 okay large change in the frequency right and the Q also has been increased from 1 over root of 2 to 5 so effect of gain band product is clearly seen the actually at the in all these cases gain at the resonant frequency is always equal to Q okay so Q can be directly measured by measuring the gain at the resonant frequency so for IQ particularly so this should have been 5 whereas you can see that it is 2.623 that is huge variation from the expected value of 5 whereas what has happened in the case of low pass let us look back the Q has that enhanced considerably okay in the second example this is the same example same frequency same Q and same K so the Q from original 5 had jumped to 48 about 10 fold increase in Q right due to gain band product here it has got decreased so we had to see the effect of this so it has got decreased from 5 to 2.623 in the case of high pass filter okay working at the same cut-off frequency or natural frequency right as the low pass filter okay and in the low pass filter case it is 10 fold increase from 5 now let us do the same thing K is replaced by K into 1- the total phase shift of the inverting amplifier plus the buffer so 2 into 1 plus K into S by GB minus this is what happens and this 1 plus K is getting modified as 1 plus K into 1 minus 2 into 1 plus K S by GB that is the only variation rest of the things remain the same as before what happens now this particular thing is having S square by omega P square S is going to be replaced by in all these approximations we are investigating at omega naught the natural frequency of the system the effect okay so this will be minus omega naught square by omega P square okay so that minus is going to make this error become plus okay so this error becomes now S by omega P QP to 1 plus this S is still there right so this S square is replaced by J omega naught square which is minus omega naught square so minus omega naught square by omega P square comes in and you can modify this and you see that this becomes 1 plus 2 K omega naught QA by GB not this in the case of low pass filter it was 1 minus it remain minus right because of T change time change caused by this this became 1 plus that means instead of lag error it is lead error that is causing this Q to decrease Q is going to decrease from its original value of QP into square root of 1 plus K okay by a factor of this so this is what has happened okay actually speaking this factor is about we have seen okay in the case of this frequency this factor is going to be 0.894 also so that therefore is going to change the Q from its original 5 to 5 by 1.894 okay which is nearly what you get in practice so it is going to be theoretically 2.63 whereas we have seen that it is about 2.623 which is pretty close approximation that gives you clear idea that two enhancement occurs in the case of low pass filter by two actual divided by 1 minus the same factor which is 1 minus 0.894 okay whereas in this case okay it is going to be decreased by QP into square root of 1 plus K divided by 2.1.894 which is 2.63 so in conclusion let us see what are the things that we have learnt today so we have seen that by increasing the constant in the denominator or increasing the coefficient of S square by omega P square we can enhance the Q by the same factor so that means if you embedded the low pass filter in an amplifier inverting amplifier and add at the input okay feedback and input then the Q gets enhanced from the original QP which is if all the resistances are made equal and capacitance are made equal in the RC filter this QP is 1 by 3 so it will be 1 by 3 into square root of 1 plus K is the amplifier gain okay which has been made 2K as far as we are concerned so that the loop amplifier gain is still K so and then omega not of the filter is going to be omega P into square root of 1 plus K so both 2 and frequency increase by a factor of 1 plus K in the low pass filter type of feedback negative feedback in the case of this is for low pass for the high pass filter Q remains same as before QP into root of 1 plus K however omega not gets decreased by the factor by which Q is increased and in this particular case GB finite GB is going to enhance Q okay this means Q will be QP into square root of 1 plus K divided by 1 minus okay 2K F naught by GB into Q actual this is the Q active so that is the enhancement effect whereas in the case of high pass it is going to 1 plus F naught Q so however they are both causing the Q to change due to the finite gain band product that means it is going to different from the design value so you have to select the gain band product of these op amp such that this factor 2K into F naught into QA by GB is much less than 1 this case it does not become unstable however in this case it is likely to become unstable if this comes close to 1 okay and therefore it is a precaution that you should take in order to make these circuits work that you should select the gain band product of the op amp or active device that you are using properly in filter design otherwise all these dangers of deviation from the design performance will be the result so thank you very much.