 We are going to cover an important topic I would consider that this is the most important topic in today's analog electronics the reason will become obvious to you as we go. Let us therefore start with what we have done in the earlier lecture 26 lecture Q enhancement the passive RC using negative and positive feedback was the topic of discussion because the passive RC network was using low Q block because it cannot increase the Q above half that we had shown how to improve the Q above half for a variety of designs that we require in filters. So that was the topic then effect of the most important parameter of the active device whether it is op amp or transistor is the gain bandwidth product in the case of transistor it is GM by C product so which actually gives the quality of the active device. So what is the effect of this active parameter on the performance of the filter that was also discussed in the earlier class we had started with low pass passive RC second order and using negative feedback with the inverting amplifier of gain K obtained omega naught the normalizing frequency of the second order system as omega P the normalizing frequency of the passive filter which is for R1 equal to R2 equal to R and C1 equal to C2 equal to C is 1 over RC into square root of 1 plus K, K is the gain of the inverting amplifier Q active is going to be Q passive which is equal to half for equal resistors and equal capacitor situation at most are in actual practice right if you do not use a buffer stage between D2 passive networks it is 1 over 3 into square root of 1 plus K. So this is the enhancement of Q by a factor of root of 1 plus K that was achieved through negative feedback and then starting with the high pass passive RC second order the same negative feedback situation in the embedded system resulted in omega naught being equal to omega P which is 1 over RC divided by square root of 1 plus K, Q active remain same as before Q passive into square root of 1 plus K both low pass plus high pass resulting in a notch output in the passive RC if used in the same embedded system with negative feedback using inverting amplifier of gain K resulted in omega naught remaining same as omega P which is 1 over RC and Q active becoming equal to Q passive into 1 plus K QP. So band pass passive filter RC passive feedback band pass if it is taken as the basis of the passive filter network RC network then in order to achieve Q enhancement to decrease the coefficient of S one has to resort to positive feedback and use a non-inverting amplifier of gain K then omega naught again remains same as omega P which is 1 over RC Q active is equal to QP by 1 minus K QP. So low pass plus high pass with negative feedback or band pass with positive feedback we have omega naught remaining same as omega P only the Q is enhanced. So this seem to be ahh sort of possibility of independent adjustment of omega P and Q which is required in design of filters. However the band pass filter with positive feedback results in ahh high sensitivity to K as far as Q is concerned because K into QP has to be pretty close to 1 in order to make the gain Q get enhanced. So that is the only disadvantage of this positive feedback situation so it is normally used for low Q applications. Negative feedback it has low sensitivity to K but sensitivity to gain band product ahh is high that is it is sensitive to the DB in the case of ahh ahh third technique that is negative feedback with notch the sensitivity is pretty low even to the gain band product even for 4 the sensitivity is low because K required is itself low for high Q realization. So independent Q adjustment and omega naught adjustment not possible in the case of negative feedback low pass prototype we had seen that Q gets enhanced by QA divided by N-QA into total phase lag error in the loop okay. So that means actually in ahh the earlier situation we had seen that this is equal to okay nothing but 2K since we used ahh gain of 2K for the inverting amplifier stage so the phase error was 2K into S by that is GB which resulted in omega naught by GB as the phase error into QA. So that is the net phase lag error in the loop so that cost enhancement of the Q by this much. So we have to select the gain band product properly in order that this quantity is very small compared to 1. Let us say F naught into QA product is a criteria for ahh low sensitivity to GB of QA. QA into delta phi should be much less than 1. In the case of ahh high pass feedback this becomes plus because there is a phase lag of negative value that means phase lead error okay. In the case of notch there is 0 phase error. So this is the easy technique to remember how ahh GB determines the Q enhancement in these feedback enhancement techniques. Now let us start with state variable filters. What are the state variable filters? These are popularly ahh filters derived starting with state space synthesis techniques. These are also known as by quad filters double integrator filters, KHN filters, Kerwin, Heelsman I think in this one yes. Newcomb filters universal active filters they are manufactured as universal active filter blocks even in IC design this is a basic building block of filters in VLSI analog VLSI. So we have to study in detail about the synthesis technique as well as understand the performance of these filters ahh starting from synthesis. Active filter design as a solution of differential equation that is some topic that has been taught ahh as early as tube electronic days when analog computer was there right. So this has the origin with analog computers what are these analog computers. The present students may not know about these computers existing as a tool for simulating ahh differential equations linear or non-linear and that is a system design concept which used to be used for by analog people as well as aeronautics and mechanical engineers in simulating their ahh mechanical electromechanical systems okay and ahh it was common practice to use this tool called analog computer what they it comprised of it comprised of integrator blocks and summing blocks and multipliers. So it used to have these as the blocks integrators summing amplifiers and multipliers. So resistors and the capacitors form the integrators okay summing amplifiers multipliers are the necessary blocks of the analog computer. It used to be a full-fledged course in ahh electrical engineering particularly in control systems for ahh control system simulation it used to be used often. Later on with transistors the size of the analog computer came down drastically and then ahh the op amp was the block that was used as the active device for building all these blocks. So today op amps ahh IC op amps are commonly available and cost of this analog computer has come down considerably but it no longer exists as an IC it exists in the form of universal active filter block. So let us see how synthesis can be done of such filters n-th order differential equation means synthesis of nth order filter. So we have a nth order differential equation linear differential equation written in terms of output voltage and input voltage. So this is the nth differential of output voltage ahh and this is the input function. So these are the n-1 states which can be derived from the nth derivative using n number of integrators. Then this equation can be written with the highest order differential on the left hand side and rest brought to the right hand side this way. And what does it say if you have the highest order differential then n number of integrators can result in the n-1 lower state variables derived from the nth derivative and to that if you add VI appropriately by selecting appropriate coefficients that becomes equal to the left hand side that is circuit wise once you have a ahh differential and derived n-1 states you can sum up all these including the input using a summing amplifier and feed it back to the input that circuit realises the nth order differential equation. So this used to be thought systematically in the earlier electrical engineering courses and this equipment was part of most of the control labs control system labs. However with the advent of digital computers all these simulations went over to digital computer and this course was removed from the curriculum unfortunately. However circuits electronic circuits it has become an integral part of one of the topics discussed however this must be also becoming part of any common basic electronics course lab. Let us therefore see how first order filter can be synthesized with this approach dv naught by dt plus k naught v naught is equal to ki vi is the first order differential equation in output voltage and input voltage this is now written as the highest differential dv naught by dt and one integrator will give you if the input is dv naught by dt integration will give you v naught here. So minus k y naught is got by the summing amplifier adding a suitable sign plus ki vi we have vi and coefficient ki summing so this is nothing but dv naught by dt that is by closing this here you are actually simulating this equation. So this is a loop feedback loop that comes into existence in order to solve this differential equation. So we first assume that this exists and obtain the lower ahh differentials by using integrators and summed up all these along with input and got the output and then fed it back. So this if you realize using ideal integrators and summing amplifier using op amp let us say you can get this characteristic at one of the outputs okay low pass and this is high pass. So actually you can see that this output is going to be high pass and integrated will result in low pass. So this is the high pass output integrate you get low pass first order high pass and first order low pass are got as outputs of summing amplifier and the integrator. So that is simulated using ideal integrator and ideal summing amplifier and one gets the characteristic like this. This can be shown to be ahh first order but over filter with cut off frequency either for low pass or high pass same point that is why it is 0.707 times this 1 the transmission here is 1 is also 1 and this is at 0.707 that occurs at roughly 1.59 for the values of ahh time constants that we have chosen 1.59 kilohertz. So this is the plot this is the op amp realization of the same thing integrator R and C. So the ahh integrated time constant is if this is V0 this will be – V0 by SCR okay this is adding VI to it. So we have here inversion – becomes plus V0 by SCR – K0 into VI at this point this is V0 is equal to 1 inversion so – V0 by SCR plus K0 into VI that is the output V0. So V0 is equal to 1 plus V0 into 1 plus omega 0 by S omega 0 being 1 by RC. So this is equal to K0 into VI so V0 over VI becomes K0 S by omega 0 divided by 1 plus S by omega 0. So that is ahh high pass filter function here it is getting multiplied by – omega 0 by S that means at this point one gets – K0 by 1 plus S by omega 0. So this is nothing but low pass filter function. So these are the ones which are plotted ahh in the ahh characteristic that we have just now seen okay. Next let us try to do the same thing with a second order system. So now we are starting with D squared V0 by DT squared. So we have that this is therefore 1 over RC okay DV0 by DT integrator that is the ahh time constant that determines this integration. So it is a positive integrator. So then we have here 1 over RC squared into V0. So this is going to be summed up here with – K1 here okay. So D squared V0 by DT squared plus K1 DV0 by DT plus K0 V0 is equal to KVS same as before the first second order differential equation. This is the highest differential these are requiring two integrators to obtain the lower states. So D squared V0 by DT squared is equal to – K1 DV0 by DT – K0 V0 plus KIVI is what is done by the summing amplifier and that is equal to V0 itself that is D squared V0 by DT squared itself okay. So we get that by connecting the output to the input feedback. So the procedure for any order is the same. So let us see how this is when simulated okay using ideal integrators and then summing amplifier results in only for the summing amplifier and op amp ahh summing amplifier is used other two are ideal integrators. So we have the output now low pass. So this is going to give you high pass here this is going to be band pass once integrated low pass in the second order. So if this is V0 let us say 1 so this will be DV0 by DT 1 over R this is actually in terms of S domain nothing but SCR into V0 1 this will be V0 1 by SCR square double integration. So and this output V0 1 therefore is equal to the – K1 times this – K0 times this – KI times that. So we will get here okay the summation of actually SCR square SCR okay into V0 1 to V0 1 plus KI into VI which will result in V0 1 by VI equal to KI SCR square which can be written as S by omega naught square divided by S by omega naught square plus S by omega naught Q plus 1 if you collect all this coefficient where omega naught is equal to 1 by RC and it is the Q which is determined by K naught okay it is 1 over K naught okay is the Q and this is what is simulated using ideal integrators and summing amplifiers using a PAM we get we design the thing at F naught equal to 1.59 for a Q of 5 and this is what is shown for the Q of 5 can see this this is nothing but the low pass goes to 0 as frequency goes on increasing very high compared to F naught at F naught it resonates so that is why it peaks at F naught okay and goes peaks slightly at lower frequency than F naught okay and because it is low pass and as far as band pass is concerned it peaks at F naught exactly that is the resonant frequency. So this is the band pass and this is the high pass goes to again a constant value at very high frequencies starts with 0 at 0 frequency that is the green color. So this is the high pass this is the band pass and this is the low pass characteristic that is at this point it is peaking and the gain is H that is K I into Q okay that will establish later when we consider these individual transfer functions. So gain go back here this is the point high pass is got this is the high pass integrate once you get only S by Omega naught naught square because this Omega naught by S is the transfer function. So this becomes band pass again integrated it becomes low pass this is the phase characteristic of low pass starts with let us say zero phase right goes on to 180 degrees at exactly this point there is 90 degree phase shift in all these things right only thing is additional phase shift of high pass to low pass there is 180 degree phase shift right. So you have that but phase variation is similar determined by the denominator polynomial numerator has a constant phase. So all the phase variations are remaining the same as determined by the denominator polynomial which is common to all the three. So band pass right you have this phase changing okay again this way okay. So this is actually I think 180 degree phase shift for the band pass according at Omega naught you can see either zero or 180 degree. So in this particular case it is 180 degree phase shift right and then the high pass and low pass are going to be 90 degree lag or lead around 180 or zero. This is the transient response this shows you that the transient response remains the same okay for a step function of voltage okay. So 100 millivolts step is given and that causes the thing to ring and the number of peaks countable the same method that we had adopted earlier in amplifier design as well as in filter design that you count the number of peaks one up to one tenth this way peak. So you have five peaks which are countable indicating Q is equal to five. So the outputs remain the same in terms of ringing whether it is low pass high pass or band pass indicating the characteristic equation is the one that determines the ringing this is the realization using op amps. So we have the first integrator followed by the next integrator with a summing amplifier this is what is called the Q determining loop this is known as one integrator another integrator a third inverter forming a loop. This loop is called a resonator block because if you disconnect this the facility for adding dv naught by dt in the equation is removed that means it is something that contains only d squared v naught by dt squared plus k naught v naught if you do not have vi also added if you break this loop here equal to zero what is this there is nothing but a harmonic equation solution of this is nothing but a sine wave right this v naught is equal to some a sine root k naught into t. So it produces a pure sine wave it is a resonator block okay just see this this is making the Q of the system infinity if you remove this connection and not have input fed okay input is grounded then this alone simulates this equation this squared v naught by dt squared plus k naught v naught equal to zero which is why this is called a resonator block this is called the Q forming loop. So it just adds a coefficient of dv naught by dt to this that is all that K1 this the RK1 that determines the Q this is Q times R if you put that determines Q. So now this has been designed for 1.59 kilohertz as f naught and Q equal to 5 is the characteristic this is what we had seen earlier with the ideal integrators also this is the op amp version this is the band stop filter. Now all this simulation op amp has been given a model of just a voltage control voltage source of gain equal to 2000 that is all as far as the integrator op amp are concerned and the Q forming loop op amp only the summing amplifier is having the realistic op amp so that it can quickly simulate all this in terms of simulation time etc it takes the least amount of time okay. So this is the band stop filter output available at this point this is band stop will exactly give you the transfer function derived okay later Q is 5 and now this is the phase plot and since the phase plot for all the other outputs also other than the notch remain the same as the other plots except the point at which it starts okay is going to be different for different outputs okay they are all going to be quadrature difference that means 90 degree different okay. So now this is for Q equal to 1 this is for Q equal to 5 this is Q equal to 9. So that one resistance that determines Q is changed from 1K to 5K to 9K and one can see how rapidly the Q varies this we had derived earlier the slope here at F naught for phase variation okay delta 5 by delta omega we have shown around omega equal to omega naught is equal to okay minus 2Q by omega naught this we had shown earlier for a second order system okay that means slope at this point depends upon the Q as Q increases the slope increases this fact about Q variation around F naught is the one that is used for tuning of filters. So a phase detector can tell exactly what the phase is on this point and thereby we can tune it exactly to F naught okay. So you can just see this that delta omega from this is equal to okay minus omega naught by 2Q okay into delta 5 that is a phase detector okay if high Q system is used a phase error of 1% for a Q of 10 is going to be detected as one tenth okay that means higher Q circuits you can see the frequency deviation more accurately with the phase detector using this principle. So please remember this later when we would like to tune the circuit to exactly a specific frequency phase detector is to be used. Now this F naught is equal to 1.99 Q equal to 5 the transient response you can see the 5 peaks this simulation also. Now what is done here is that input of a square wave at F naught is fed the filter is having resonance occurring at F naught and these are the different outputs okay this is the output corresponding to the notch. So the square wave comes without the fundamental because it has been tuned to F naught and F naught is the frequency at which the square wave comes the fundamental is removed okay because it is peaking at F naught. So the fundamental component is removed that means square wave minus the fundamental is like this. Please remember that you have studied this for the square wave of VP the fundamental is 4 VP by pi that means fundamental amplitude is more than that of the square wave. So if you subtract this you will get it as just this this will be reverse this will be reverse like this. So you get this kind of thing after removing the fundamental. So this is exactly tuned to the incoming frequency of naught and this is the fundamental component which predominantly appears because it has a Q of 5 that means it is peaking with again of H naught into or Ki into Q right Ki is same as H naught that is okay the center frequency gain is H naught or Ki okay into Q the low pass frequency low frequency gain is H naught high pass high frequency gain is H naught and at omega naught gain is H naught into Q that we will see when we look at the transfer function later. So as far as the high pass output is concerned there is this sharp increase so it cannot remove the harmonics this is the effect due to the odd harmonics present in the square wave. So summation of effect of all the odd harmonics is a jump whenever the square wave jumps there is a jump you can see the square wave jumping that is it produce both at the notch as well as the high pass and that is removed from low pass and band pass because band low pass permits only the fundamental and all the harmonics are removed and band pass selects only the fundamental okay so this is the thing this is the low pass and this is the band pass they almost look alike except for a phase difference of 90 degrees and between the low pass and the high pass also there is a phase shift of 90 degrees okay. Simulation of second order filter with op amps at 15.9 kilohertz so the frequency is increased to 15.9 from 1.59 10 times Q remains the same. Now you will see the effect of the gain measured product this will quantitatively determine in the next class however you can see this difference the frequency is almost the same 15.9 kilohertz from 1.59 10 fold increase however the Q into H naught or Q into Ki Q is 5 this should have been 5 whereas it had remained 5 when we worked at 1.59 now it has changed over to 6.658 does it change why should it increase like this and the notch should have been 0 but there is 338 milli that is it should have been 1 at low frequency and high frequency and it should have been 0 at this point whereas there is a transmission of 0.338 occurring here why does it take place is what we are going to see in the next class right qualitatively. Now you can see the effect of the Q has got enhanced that is what we saw in H naught into Q effect in the frequency response it has gone to 6.6 now here the Q is showing 5 peaks as we saw earlier at low frequencies it is showing 1 2 3 4 5 6 almost 7 peaks okay so the Q has got enhanced a third order filter same procedure can be extended for any order but you will see why this kind of extension using this kind of methodology is not recommended for higher order filter design and how higher order filters can be designed using cascading of second orders and first order or only second orders right it is safe. So however I am giving you a procedure the third order differential is dependent upon the rest of the states this way you put all these things on the right hand side along with the input just as before and use 3 integrators to get the lower states from the third order okay you get the lower states dv naught by dt square rc 1 by rc square dv naught by dt 1 by rc cube v naught. So you sum up all this k naught k1 k2 and ki for vi using a summing amplifier like this as before and get the output connected to the input which is solving the third order differential equation that is simulated. Now we design a Butterworth filter using that actually k1 and k2 are appropriately chosen so that the coefficients here are 2 and 2 okay. So that is design third order Butterworth these equations are given in any book on filters or circuits and not circuits the third order normalized Butterworth design. So that is simulated and you can that is done for same frequency 1.59 hertz maximally flat magnitude response both for low pass and high pass this is the high pass output as before this is going to be the low pass output. The op-amp realization involves 5 such op-amps in one loop and the other additions take place using this. So we have here 1, 2, 3 integrators and the summing stage and the final inversion for the negative feedback. So this is the characteristic using op-amps as the summing amplifiers and when we increase the frequency from 1.59 to 15.9 you can see the high pass getting distorted considerably the distortion in low pass is not visible it also is different from the theoretically predicted value. This is because of the effect of gain mandate product which we are going to investigate in the next class. So this is the realization that is when the same third order filter is realized using the above technique earlier mentioned technique with the frequency shifted further from 15.9 up to f not equal to very nearly equal to 186 kilohertz close to the gain mandate product of the op-amp the maximally flat changes to this kind of ripple within the pass band indicating the effect of a higher order system this is one first order low pass and this is a second order low pass characteristic combined giving the third order effect because of the gain mandate product. And one can see that if you cascade second order with first order it almost retains the general second order response of just peaking once okay. So this is the effect of a second order itself right even though the effect of gain mandate is seen as increased Q okay for the second order so the peak is increased otherwise it should have been maximally flat. So these are the observations one can make higher even other filters can be realized by cascading second order filters higher order filter is realized using cascading first order with second order this is the easy method of design that is why the second order universal active filter is a basic building block in most of the filter designs and it is available as an IC later on identify the ICs that are available as these building blocks. Direct realization greater than 3 using op-amps will lead to inferior performance due to the cumulative phase error in the feedback loop. So this is normally avoided in most of the feedback loops higher order outputs at different points in a second order filter. So let us now go back to the original circuit with 3 op-amps in a loop W integrator loop and we see V0 1 by VI this is what we had derived earlier is the high pass filter H0 into S2 by omega naught squared divided by S2 by omega naught squared plus S by omega naught 2 plus 1 high pass filter okay. So this is the characteristic that we had already seen and it is close to 5 you can see at 1.59 this is 1.6 kilo hertz and it is becoming equal to 1 at very high frequency this is the high pass output. So H0 is equal to 1 in our circuit that we had shown earlier H0 is equal to 1 H0 is same as Ki. So this is same as this this is Q times R that Q has been adjusted to be 5 and this is same as this makes Ki equal to 1 or H naught equal to 1. So that is what has been chosen in this characteristic so that is depicted here this is H naught into Q 5. So it fits with the theory exactly this is the band pass filter output can see that characteristic is speaking at exactly F naught which is 1.591 and H naught into Q is 4.949 same as what we have expected and it goes to 0 on either side this is the characteristic of the band pass this is the low pass H naught by this and at frequencies low frequencies it is H naught which is 1 in this case at omega equal to omega naught it is H naught into Q that we will see. So this is nearly equal to 1 and this is going to be 1.569 slightly reduced to because of the low pass action it is speaking is always at a low frequency than omega F naught okay for the high pass it is slightly at a high frequency than F naught only for band pass it is exactly at F naught. So these are the observations that we have made as far as notch is concerned okay this is the output at the 4th of amp okay and H naught into 1 plus S squared by omega naught squared indicating 0 transmission at omega equal to omega naught A i is equal to 0 because of this and it is supposed to peak the peak is killed at that point okay because it is getting multiplied by 0. So normally something that was speaking at low pass high pass band pass that is suppressed because of the 0 of transmission. So the high Q filter has a 0 of transmission and it becomes like that means with increasing Q it is going to be like this this is what we will see here. So this is for Q of 5 and if you increase the Q further later on see that it is going to become narrower and this point at which the transmission should be 0 is not exactly 0 is about 12 milli that means this is 1 this is 0.012 okay transmission so it is not yet gone to 0 why it has not gone to 0 is something that we are going to investigate because of the active device finite gain or gain menu product. So both effect this 0 okay and this has gone to nearly 1 994 992 on this side so that is the notch output by adding V naught 1 to V naught 2 and V naught 3 V naught 1 is high pass this is high pass this is V naught 2 is band pass with the negative sign this is the low pass. So alpha 1 is positive alpha 3 is positive alpha 2 is negative you get A S square plus V S plus C you can make add it as positive or negative or 0 depending upon which output you take and add okay and we can locate the 0 of this polynomial anywhere on the S plane whereas the poles are always located on the right half as Q is positive okay and therefore pole in the S plane the poles of this system will always lie on the left half plane okay as complex conjugate pair or on the negative real axis but the zeros can lie anywhere on the S plane as complex conjugate pairs okay or on the negative or positive real axis. So that is the facility of a generalized synthesis and for example of all pass filter design requires that what is null pass filter the magnitude of this should remain independent of frequency that means it should be equal to H naught at all frequencies so and only the phase should change. So for that it is enough if you make the odd ones have negative polarity as for the coefficient is concerned compared to the denominator. So that means if the poles are like this on the left half plane for stability you have to have a pole always located on the left half plane zeros can be just mirror image of poles that is the pole 0 location for an all pass filter always any order right you might have the poles like this then the zeros should be exactly like this mirror image this is for higher order is for the second order it is like this. So then magnitude is H naught at all frequencies phase is going to be contributing lag here by this much amount and another lag by this turning to minus. So minus 2 tan inverse omega by omega naught 1 minus omega by omega naught square and if you actually obtain the delay function delta phi by delta omega minus okay that will be actually maximally flat that maximally flat delay function type of filters are called Thompson's filter this I had mentioned earlier okay and that maximum flatness occurs for a Q of 1 by root 3 for a second order system that you can prove by taking delta phi by delta omega and making it maximally flat. Delta phi by delta omega will be having the omega by omega naught function okay coming out as this 1 plus omega by omega naught squared divided by omega by omega naught to the power 4 plus what is that minus 2 plus 1 over Q squared this please derive it squared plus 1. So you have to for maximum flatness the coefficient of omega squared by omega naught squared should be same as that of the numerator so 1 is equal to minus 2 plus 1 over Q squared or Q is equal to 1 over 3 that is the Thompson's filter. So this is an all-pass filter design that gives you only delay and it is used as a delay compensating mechanism right it does not affect the amplitude it only affects the phase. So this is the phase characteristic for Q equal to 1 Q equal to 1 is amplitude character will be constant h naught 1 throughout. So in conclusion we have covered systematic method of synthesis of state variable filters and we have obtained the circuit of what is called universal active filter block. Why is it called universal active filter block? It is a double integrator block or a resonator block with Q forming loop separately okay by taking the combination of low pass plus high pass plus band pass outputs available one can formulate any second order system with specific omega naught and Q which is sufficient to derive any nth order filter okay required for any application that is why it is called universal with facility of one more adder one can add this alpha 1 alpha 2 and alpha 3 kind of outputs and obtain a generalized second order system with zeros located anywhere on the S plane and poles located only on the laptop plane as complex conjugate pair or on the negative real axis which obtain stable filter systems for use. Also seen that if Q of the resonator block is Q of the system is made infinity it becomes a double integrator oscillator which oscillates at omega naught that is why it is called a resonator block or a double integrator loop. So in the next class we will see that this double integrator loop is really similar to inductor getting simulated across the capacitor forming a tank circuit just like the inductor simulation filters right. So it is no different from the inductor simulation circuit that we are discussed earlier topologically it looks different from the garter this will be established in the next class. Thank you very much.