 Today we are going to have the 33rd lecture on trans conductor based oscillators. Let us see what we had done in the last lecture. We started with L C oscillators that is we started with oscillator as nothing but harmonic equation. This means we had this T squared V naught by DT squared plus K V naught equal to 0. So we took L C network which really solves this if this is taken as V naught that is the differential equation governing this. K is equal to uhh root K is the frequency of oscillation which is 1 over root L C. K is equal to 1 by L C. So then we said the last component is the parallel resistance and that can be compensated by a negative resistance that is what the active device does. So this is the total story of an oscillator that the admittance of the device it is basically a 2 power device which is equal to 0 at the oscillating frequency omega naught. So this is the condition for oscillation that the positive resistance should be exactly compensated for by negative resistance so that effective resistance is infinity or Q is infinity. And replacing this inductor by a garator based circuit one can therefore simulate the inductor using RCs and a PAMP active device or RCs and transistors and therefore convert this into an RC oscillator. Then we had another view point of oscillator that oscillator is something where you are having a band pass network for example which will peak at a certain frequency omega naught the Q is high now okay. This particular thing is going to be peaking at omega naught equal to 1 over root L C and this is going to be equal to in this case if it is RP and RP let us say this is going to be half. And one can provide a boosting up of gain to 2 so that loop gain the overall loop gain becomes equal to 1 exactly only at omega naught equal to 1 over root L C. That means another concept of oscillator is that if you have the loop gain okay so this is now going to be closed if the loop gain can become equal to 1 exactly at omega equal to omega naught then that is also a candidate for oscillation okay. So that means basically I am closing the loop like this so here the loop gain equal to 1 at omega equal to omega naught that concept is used in making it an oscillator. For that you can actually represent this as a band pass filter which is peaking at omega equal to omega naught that means gain loop gain is reduced to less than 1 on either side of omega naught. And the phase shift is exactly 0 that is why you close it output is equal to input it can sustain oscillation. So apart from this concept of admittance at any port being equal to 0 now we have a feedback network where loop gain is equal to 0 is equal to 1 okay at omega equal to omega naught. Now then we went over to quadrature oscillator or phase shift based oscillator where we have if you actually open this loop here it is nothing but solution of a differential equation. So the same old differential equation so you just have done it in the filters this is how you write it. So d square V naught by dt square integrated d V naught by dt integrated my is going to be this is minus d V naught by dt this is plus V naught and this is minus V naught. So when you close the loop it again becomes harmonic oscillator and then we had seen this that again the loop gain okay is equal to 1 only at omega equal to omega naught that is what the loop gain is minus this is from here to here minus 1 by SCR that into another minus 1 by SCR square that into minus 1 is equal to 1. So this happens exactly at omega equal to omega naught equal to 1 by RC. So this also follows okay even though it is a phase shift and magnitude adjust so that loop gain becomes equal to 1 okay and the phase shift is 0 this gives 90 degree this gives 90 degree okay 180 degree and then this is an inverter 180 degree so 360 degree phase shift 0 degree right. So again loop gain becomes equal to 1 at omega equal to omega naught and this also can be viewed that I had already done if you consider this capacitor this is the capacitor voltage VC and what does the capacitor C. So this is VC this voltage is VC this is minus VC inverter so this minus VC comes here and this becomes plus VC by SCR and the current in this is plus VC by SCR square. So the current that is flowing out of this when the voltage across it is VC is VC by SCR square that means it is seeing an inductor across the capacitor. So the same thing gets valid that concept of loop gain equal to 1 also can be viewed as admittance between these two becoming equal to 0 at omega equal to omega naught because this is a capacitor and it sees an inductor of CR square across it. So again this concept of oscillator is all pervading right. So a capacitor in any oscillator sees an inductor that is if it is a system oscillator then the capacitor must see an ideal inductor okay for it to oscillate. What does the resistor see if you post that question right here you can take the voltage across the resistance here as V okay again minus here plus here and this voltage is integrated okay V by SCR again integrated okay V minus here and plus V by SCR square okay and that divided by S square C square R square. So that okay divided by R S 1 by S square C square R square by R okay S square C square R cube. So that is a frequency dependent negative resistance okay. So again the total admittance becomes equal to 0 when omega equal to omega naught is VC that is the same thing that means a resistance sees a frequency dependent negative resistance so that net admittance at that frequency is going to be 0. So the concept of oscillator okay even though it might be attacked in terms of loop gain or admittance it will mean one and the same in any oscillator which is a second order. Now this is a third order oscillator that means this is going to be a third order differential equation that is getting solved. So what should happen in this case if it is a differential equation again d cube okay by dt cube plus K2 d square V by dt square plus K1 dV by dt plus K0 equal to 0 that means this can form this coupled with this can form one harmonic oscillator this along with this forms another harmonic oscillator if both equations have similar coefficients for K then it is a harmonic oscillator that means both of these okay this pair and this pair. So go to 0 with the same value of frequency of oscillation then it becomes a harmonic oscillator. So that principle is utilized in what is called phase shift oscillator or ring oscillator concept wherein each inverter here as again of –2 okay that you can actually assume that it is some K okay and then divide by 1 plus this SC into 2R. So this whole cube this whole cube becomes exactly equal to 1 when this is the real part – 8 okay divided by 1 – this okay becomes equal to 1. The imaginary part should be independently 0 okay and it should happen at the same frequency omega equal to omega naught as this okay becoming equal to 1. So this becomes equal to 0 at omega naught equal to root 3 by 2 RC that is the frequency of oscillation and that frequency if this becomes equal to 1 okay that is why the gain is chosen as – 8 okay and that is what happens this becomes 9 so that 1 – 9 becomes – 8 so – 8 by – 8 is 1. So that is the famous ring oscillator the CMOS ring oscillator which is nothing but this basic structure this is just one stage let us suppose we have this okay this is 2R this is C and this is R okay this is plus V this is – V. So this is one structure three such structures couple together and forming a ring will give you the oscillation frequency omega naught equal to exactly root 3 by 2 RC. So if the gain exceeds that right if you do not put this 2R etc the gain is very high and therefore the whole thing goes into a large amplitude oscillation where the waveform may look something like a trapezoid not any longer the sine wave okay and the frequency is going to be far different from this much lower than this. Now we come to an important topic regarding the frequency stability of oscillators. Let us look at what we did in terms of RLC oscillator we could view this oscillator with the loop like this and op amp with gain of 2 this is R1 and this R1 so the gain of this is 2 and then we had put that resistance across it which is RP we had called it so that it could simulate – RP to compensate for the losses occurring across this L and C. So this setup is now redrawn to form a loop and we saw that the gain of this from here to here is 2 okay this is V this is 2V and this 2V comes here the Tavernance equivalent will be again RP RP so V 2V into half so that is what and then Tavernance resistance of RP by 2 and L and C so we have V Tavernance equivalent of RP by 2 L and C. So this is going to be resulting in again S squared L C L divided by R by 2 RP by 2 plus okay 1 okay and the numerator will be SL divided by RP by 2 so band pass characteristic so obviously the loop gain becomes exactly equal to 1 when omega equal to 1 over root KLC this cancels with this and this SL by RP by 2 SL by RP by 2 and this 2 into half make it equal to exactly equal to 1 so that is the loop gain concept. So now what happens let us see this is an op amp is giving a gain of 2 so as far as this phase shift from here to here is concerned it is 0 in the non-inverting ideal amplifier right the phase shift is 0 and then this network alone has a certain amount of phase shift which is going to be like this that it is initially having a phase of plus pi by 2 it will go on and go to minus pi by 2 band pass and exactly 0 phase shift for omega equal to omega naught which is 1 by root KLC. The important part to note here is that the rate of change of phase here okay is delta phi by this is phi this is omega so delta phi by del power we have shown it to be equal to minus 2 Q by omega. So it is directly proportional to Q if the quality factor of this which is Q of this is nothing but it is RP by 2 divided by omega L. So that is the Q of this LC circuit so this is nothing but S by omega naught Q so that is what governs the rate of change of phase here higher the Q higher so what happens if there is a phase shift from here to here due to the amplifier why would that be a phase shift because this is not really infinite gain it is dependent upon it is gain bandwidth product GB by S that is what we have seen and if it is an amplifier with gain of 2 then it is actually 2 divided by 1 plus this 2 S by GB this is the phase error so that means the there is a lag from here to here because of the finite gain bandwidth product. So the oscillation however occurs only when the phase shift from here to this point in the loop okay the output here should be in phase with the input and equal in magnitude that is the thing. So if there is a phase lag error in this okay then this network will not be actually oscillating at omega equal to omega naught equal to 1 over root LC but at a slightly lower frequency such that at that frequency the lag of 2 S by GB okay or 2 omega by GB or 2 omega naught by GB lag is compensated by a lead here. So if the slope is very high you will see that the deviation in frequency that is necessary for the passive network embedded okay is very small if the Q is very low the deviation in frequency of oscillation should be considerable from the ideal value that is what causes frequency to be unstable the embedded network the passive network if Q is high then for a small change in the error here due to variation in GB due to temperature etc the frequency that has to be changed okay in order to accommodate the phase error so that overall phase shift is equal to zero in practice is going to be very small. So frequency stability really depends upon not just the condition that the phase has to be zero somehow but the fact that the contribution okay of what is that phase variation by the passive network to accommodate the phase variation okay opposite phase variation caused by the active device parameters is very very small okay. So this particular thing therefore requires another example to work out that means what is the let us say phase error here this is going to be lag by 2 omega naught divided by GB that is the delta phi lag error minus it becomes so this should be giving you a lead error of plus delta phi okay that means what is this we know that the slope at that point is the delta phi by delta omega is equal to 2Q by omega naught. So we have here delta omega frequency deviation equal to so important thing omega naught divided by 2Q into delta phi the delta phi here is 2 omega naught by GB. So this is omega naught divided by Q is nothing but RP by 2 divided by omega naught L. So that is the change that is required so the Q being equal to this. So higher the Q higher the value of RP better it is for frequency stability. So LCR slaters better still if the whole thing is replaced by a crystal then the frequency stability that means any device phase shift error is always accommodated by a very small deviation in sort of frequency because of the high Q. Frequency stability of oscillators the continued now with passive network which is nothing but not RLC but RC network. And we know that RC networks like this whether it is in bridge or the other network that we had considered for band pass earlier in the last lecture which could be just this both of which have a limited Q at best the Q can be equal to half. When the resistances are equal and capacitance are equal Q deteriorates to 1 by 3 okay. So the transfer function is SCR by 1 plus 3 SCR K SCR square for this network from here to here. As far as this amplifier now is concerned it has to accommodate this loop gain to become equal to 1 then it has to have a gain of 3. So at ideally omega equal to 1 by RC this cancels with this so the loop gain is 1. So that is the concept conceptually it is alright it is going to work as an oscillator with omega not equal to 1 by RC. However if you now have an error phase error here what is the phase error here from here to here now okay it is a gain of 3 that means 3 divided by 1 plus okay 3S by GB that is the phase error okay. So that means delta phi in this case is okay a lag of 3 omega not by GB this is the lag error. So that means this passive network whose phase shift is exactly like that it is band pass okay because of this again pi by 2 it goes like this but the Q being low 1 by 3 the slope here is okay not so steep as the LC oscillator case right so it is less steep that means the frequency deviation is large. So what is it the frequency deviation delta omega in this case is going to be nothing but delta phi okay into omega not divided by 2Q and delta phi here is 3 omega not by GB and Q is 1 by 3 so the whole thing now boils down to 9 by 2 which is 4.5 okay omega not divided by GB. So the frequency is going to decrease by that extent that means actual omega not into 1 minus 4.5 omega not by GB simple it is to calculate therefore these are the oscillators are not at all favored if frequency stability is of concern. So these wind bridge oscillators should not be used any longer okay as practical oscillator systems they do not have any advantage over the other well known LC oscillators with high Q. So that is demonstrated here what all we have discussed is given here okay and this is the phase shift of the op amp 741 okay with gain banded product of 1 megahertz okay used as gain of 2 for the LC oscillator case and gain of 3 for the wind bridge oscillator case. However this is the variation of what is that phase with respect to frequency okay the slope here is 1 by 3 for the LC oscillator case and it is nothing but 2Q by omega not for the LC oscillator. So you can therefore see that frequency stability is going to be of great concern to us okay in oscillator building and therefore this is roughly the way the calculation is made this is due to the band pass passive network embedded okay with the amplifier K is equal to 3 for the case of wind bridge K is equal to 2 in the case of LC oscillator okay. So this is the error in frequency due to this kind of what is that phase error caused by the amplifier embedded in the passive network. So this can be again done the same way we did it for the filters this is originally a second order band pass that can be modified as a second order band pass okay with modified Q and modified frequency. So you simply multiply by 1 you get this S squared by omega naught squared I have indicated in terms of phase error why frequency stability is poor here again mathematically making the approximation KS by GB is a small quantity which is going to result in additional terms here this is going to be adding KS divided by GB here to the S coefficient this is going to add to the S squared coefficient KS squared by omega naught Q GB okay and then this is going to make it a cubic term KS cubed okay divided by GB omega naught squared okay. So now what happens you have to make approximations so as far as this is concerned this error is going to only change coefficient of S squared coefficient of S squared so coefficient of new S squared will now become S squared by S squared I think you should write it here S squared by omega naught squared omega naught squared into 1 plus okay K divided by omega naught Q okay S squared by omega naught squared so this will be cancelling with 1 omega naught there divided by GB. So the coefficient of omega naught squared is going to change to omega naught squared into 1 plus okay omega naught squared divided by 1 plus K omega naught by Q GB which is the same as what we had earlier indicated so omega naught becomes equal to okay earlier omega naught actual becomes equal to omega naught divided by 1 plus square root of K omega naught by Q GB is going to result in K by 2 omega naught divided by Q GB this can be represented as 1 minus K by 2 omega naught by Q GB. So that is the modification of the coefficient of S squared this is going to contribute to okay this and this will contribute to change in coefficient of S which is the one that we have discussed in the case of filters as enhancement of Q. So Q enhancement will require only different value of gain okay than earlier so it is going to require a different Q okay to make the loop gain equal to well different K2 in magnitude to make the gain equal to loop gain magnitude equal to 1 that is all as far the frequency stability is concerned that is only governed by the coefficient of S squared which is modified as this. So this is the discussion that we have made earlier so now coming to another important topic that is voltage control oscillators. We had seen how voltage control filters can be built in our double integrator filter or universal active filter block which is just two integrators we converted each integrator into a multiplier followed by an integrator so that the frequency omega naught of the integrator this transfer function is omega naught divided by S minus omega naught is equal to 1 by RC this changed over to by this arrangement VC by 10 RC where this is a multiplier with VI as input VC as another input so it is VC VI by 10. So in which case it is equivalent to changing omega naught to from 1 over RC to VC by 10 RC. So that is how omega naught became controllable directly by VC in the filter case same thing if it is an oscillator again the frequency of oscillation omega naught can be changed to from 1 over RC to omega naught equal to VC by 10 RC by this arrangement of replacing each one of the integrator by voltage controlled integrators. So RC oscillators gets replaced by RC into VR by VC VR is 10 volts for us right and therefore this becomes from 1 over RC VC by 10 RC. Such a VCO should also use amplitude stabilization the like of which we had used in the last class. Let us see this interesting quadrature oscillator how it is amplitude stabilization could be made differently different from that of what we used earlier. For detecting the amplitude of oscillation we had used a squaring circuit or it may be just a diode rectifier also we used a squaring circuit only to demonstrate that ideal non-linear elements like multipliers give quick idea about the design and its dynamic range okay and practically realized systems like that okay the non-idealities do not get shown drastically okay because the ideal devices okay and the practical devices have very close performance factors right. So voltage controlled oscillator if it is designed this way it has to have an amplitude control that is necessary. So how do we do the amplitude control we will do the amplitude control later and also show it getting converted as a VCO. So if you do the VCO design like this an important parameter of the VCO is KVCO which is delta omega naught by delta VC what is a VCO communication wise VCO is known as FM generator or FSK generator this is nothing but the good old linear VCO. Therefore if you have a sine wave VCO and frequency is directly proportional to control voltage then if VC is equal to VCQ plus some V modulation sine omega modulation is the modulating frequency so this is the modulation component then the frequency omega naught is going to be omega naught Q which is dependent directly on VCQ and the VM sine omega MT is going to be modulating this carrier this is not now called the carrier. So this is going to be nothing but the delta omega deviation sine omega MT. So the delta omega deviation is also proportional to VM directly proportional to VM okay. So it is this now that is determined by what is called the sensitivity delta omega naught by delta VC which is the determining the frequency deviation. So if you have VC changing from VC1 to VC2 then the frequency changes from omega 1 to omega 2 okay and that is called FSK generator. So these are the ones that are used as part of modem modulator modulator contains nothing but a VCO which gets modulated by digital data if it is FSK transmission and analog data if it is FM generation. So it is an important unit of today so let us understand more about it the important parameter associated with it is if it is a linear VCO delta omega naught by delta VC is 1 by VR which is 10 here into RC. So omega naught divided by VC radians per second per volt is the dimension of the KVCO which is one of the important parameters. Now look at the circuit we had already discussed this the amplitude stabilization scheme of this double integrator oscillator that earlier and shown it working as amplitude stabilizer. So that makes use of the information that in the last class we saw that you take one integrator output square it so you get VP sin square omega t here and VP cos omega t here becomes VP square by 10 cos square omega t this is VP square by 10 sin square omega t okay and these facilities equal resistive facilities making the whole thing purely a DC of VP square by 10 R2 current and that is received by this voltage reference as V reference by R2 so this current okay DC current equals this current okay and that is what makes the amplitude get stabilized at whatever value which is V reference into 10 square root of that that we had seen in the last class. So such an amplitude stabilization loop is necessary now I want to make that amplitude stabilized oscillator become voltage controlled oscillator. So I have replaced each one of these integrators by multiplier followed by the integrator multiplier. Moment I do that F naught becomes VC by 10 2 pi RC so VC by 20 pi RC. So VC here we have arranged it such that it is 5 volts DC plus 2 sin 2 pi 20 pi okay in so many volts. So it now becomes F naught equal to some quiescent frequency carrier plus or minus the deviation okay and this deviation is to make to occur at 10 hertz okay. So this is shown here this is the FM it is getting generated slowly and stabilizes at the voltage of 1 volt because we have a V reference equal to 0.1 we had earlier seen that is stabilized at 1 volt accurately. And then the frequency is varied because VC is being varied as sine wave at another lower frequency so you can see clearly this portion is low frequency this is lowest frequency it is wide and this is compressed so this is the highest frequency okay. So this is nothing but an FM waveform okay this is FM okay this is one of the most interesting analog projects one can do right which involves multiple feedback systems right. Now this is FSK generator VC is changed from 6 volts to 3 volts 6 may be 1 3 is 0 so the digital data is modulating the oscillation frequency at 10 hertz so you can see here that this is high frequency okay maintained constant this is low frequency maintained constant. This is the way modems work today and that is the modulator FSK. So you can transmit this FSK signal through the telephone lines at different 1s and 0 frequencies and receive at the receiving end and detect it the detector mechanism will discuss later that VCO is put in a feedback network that is the PLL. So LC oscillators now let us see again this conductor as active element now what is a trans conductor we have a voltage so this is the ideal 3 terminal trans conductor this is GM into V so this is the simplest model representing either MOSFET or a JFET or a bipolar transistor there may be a resistance across this that is all. So this can represent all those oscillators that can be using that as the active element. So what have we done we have discussed that in a 2 port network with an embedded passive network let us say you will have some complex passive network here and a complex passive network here and another network here. So that is incorporating may be resistors inductors and capacitors resistors inductors and capacitors here resistors inductors and capacitors so that forms one of the most generalized trans conductor oscillator circuit so this may be the ground. So now what happens here so we have this as say Y1 this as some YF and this as some Y naught this may represent the load resistance plus whatever inductor or capacitor that is parasitic as well as what we are putting there for making it an oscillator. Now similarly this can become combination of RL and C this also could be. So if such an embedded network is there we can write in our 2 port the 2 port parameters let us say in this case since all of these are shunting okay we will represent it in terms of its Y parameter. So then we can formulate the 2 port network and what this oscillator says is that the determinant of the matrix has to be 0 at the oscillating frequency. So that determinant is characterized we have discussed it elaborately in the 2 port network by what is called as PI into P naught okay and 1 minus GL which is PRPF by PI into P naught that was the loop gain. So in our case therefore if it is to become an oscillator this 1 minus GL should be equal to 0 at omega equal to omega naught. Again we come down to the same thing loop gain equal to 1 that is in magnitude and there should be no phase shift in loop gain at that frequency. So that is the same condition that we have repeatedly used earlier for oscillation synthesis and it also tells us that admittance at any port of the network input port or output port is nothing but okay please remember recollect this okay so this is the loop gain okay. So again either the admittance equal to 0 at omega equal to omega naught or loop gain equal to 1 at omega equal to omega naught or the determinant of the matrix between 0 at omega equal to omega naught all these mean the same thing that is the oscillation. So this is what is understood so now we will write down the why port admittance here incorporating everything at the input port everything at the output port everything between the input and output that is the feedback. So why I plus YF Y naught plus YF okay is the short circuit admittance the short circuit YI plus YF short circuit this Y naught plus YF and this one is the short circuit current at the output for input VI so GM into VI minus YF into VI is the short circuit current and when the input is shorted it is minus YF alone. So the determinant becomes just this so is YI Y naught plus YF into YI plus Y naught plus GM equal to 0 that is the unique way you can select topology for oscillator here it can be shown that YI and Y naught should be of the same type and YF okay has to be of the opposite type that means if these are inductive this is capacitive and if this is capacitive this is inductive this results in two of the most popular LC oscillators which is Hartley and Colpitt's oscillator. So let us look at it so this is the Hartley oscillator we have the YI okay as 1 over j omega L1 Y naught as 1 over j omega L2 plus 1 over R2 and YF as okay the j omega C so you have this written there you can if you write you will see that it as real part and imagine the part okay both should be simultaneously going to 0 and you will get the frequency of oscillation as nothing but the resonant frequency of this right independent of the embedded network so it is omega naught is equal to 1 over root L1 plus L2 into C. So condition for this to really become an oscillator is L2 by L1 is equal to GM into R2 this is what you get by equating that equation of earlier okay to 0. So what is it ultimately you can also show here that if it is L1 plus L2 by C then the resistance across this inductor okay becomes infinity exactly at condition L2 by L1 is equal to GM into R2. So GM is such that it is 1 okay that is bringing about a negative resistance effect across this tank circuit which will make the positive resistance get cancelled here overall so that the effective equivalent of this whole thing is just the resonant circuit L1 plus L2 shunted by C with shunting impedance resistance going to infinity under this condition. So all of them are considered as LC oscillator with negative resistance compensating for the positive resistance across the whole tank circuit. So it is no different from the LC oscillator that we have discussed here right. So only this active device GM is bringing about a negative resistance in a such a manner that the whole thing becomes a purely ideal resonant circuit L1 plus L2 shunted by C called Pits oscillator. Now here it is just that the feedback YF is now 1 over J omega L and this is J omega C1 by I and Y naught is J omega C2 plus 1 over R2. So then again the frequency of oscillation remains the same L shunted by C1 in series with C2 which is omega naught is equal to 1 over root L into C1 C2 by C1 plus C2 and the condition for oscillation is C1 by C2 is equal to GM into R2 that is the condition at which the effective resistance across the tank circuit goes to infinity. So the loss component across C2 is compensated for by appropriate negative resistance across this which is brought about by the active device trans conductor GM. Crystal oscillators crystal anyway we had indicated this in the passive network discussion and there we had shown that this is a series resonant formed by L1 and C1 and parallel resonance formed by L1 into C1 C naught by C1 plus C naught okay that is the parallel resonance frequency. So the Q is determined by R1 so the actual equivalent of the crystal is the series resonance that means this is the zero in the numerator of the impedance function and this is the pole in the denominator the Q of QS and QPR very close to one another omega S is close to omega P. So it can have both series resonance effect and parallel resonance effect around the same frequency okay and that is the beauty of this network. This network therefore can be used as inductor or capacitor or purely resistance of high order okay all at the same frequency that is the crystal frequency. So Q's are really very high that is why it brings about frequency stability. So it is typical use of this as a colpitz oscillator is seen here therefore this is acting as an inductor now close to the crystal frequency okay and this is the these are the two capacitors of earlier situation right. So in conclusion what we can see is that oscillator has been treated exactly similar to filter that means let us look at the discussion that we have had negative feedback amplifiers okay feedback amplifiers then. So these are all actually considered as low pass filters okay basically we went from first order to second order we said second order is best suited for this okay negative feedback by amplifiers systems then we discussed. So this could be treated as nothing but again design of high Q filters okay this also could be treated as high Q filters and Q of 1 is recommended for the second order okay. So if it is maximally flat Q of 1 over root 2 if it is transient response Q of 1 that means these are all low pass filter designs of second order. Negative feedback systems again so it is a PID control or something they are treated as second order low pass filter again Q of 1 if it is the transient response and Q of 1 over root 2 if it is the maximally flat frequency response steady state response. See all these have similar conclusions that are leached then we went over to filters proper where we could design it for any Q and any omega naught okay then oscillators Q equal to infinity oscillation frequency natural frequency of the system omega naught. So in all these topics things are mostly the same common thing that we have discussed these are coming about due to the parasitic effects and the active device parameters and we can exploit these for a better design of amplifier or system negative feedback system here we are designing it for specific Q and omega naught so that the overall filter characteristic is what we desired oscillator we are now making the frequency of oscillation stable Q equal to infinity and we require amplitude control okay and frequency stability in the case of these which themselves are amplitude control and frequency stability. Now all these things what we have learnt about filters is becoming useful so I would consider that all these topics have an important topic that has to be understood that is filters coming into picture. So please remember that is this topic is something that is fundamental to analog design so if you understand filters you can design good amplifiers good oscillators and good negative feedback systems thank you very much.