 Today we are going to have the 32 lecture number 32 quadrature oscillator and ring oscillator or the topics to be covered LC oscillator effect of non-ideality is going to be also covered. Let us see what we have done in our 31st lecture we started with basic sinusoidal oscillator topology that is called harmonic oscillator for that we just said it has to be represented as a differential equation d squared V naught by dt squared plus k V naught equal to 0 is the harmonic oscillator equation well known to everybody okay in terms of simple pendulum motion right simple pendulum and the frequency is root k. So the solution of this is in network form realized by using L and C and this differential equation written for current or voltage. Is going to result in this second order with first order coefficient being absent and the frequency of oscillation is going to be 1 over root LC because this k will be 1 by LC. So root k is 1 over root LC that is the radian frequency omega naught radians per second. Now this one can say that is the ideal situation if there is loss due to inductor having some series resistance and capacitance having some leakage or series resistance can be represented this is this particular thing will be discussing in today's lecture the total loss of L and C can be represented as R parallel. So this R parallel of course has to be compensated for by a negative resistance and this negative resistance should be exactly equal in magnitude to the positive resistance RP then only the oscillation is sustained to start with RP dash magnitude should be less than RP so that effectively there is negative resistance across it so the oscillations can grow. So this is required for in practice self starting of the oscillation. So these are the things that we discussed in the last class and as the amplitude required by us is approached the RP dash which is going to be simulated by the active device has to keep changing such that at the required amplitude it is going to be exactly equal to at the required amplitude RP dash is going to be equal to RP then the amplitude stabilizes and this can be done by a crude method of bringing in resistance after a certain amplitude is reached by may be using a Zener diode after a certain amplitude is reached another resistance is brought into picture so defective resistance becomes equal to RP negative resistance magnitude becomes equal to RP. So that is the technique of amplitude stabilization and then we could also do it using a control circuit to change the value of RP dash in such a manner that it exactly becomes equal to RP. So this method of automatic amplitude stabilization was discussed in terms of changing the gain of the loop so that the loop gain becomes equal to 1 and that kind of basic concept is used in most of the communication receivers at the front end for again amplitude stabilization of the received signal IF or RF maintaining it constant at a certain value to avoid the effect of fading etc of the signal. So that is demonstrated by applying a voltage VP sin omega t to a multiplier which is now going to be treated as a voltage controlled amplifier whose gain is going to be varied by this control voltage so VC by 10 is the gain okay that into this VX if this is VX this is VY so VY is equal to VC so VC by 10 is the gain that is varied VC can go to plus or minus 10 volts okay in this case it is 10 volts plus 10 volts by 10 volts or 1 so it can be varied from 1 to 0 to minus 1. Let us consider that it is always positive here so then the gain can just change from 1 to 0 so output has to be maintained constant so this output is sensed and this is AC to DC converter very sophisticated one where the AC VPO sin omega t VPO being equal to VP into VC by 10 so VPO is VP into VC by 10 so by changing this you can change VPO so VPO sin omega t is the output of this so that VPO is squared so you get VPO squared sin squared omega t here which is nothing but VPO squared into 1 minus cos 2 omega t by 2 so if you get rid of this by using a filter the DC alone will respond to this comparator here it is being compared with minus V reference here okay so this is positive so this positive voltage has to become equal to V reference in magnitude so that this current is the same as this current so the current does not go into the capacitor so the capacitor hold sound to a constant voltage if the gain is such that output is less than what we have set it then gain will increase if output is such that it is more than what we have said the gain will decrease that is the negative feedback arrangement that is called AGC or AVC so this was demonstrated okay and here the transfer what is called sensitivity of voltage controlled amplifier is determined by VC by 10 so it is output gain okay is going to be VC by 10 into VP so what happens this now becomes the sensitivity factor that means this voltage VPO okay delta VPO by delta VC is going to be just VP by 10 so this is the quiescent voltage okay and therefore sensitivity factor is VPO delta VPO by delta VI which is VP by 10 similarly the sensitivity of the AC to DC converter is going to be output is VPO squared by 2 input is VPO so you have this output DC okay so that change in output DC for a change in input is going to be just VPO itself this will be 2 VPO by 2 that means it is VP for itself so it is necessary therefore to have a finite magnitude of input in order to make this loop gain pretty high the overall loop gain is the sensitivity factor of this into this into the integration which is 1 over SCR this is the transfer so the loop gain is made up of these 3 sensitivity factors so if it is efficiently high then for a any change here corresponding change will occur here in the opposite direction so that output follows the input okay so this if it is constant this will remain constant so will this so that is part of the amplitude stabilization loop that was discussed in terms of AGC and EBC now let us go back to the other non idealities of this LC oscillator normally this coil that is put as an inductor in the tank circuit of ours as inductor in series with the resistance and this capacitance is fairly normal high frequency application right it is relatively lossless okay and therefore the primary non ideality that comes in the so called oscillator circuit of ours LC oscillator is the series resistance of the coil so this series resistance of the coil we have represented as RS okay can be converted to an equivalent parallel resistance this is normally covered in your network scores however it is very rarely remembered when applications in RF etc design become necessary so let us therefore consider this non ideal inductor or the coil with the coil resistance is the DC resistance normally right and what is it when you convert it into a inductor in parallel with the resistance if the Q of the coil is high okay what happens when the Q is low what happens we like to see so 1 by RS plus G omega L that is the series resistance with the coil okay that is the admittance is equal to again write it as RS minus D omega L divided by it is a mistake here RS squared plus omega squared L squared so this is the conductance part okay and this is the susceptance part okay so RS divided by RS squared plus omega squared L squared my minus J omega L divided by RS squared plus omega squared L squared so that is the inductive susceptance that can be written as 1 by RS is 1 divided by 1 plus omega L by RS is called as the Q of the coil it is important inductive reactance compared to the series resistance which should be ideally speaking going to 0 so the quality of the inductor is dependent upon whether this is going to infinity or no that is RS is tending to 0 so 1 by 1 plus QC squared this can be written as so that means the shunt resistance now is equivalent to RS into 1 plus QC squared so this is the thing that shunt resistance equivalent shunt resistance at a frequency omega is written as 1 plus QC squared please remember QC depends upon the frequency so and similarly the inductor can be written as 1 by J omega L okay into 1 by 1 plus 1 over QC squared so for high Q coil this can be approximated by 1 by QC squared into RS this one can be ignored and here the inductor can be represented as 1 by J omega L the admittance so effectively an inductor in series with the resistance RS is represented as the same inductor shunted by QC squared into RS as the parallel resistance when the Q is high so please remember this fundamental aspect of conversion so another non-ideality that comes into picture in this LC oscillator circuit is that of the finite gain might be product this is the active parameter the earlier one is the passive parameter non-idealities okay and this now we are concerned with active parameter which is the gain bandwidth product which we had shown the gain of this stage is approximately equal to GB by yes that means this acts really as an integrator. So in feedback loop okay with this what happens to it that is the one non-ideality which is due to the active device the other important non-ideality that comes about okay is we have to simulate the negative resistance using the up amp that means here the gain of 2 amplifier is used just as in the previous lecture so R and R make the gain from here to here to and this RP dash connected between input and output of that amplifier gets reflected as minus RP dash. However that is valid only as long as the output of this up amp does not go to saturation obviously this output cannot go above plus VS or minus VS assuming that it is swinging from rail to rail. So the maximum limit is plus VS on one side and minus VS on the other that means this after VS is reached or below minus VS okay can be considered as a constant voltage then what happens at this point the voltage drives a current in the normal direction as a positive resistance so this current will be simply early up to that point when this is in active region it is if this is V this would be 2V so it is being reflected at the input as a negative resistance but beyond this voltage it is a positive resistance of the same magnitude because the current is V minus okay VS divided by RP indicating that the slope is 1 by RP here the slope was 1 minus 1 by RP indicating that it is negative resistance again same way when the voltage here reaches minus VS again it becomes constant then the slope is V plus VS by RP so then again it is 1 over RP the slope here. So in all this regions beyond this and beyond this at that point of time when this is going to plus VS or minus VS this will go to plus VS by 2 or minus VS by 2 that means the range at the input for the tank circuit over which this will behave like a negative resistance compensating for the positive resistance RP here is limited to an amplitude at the input corresponding to plus VS by 2 and minus VS by 2. So because of the practical limitation of the active device as negative resistance this negative resistance is called n type of negative resistance okay it is mainly because actually while measuring this negative resistance in terms of input current and input voltage okay one can do it okay by applying constant voltages sources when you apply a constant voltage source this positive feedback is nullified for the amplifier that is only negative feedback so it is a stable system okay. So you can actually obtain these characteristics by using voltage sources because they will be having single valued currents whereas if you use current source then it has multiple valued currents voltages and therefore it is unstable. So now at this temperature I would like to see show you the other type of non-ideality that can arise the same circuit the earlier circuit was that this was positive and this was negative and RR gain of 2 okay and therefore one could connect a voltage source when you connect the voltage source what happens the effective resistance from here to here by the voltage source is zero. So when I connect this resistance RP dash it is now capable of accepting a current which is this will be 2V this is V so it will accept a current VS by RP dash into it. So it can actually see the negative resistance the voltage source can see the negative resistance the amplifier does not see anything happening here because the voltage source okay forces the voltage all the time to be VS for the amplifier so this output feedback is not going into the amplifier is going as current into the source so that is why it is a stable negative resistance. So this is called short circuit stable and open circuit if I open this right what happens if I try to drive a current here I will not get a negative voltage here inverted okay developed that is because it is effective is open so effective positive feedback dominating over the negative feedback so the output this is like a Smith trigger the output will go to either positive saturation or negative saturation. So this is not an active device as far as we are concerned because it is not behaving like an amplifier okay the concept of nullator is not valid if this is driven by a current source so that is why it is called open circuit unstable whereas it is short circuit stable. Now if I just have that is called n type of negative resistance if I just invert this and have the same circuit R and R as feedback so and then put an RP dash then it is open circuit stable because full negative feedback is there only half positive feedback is there so effectively it has negative feedback still coming to picture so operating point is perfectly stable however I have to keep driving it only by means of current source okay because this has to remain full okay so if this current I enters this this I will go through this develop a voltage plus minus now this is an active device so this voltage is same as this voltage that means this drop I into RP dash will come across this as also drop I into RP dash same so what will be the drop across this will also be plus minus which is equal to I into RP dash because the same resistance same current is flowing through both these resistances so the voltage here is the same as voltage here so the voltage that is developed from ground is nothing but minus I RP dash so my current source now sees a negative resistance of minus RP dash so if you really plot the characteristic it will now look like just negative resistance alright but instead of coming if this is voltage and this current right it will go like this and become positive okay so this is what will happen to it okay instead of going like this and becoming n type it goes like this and becomes s type now that means only for specific values of currents there is a single valued voltage okay and for voltage source drive it is unstable so that is the difference between n type and s type a classic example of s type is nothing but the diac okay so this is in terms of the dynamic range of operation of this negative resistance as far as the other non idealities are concerned it is to do with the finite gain band product of the op amp which we have considered yeah GA is equal to GB by s so let us see what happens due to this because of the gain stage being 2 just as we did in the filter case or amplifier case we know that the gain is going to be now 2 divided by 1 plus 1 over loop gain or the loop gain GB by s into half so 1 over loop gain is 2S by GB so there is a phase error time and again we are bringing in this phase error and we had also discussed in filter how it can be compensated in a loop now what does it do to the oscillator RP dash that gets simulated at the input okay admittance point the input admittance now is going to be 1 over RP dash divided by 1 minus the gain the gain has become 2 divided by 1 plus 2S by GB or it can be approximated just as we did earlier 2 into 1 minus 2S by GB so one gets the simulation as RP dash 1 by RP dash divided by 1 minus 2 into 1 minus 2S by GB which is RP dash okay that is 1 by RP dash divided by minus 1 plus 4S by GB which is equivalent to minus 1 by RP dash the negative resistance that we expect apart from that there is a capacitance this S into C the C value is equal to 4 by GB into RP dash that is the capacitive effect that gets reflected due to the finite gain bandwidth product GB is infinity this is 0 so that is the contribution of the active device okay so this capacitance will be in addition to the capacitance that we had already put in the oscillator so what does this do this will change the frequency to 1 over root L into C plus 4 by GB RP dash that is the variation that is cost so the frequency has now changed root means you can take out the C so that is 1 over root LC the ideal value in the case of LC oscillators the C has come out so this is 4 by GB RP dash C okay and because of the square root approximation this 4 becomes 2 half and I multiply by omega n omega n there then what do I get 2 omega n divided by GB into omega n RP dash C and for oscillation we want RP dash to be equal to RP in magnitude this is the negative resistance minus RP dash is to be compensating this RP so you can write this as Q of the original tank circuit so this is Q omega n RP into C is the Q of the circuit so the deviation is 2 omega n by GB into Q so this is an important aspect this is the non-ideal contribution 2 omega n by GB is the non-ideal contribution in terms of the phase shift error okay that divided by Q of the coil so that is the deviation in frequency higher the Q smaller is the contribution due to the active device higher the Q of the original tank circuit smaller is the contribution due to the non-ideality of the op amp that is an important point to note that the original Q of the tank circuit should be very high in order to have this we will discuss again this aspect of non-ideality in other oscillators all of them will be coming only in terms of this and the Q and we will notice that the frequency stability an important aspect of design of oscillators is dependent upon the Q of the network that is embedded with the active network okay so that the active device non-idealities do not pay a major role in varying the frequency because this GB is dependent upon the design of the active device temperature time etc so one of the basic LC network which has very high Q this we had already presented when presenting passive devices okay is the crystal that is a crystal oscillators are the ones that Q the best frequency stability of all right and it is equivalent is L in series with CS of the order of picofarads picofarads 0.0 to 3.5 picofarads typically 90 kilohertz crystal and RS of this magnitude L is 1374 okay 3.5 picofarads it may be the lead what is that parasitic capacitance shunting the terminals crystals of the order of 5000 to 5000 Q okay are available okay they are the ideal elements for stable oscillators. Let us see the negative resistance simulator in general so we have this topology that we had just now seen for obtaining negative resistance where this was R and this also was R so the gain one from here to here was 2 and we had put another RP dash here so that gets reflected as minus RP dash now if you replace these by Z1 here in feedback okay and Z2 here and Z3 here the gain of this becomes 1 plus Z2 over Z3 instead of 2 it is 1 plus Z2 over Z3 now and this is V and this is V plus V Z2 by Z3 that means this current through this is going to be this minus this so this gets cancelled so you get minus V into Z2 by Z3 that divided by Z1 is the current so effectively if you see the voltage at the input divided by the current it simulates a negative impedance which is of the magnitude equal to this Z1 Z3 by Z2 Z1 Z3 by Z2 this is therefore called as negative impedance inverter come converter so if Z2 is the impedance and Z3 Z1 are resistances then it is an impedance inverter but negative okay that means if you put a capacitor here and these are resistances it gives you a negative inductance you put an inductor here it gives you negative capacitance this is an interesting building block if Z3 for example here itself is negative then it gives you Zi seen as positive impedance inverter that means if you now connect Z2 for this as capacitor it gives you Z1 Z are resistances it gives you an inductor or if you put a capacitor here it is an inductor if you put an inductor here it becomes a capacitor if you put a short circuit here it is an open circuit if you put an open circuit here it is a short circuit so this is a positive impedance inverter when you are terminating by capacitor and these are resistances it is called a generator we had already introduced you to generator for inductor simulators but this is a more general aspect of impedance inversion and conversion. Let us see how it is a positive impedance inverter so this is the first one where now Z itself is replaced by okay a negative impedance inverter minus Z3 Z5 Z3 Z5 by Z4 so effectively if you now see Zi it is Z1 Z3 Z5 Z1 Z3 Z5 divided by Z2 Z4. So it becomes now eligible for two garator circuits either Z2 can be capacitor or Z4 can be capacitors and all the others could be resistors and Zi then becomes SCR square so all of them are resistors of R and only one either Z2 or Z4 is capacitor then it is called garator that means the same topology gives you two garator circuits please remember this so I am telling you that starting from somewhere you can arrive synthesis most of these circuits once you get a circuit okay for a particular operation you can multiply the number of such circuits that you can realize topologies different topologies using a PAM for example the active device by using the powerful concept of nullator and norator. So we know we have already defined okay what a nullator is an op amp input is a nullator an op amp output is a norator so we can replace this by these ideal elements okay active elements nullator, nullator, norator, norator. So this topology itself is responsible for two garators two topologies of garators. Now I am redrawing this entire structure in terms of only nullator norator and these admittance impedances okay we have Z5 grounded then Z4 going to the norator and Z3 is connected and it is going to Z2 and it is this terminal is going to another norator and Z1 coming to the input input and the other point of Z2 is connected through in a later this and this are also connected through in a later. So let us therefore draw that circuit that way this is drawn that way from the input okay so we have Z1 in fact it is Z5 that is actually this is good as Z1 there in fact it solves a little bit Z1, Z2, Z3, Z4, Z5 okay so it is drawn that way so Z1 is grounded then Z2 then a norator Z3, Z4 another norator Z5 to the input this is the input point okay nullator from input to the Z1 and another nullator from this Z1 to junction of Z3 and Z4 that is how it is looking like. So we had identified one nullator with another norator right so this nullator is coupled with this norator in the original circuit and this nullator is coupled with this norator why should we do that? Once you have nullators and norators the pairs can be just paired in any manner you want. So this now can be paired with this okay and this can be now paired with this that results in to other topologies for the garator each of these topologies again will give you by making Z2 or Z4 capacitor okay you have two other garators that means this paired this way results in okay two more garators so 2 plus 2 4 you have now each of these topologies okay results in two garators please remember either making Z2 or Z4 this pairing itself is now different okay next just see from here to here the voltage is zero from here to here the voltage is zero now what I can do is I can put another nullator from here to now see this can be removed okay I can put a nullator from here to here that is what I have put and this original nullator can be removed because this whole three form a loop so one of it has to be removed because nullator norators occur always in pairs. So now you have a new topology with two pairs of nullator and norator so this is one pair this is another pair or you can actually identify this is one pair this is another pair so that means you get two more topologies each of which will contribute to okay two other right so you get again four garator circuits okay why should you remove this you can remove this and retain this that is what has been done here remove this and retain this you get four more right so you can in all you get 12 topologies for the garator circuits that is the power of nullator norator it is so simple okay and therefore in the era of active devices becoming pretty cheap that is you can use it without increasing the cost but better performance can be realized perhaps for the same structure okay for the same function okay by growing more nullator norator pairs okay these are the two garators using the first topology okay this is Z2 and this is Z4 okay and out of which we had chosen this okay even in the case of filter to simulate the inductor so out of the 12 topologies topology which were became well known as Antonius garator has the best performance because the loop has zero phase error whereas all the other things have phase error causing the enhancement of Q so that is the advantage of nullator norator concept that Antonius garator out of the 12 topologies the best okay what that topology is I would like you to decipher from the performance point of view okay how the phase error is compensated in Antonius garator the method is the same one that we had adopted in filter design so this is the inductor simulator circuit so if we replace the LC oscillator with the garator simulating the inductor then we get an RC oscillator that means this simulation this thing inductor ideal inductor okay and therefore I simply connect a capacitor across it and I might have to compensate for the non-ideality of the op amp etc so I also connect a negative resistance across it because this is a gain of 2 amplifier at the input so it can also simulate a negative resistance to compensate for the losses of the inductor due to finite gain and finite gain bandwidth product we had already shown results in a negative resistance which depends upon the gain bandwidth product okay the phase error so it does the Q enhancement of this right that will also effect the oscillator we will see that later so the if in the ideal case the inductor and capacitors resonate then omega n okay is equal to 1 over root LC L is equal to C squared R so the oscillation frequency is 1 over RC if RP is equal to RP dash so that becomes a very nice RC oscillator circuit and this one aspect of migrating over to RC oscillator we will see other RC oscillators okay presently another viewpoint of oscillator in our previous oscillator circuit we had this RP dash okay and RP dash okay L and C okay let us say this is RP dash and this is RP dash so when LC resonates okay this results in a transfer function of half because this becomes infinity okay and this is purely resistive R dash so this results in an attenuator of half so if I preceded by means of an amplifier with gain of 2 then overall gain is going to be 1 at only resonance on either side of resonance the attenuation is more so it is a band pass characteristics I am getting with this transfer function being half okay into 2 by R 2 by R dash plus SC plus 1 over SL so it is SL by R S squared LC plus 2 SL by R dash okay actually it is R dash 2 SL by R dash plus 1 so what do you get here at resonance omega equal to 1 over root LC this gets cancelled with this SL by R dash divided by 2 SL by R dash it is half so if now a non inverting amplifier of gain 2 is used loop can be now formed which has a loop gain of 1 only at resonance omega not equal to 1 over root LC otherwise the gain is less than loop gain is less than 1 so it sustains okay output equal to input before closing the loop so if you now put an amplifier of gain 2 here so this is VI this will be V naught will be exactly equal to VI both in magnitude and phase so now you can close the loop and sustain the oscillation if there is initially some energy put into it so that is the way another way of looking at oscillator as 1 where the loop gain becomes equal to 1 at omega equal to omega naught resonance which is 1 over root LC this is the complete circuit with the loop closed so a band pass filter this is nothing but a band pass filter with maximum gain of half okay so but this can have high Q you can actually replace this whole thing with a low Q band pass filter that is RC why I am saying this low Q right that we have shown that the Q of a passive network cannot exceed half okay so I replace this with a band pass this will remove low the high frequencies this will remove the low frequencies so this is effectively a band pass this is RC this is RC then this also has a characteristic like this if a shift is 0 exactly at this point which is omega equal to 1 by RC and this will be 1 by 3 okay its transfer function is SCR divided by 1 plus 3 SCR plus SCR whole square okay either this or this can be used both have the same transfer function and resonate at the same point same transfer function for both so even though topologically they are different circuits they have the same transfer function so either this or this can be used in place of the RLC network only what we have to do is instead of gain of 2 we have to use gain of 3 okay because only when this is 3 SCR by 1 plus 3 SCR plus SCR square the loop gain becomes equal to exactly equal to 1 and phase shift becomes equal to 0 so this is what is done by closing the loop with this kind of thing however the Q of this is low what is the consequence of low Q okay and why this kind of structure is never preferred in RC oscillators okay is going to be discussed after we discussed the topic of frequency stability in such structures the low Q structures have more error that I have already pointed out in our LC network so let us go to other types of RC oscillators phase shift oscillators okay again these are RC oscillators RC network with active device as op-amp quadrature oscillator and then ring oscillators these are some of the popular oscillators which are used in present day electronics analog electronics they have been earlier also used is one of the most important structure this has come down from the analog computer era okay and now it is popularly known as quadrature oscillator it has a distinction of giving two outputs which are always in quadrature at the frequency of oscillation that is nothing but simulating the second order differential equation I had already mentioned about this resonator block please understand this is nothing but the resonator block one integrator followed by another integrator and a inverter so this is going to solve you this squared V naught by dt squared equal to minus KV naught which is nothing but the harmonic equation okay so omega naught is equal to 1 by RC that is the frequency of oscillation and these outputs will be in quadrature with one another okay so one can actually because of finite gain we had seen this right the oscillation will be finite DC gain oscillation will be damped okay the damping will be less when the gain is high when the gain is infinity no damping okay finite gain bandwidth product will result in worth the thing oscillating okay in spite of having finite gain okay at certain frequencies or beyond certain frequency so this is going to be definitely requiring some way of coaxing it to oscillate if it is stuck at low frequencies maybe so one can actually give a coefficient here okay because finite gain will result in a positive coefficient for DV naught by dt coming into picture which is the coefficient of which is inversely proportional to A naught okay so you can compensate for it by giving a positive feedback like this okay so if you invert it and fit it here the poles will be located on the left half side if you do not invert it poles will be located on the right half side the poles get located on the right half side the oscillations can grow okay and come to the required amplitude at that amplitude this has to go come out of it that means this feedback should be removed okay so that means actually we can control this okay feedback by using for example a multiplier here okay changing the value of the resistance making it positive or negative based on the control voltage going positive or negative so it can be made to have initially poles on the right half plane and gradually the poles can shift towards the left half plane by reducing the value of VC okay and then ultimately it can come to the j omega axis at the required amplitude this is the amplitude stabilization scheme that can be incorporated into this okay and that is done like this there is a very nice way of sensing the amplitude this is when it oscillates if this is VP sin omega t this will be VP minus VP cos omega t okay because omega naught RC is equal to 1 that is a frequency is determined by omega naught RC becoming equal to 1 or omega naught equal to 1 over RC so if you square these and this and then add using equal resistances then it will be a pure DC so what happens here is if you use a multiplier for example if this is VP sin omega t this is VP cos omega t this becomes squared VP squared by 10 sin squared omega t this becomes VP squared by 10 cos squared omega t so addition of this will result in VP squared by 10 because this plus this here so effective current flowing through this is VP squared by 10 divided by R2 that must be made equal to V reference okay divided by R2 so V reference into 10 is VP squared VP is square so VP is square root of 10 times V reference and this comparators is to it that this VP is brought up to that point so that is voltage remains constant at VC right appropriate to maintain it okay at an amplitude of this that means this voltage here should be nothing but 0 virtual ground so this current is the same as this current which is flowing here so that is the amplitude control loop and you can see that we have done with V reference equal to 0.4 VP equal to 2 this has been simulated and you can see the quadrature outputs okay at these two points okay is exactly equal to 2 volts and the frequency omega is equal to 1 over Rc then we have VP is equal to 2 volts VP reference is 0.1 square root of 0.1 into 10 is 1 VP is equal to 1 so you can see that is exactly equal to 1 so a precision amplitude fast amplitude stabilization scheme can be done that means this is one of the most interesting but pretty complex feedback system that can exist in analog this itself is a negative feedback system integrator inverters summing amplifiers okay integrators and all and these are now put in a feedback loop okay now for amplitude stabilization you have a separate feedback loop okay so you can see the great thrill of watching it gets simulated or actually implementing it hard hardware and seeing it working satisfactorily now the last topic for the day is the ring oscillator this is the famous CMOS inverter type of oscillator which is made out of so this is also nothing but the Rc phase shift oscillator so if you use a network like this it is nothing but first order low pass okay its gain is DC gain is 2 and it is time constant here please remember this is nothing but this is actually should be 2 that is 2R cross this okay so this being 2R you will have this 2R whole square coming into picture okay so this gain of individual stage is 2 divided by 1 plus 2 SCR okay you have 3 such inverters effectively therefore minus 2 cubed by 1 plus 2 SCR whole cubed okay so you can actually evaluate this okay is going to be okay minus 8 okay and divided by 1 minus there is a sort of square term here which results in 3 omega naught square C square okay to 2R square okay that part should be equaling 1 and there is imaginary part in this which is 3 omega C into 2R okay then this 2R whole cubed okay so we have this 2R whole square okay omega C square C square 2R whole square becoming equal to 3 or omega naught is equal to root 3 by 2R C okay there is a frequency and this particular thing happens okay to have a loop gain of 1 at this particular frequency and this condition gets satisfied okay so it requires gain of 8 to make the loop gain equal to 1 because this becomes okay 9 okay so 1 minus 9 it is minus 8 so the loop gain is 1 exactly so this is the ring oscillator one can build this and see for yourself how CMOS ring oscillators work only as an analog oscillator but of course if you are operating up to the supply okay this whole thing cannot be assumed as linear so it becomes non-linear it deviates from this frequency okay so we have discussed RC oscillators starting from LC oscillator replacing GAL with inductor LC replacing GAL with simulated inductor that is gui-rater we had also seen how gui-raters can be synthesized starting from one gui-rater okay which is combination of two negative impedance inverters becoming positive impedance inverter or converter so using some gui-raters next we went over to phase shift oscillators under the category we have put quadrature oscillator which is one of the important oscillators of today and the what is that CMOS inverter type of oscillators derived from op-amps okay and these oscillators are common now only as ring oscillators okay and the old phase shift oscillator is out of use it has no practical utility whatsoever except for the ring oscillator which is based on the phase shift oscillators of the olden era. Thank you very much in the next class will be discussing about the non-ideal effect and frequency stability in oscillators and going over to LC oscillators like Hartley oscillator, Kristall oscillator, Kalpitz oscillators etc.