 Today in lecture 31 of hours we will be starting with discussion on oscillators waveform generators both sinusoidal and non-sinusoidal. Before we go to that let us review what we have done in 30th lecture phase lock loop basically what is called as self-tuned filter a filter second order filter with high Q low pass output phase is compared with that of the input and it should be 90 degree so the loop gets locked to 90 degree phase shift at all the times as the input frequency is changed phase remains constant at 90 degrees that is what is called phase locking and that facilitates automatic tuning of the filter to the reference frequency which is the incoming frequency. So the negative feedback with loop gain high maintains this lock lock of phase to 90 degrees very accurately over a wide range of incoming frequencies. Now the lock range for example is sort of that is decided by the there are three blocks one is phase detector and then there is this amplifier plus low pass filter and then what is considered as the voltage controlled filter goes output goes here. So that is the phase lock loop the sensitivity of this is called KPD the sensitivity of this is A naught divided by 1 plus S by Omega LP the sensitivity of this is K VCF converts voltage to phase difference Phi which is dependent upon this VC. So the loop gain is nothing but KPD A naught by 1 plus S by Omega LP into KVCR okay if it is an integrator this one is neglected compared to S by Omega LP that is all that happens with the integrator okay so that lock range is therefore governed by any one of these limiting the range over which they function that means this may go to saturation or here the loop gain can go to 0 I mean it KPD can go to 0 and here op amp can go to saturation VCF can reach the limits of operation okay of its Omega naught. So any one of them whatever is lower is the one that limits the lock range capture range again before the loop gets locked okay this is at a cohesant state hopefully and this filter is tuned to some frequency and let us say this is the low pass filter output as we had taken okay so the low pass filter output decides what the KPD of this because it depends upon the magnitude of both these inputs this is the multiple type of phase detector so the amplitude of these low pass filter output also governs the KPD of this. So if there is no amplitude there is no chance of this locking okay so one has to come very close to the VCF Omega naught Q let us say we call it Omega naught Q at which some output can be got so that this KPD can be substantial in order to make the loop lock itself okay as I capture range is always less than the lock range capture must occur before lock capture range is always less than the lock range lock range decided by the limits of any one of the blocks in PLL capture range occurs always close to the cohesant state where the loop can gain can be high that is the way the automatic locking occurs phase to 90 degrees design then was discussed second order filter blocks can be cascaded together to form a complex higher order filter second order okay then okay first second third second order so on then there may be a first order if it is odd so that is how you get the filter higher order filter design done poles of the second order filter peak around the bandwidth okay this if one has to decide to design let us say low pass okay or band pass okay with center frequency like this then one knows that the basic second order is going to be like this at the center frequency and this is the basic thing then other filters must be located okay with higher and higher cues on either side of this okay so that this fall in gain can be compensated for by a peaking so we have these other filters which are built around the pass band H okay so that it becomes maximally flat so these gains will be higher okay and cues will be higher and with the resultant effect one can get maximally flat structure okay and if you want to get it the specific frequencies there must be zeros located there and those zeros okay bring about some variation in the pass band so that is again compensated for by more peaking by these higher Q filters around the center frequency near the pass bandage so these are the intuitive feelings about the filter design so one can actually really do the filter design by a locating the second orders suitably just by observation first locate the zeros wherever narrow band noise is present then locate the poles with peaking near the pass bandage if it is a low pass filter okay or band pass filter higher Q poles with the higher frequencies get located closer to the bandwidth this is for the low pass okay effect of gain bandwidth product is to cause Q enhancement which can be compensated so all these factors are important in design okay the Q enhancements caused by the finite gain bandwidth product how it can be compensated without increasing the number of op amps or active devices now waveform generation drastically different from the filter design first let us know why we should generate waveforms waveform generation is needed primarily for testing and testing is an important part of product manufacture or IC manufacture so testing and operation of analog and digital systems require a very pure sine wave to find out the distortion of a given particular stage if it is linear system okay and also to obtain the frequency response of a given system the sine wave is important so whether it is testing of for your DAC or ADC or anything these test waveforms okay have to be precisely generated with a stable frequency and stable amplitude okay clock this is an important thing it is like the heartbeat of an electronic system so the clock generation is a very important part of today's system design it is necessary in digital systems also it becomes necessary in switched mode analog systems right mixed mode systems square wave there is one of the waveforms that are necessary as clock for example even rectangular wave with pulse with modulation okay generation so these are now becoming more and more useful okay in terms of efficiency because the active device is mainly used as a switch in these sawtooth waveform generation is mainly for displaying waveforms in oscilloscopes and also in cathode tube television receivers triangular waveform this is again an important version of symmetric waveform like sine wave okay which has rich harmonic content in it arbitrary waveform yeah for some applications arbitrary waveforms are required so that in one shot one can generate get gather multiple information about the characteristics of the system in all these things constant amplitude as frequency varies is a must amplitude must be independent of frequency okay variation precise adjustment of amplitude is necessary frequency stability is another important aspect in oscillator design so all these we will touch upon in terms of detailed discussion as to how to achieve these so these are the typical waveforms this is a sine wave this is the clock wave this is the triangular wave this is the sawtooth this is a square wave this is the arbitrary waveform so in all these things again the amplitude and the frequency become very very important part of design stability of these coming to the fundamental aspect of sine wave oscillators this is something that you have studied in your plus 2 most everybody knows that this is what is called harmonic oscillator equation del del squared X by del delta squared T squared okay this is here is this plus K X equal to 0 so solution of that is X equal to A amplitude sine root K is K forms the root K forms the radian frequency of the waveform plus this is the phase A and 5 depend upon the initial conditions that we know in the case of electrical oscillation X is replaced by either voltage or current that is all the differences so in a network whatever voltage or current exist at any point is governing this kind of equation if it is an oscillator so what is it actually what is the network this we have already seen earlier when we introduce you to the inductor and the capacitor this is a storage element this is also a storage meant this source energy in the form of electromagnetic form in this energy stored in electrostatic form so it keeps on changing from one form to the other right switching from one to the other that is how the harmonic oscillation works now that is it by equation indicated as V is equal to minus L DI by DT is equal to 1 over C integral IDT so this now forms the second order differential equation with DI by DT term being absent that is why it is called harmonic oscillator D squared I by DT squared plus I by LC equal to 0 the solution of this I is equal to some magnitude IP existing at the initial conditions IP is determined by the initial condition IP sine T by root LC it is root K 1 over root LC is the coefficient K squared governing this so this is the frequency omega equal to 1 over root LC plus 5 IP sine omega NT plus 5 IP and 5 depend upon the initial condition omega n is the frequency which is 1 over root LC if the capacitor is initially charged to 1 volt let us say at T equal to 0 and inductor current at T equal to 0 is 0 then I is equal to IP sine omega NT 5 is equal to 0 because at T equal to 0 I is equal to 0 so 5 is automatically 0 it is a sine wave starting with 0 reaching IP after quarter of a cycle DI by DT therefore the slope at T equal to 0 however is IP by omega n cos omega NT it is a cosine wave and L DI by DT multiplying this by L okay you get the voltage that is present okay across the inductor okay L DI by DT okay so omega n equal to 1 over root LC which is also the voltage across the capacitor natural frequency of the system okay if L DI by DT is equal to 1 volt okay IP is equal to omega n amperes that is how the IP gets fixed by the initial conditions so this is what is simulating it L is equal to 1 milli henry C equal to 0.1 micro farad gives you time period for the waveform of okay 63.61 micro 6 so voltage across C at T equal to 0 is 1 volt and current through the inductor at T equal to 0 is 0 so we have this this is the current through the inductor the voltage across the capacitor this is the sorry current through the inductor and the voltage across the capacitor starts with 1 volt goes on like this so that is sustained throughout and this frequency is exactly what we have calculated 63.61 now if there is a resistance across this the equation turns out to be V by RP total current plus this is actually now being called as RP plus 1 over L integral V DT that is the current in the inductor C DV by DT is the current through the capacitor so summation of all these currents should be equal to 0 that gives you second order equation it is this squared V by DT squared plus 1 over RPC DV by DT plus V by LC equal to 0 which gives us normalized representation of this this as omega n squared so this is called Q of the tank circuit this is called the tank circuit because it can store energy okay if the Q is infinity it perennially stores energy in one form or other or it is an oscillator so Q is equal to omega n RPC here RP into root of C by L omega n being 1 over root LC if you replace this is the Q of this tank circuit of parallel resonant circuit omega n is the natural frequency Q is known as the quality factor of the resonant system its ability to stock energy it indicates Q equal to infinity means it is perennially storing energy. So the circuit is popularly known as tank circuit because of its ability to store electrical energy okay now writing down the whole thing in the Laplace transform and solving for the roots of this you get S equal to minus omega n by 2 Q okay plus or minus omega n by 2 Q square root of 1 minus 4 Q square for Q less than half there are 2 poles on the negative real axis okay this is omega n by 2 Q minus this is sigma this is j omega axis of your S plane so this is minus omega n by 2 Q plus omega n by 2 Q square root of 1 minus Q square this is minus omega n by 2 Q square root of 1 minus 4 Q square so these are the complex conjugate I mean these are the poles on the negative real axis they become complex conjugate pairs for Q greater than half then the poles are going to be located like this let us say so it is minus omega naught by Q to Q okay for higher Q Q it is going to be closer to the j omega axis at this distance is decreasing and this is the complex conjugate pair of poles so for varying Q you get it in a if you put it in a normalised fashion where you put as sigma by omega n as j omega by omega n then it becomes a unit circle otherwise it is going to be located around this point. Now typically for this circuit for the same value of L and C R is put as 1 kilo ohm so that Q is 10 what is the physical significance of this there are therefore it starts with 1 volt nearly 1 volt okay and then you go on up to 0.1 one tenth and then you can count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 such peaks before which it becomes after which it becomes less than 0.1 so there are 10 such peaks appearing okay as far as this thanks circuit is concerned this way you can quickly estimate the Q of the system now what can we do to enhance this Q so that it can oscillate. So this loss which may be due to combination of capacitor loss and inductive loss combined to single resistance in parallel can be compensated by a negative resistance which is going to be simulated by the active device. So it is this negative resistance which is the active device simulated one in any oscillator. So you can view any LC oscillator whatever be the type okay as basically having some loss component compensating compensated by the negative resistance. So there is hardly oscillator the Colpit oscillator or any crystal oscillator any of these belong to this category of LC oscillators and we will see later how it can be thought of as bringing about a negative resistance. So this is the basic principle. So how to simulate this negative resistance this negative resistance of course can be simulated using active device in our present case we are taking an off ham it could be a fat or a bipolar transistor or a MOSFET or a junction fat any of these can act as active devices to simulate this negative resistance. So we have LC and the R so this circuit is the one that simulates negative resistance we had already discussed it in our filters case right please recollect that this entire circuit with a gain of 2 for example simulates between its input okay and ground a negative resistance of minus RP dash how does it happen RP dash is connected between input and output of a gain of 2 stage because it is having R and R here the system is nothing but the non inverting amplifier with a gain of 2 is 1 – G okay this means this is minus RP dash. So whatever resistance is connected here in the gain of 2 situation that gets reflected as a negative resistance. So that is how we simulate and effective resistance therefore is the original parallel positive resistance the loss component shunted by minus RP that is it is a shunt combination of RP and RP dash which is the effective resistance actually it is 1 over effective resistance please remember that 1 over effective resistance therefore is equal to 1 by minus RP dash plus 1 over RP that is the admittance that should be negative so that means RP dash should be less than RP in order to make it negative if RP dash is greater than RP it is positive effective resistance. So then the oscillation amplitude will progressively increase instead of decreasing in our earlier situation we have seen by putting 1K resistance across the LC finite cube makes it decaying sine wave whereas if it is having effective negative resistance it will be an exponentially growing sine wave as it is shown here. So in order to be self starting an oscillator has to be self starting what are the requirement first of all this condition should be valid that means the initially that should be a negative resistance effectively across the LNC which facilitates growth of sine wave oscillation from a initial energy that is just some little bit of noise has to be there okay and then it has to be having effective negative resistance then it will be self starting and it will keep on growing then as it is keeping on growing you must make sure that at the required amplitude of oscillation the effective resistance should be infinity so that Q of the system is infinity so at that amplitude okay the oscillations get sustained once again let us understand initially it has to be negative so that there is a growth of oscillation at the required amplitude of oscillation this effective resistance okay should become infinity so that the Q becomes infinity so that at that amplitude it is sustained. So the resistance RP dash must be made non-linear such that it increases in magnitude as voltage across it increases to a given amplitude so R should be dependent upon the voltage or current right in such a manner that R at a given amplitude only becomes equal to RP that is this is RP dash I should say magnitude of RP dash should be voltage dependent and it should be less than RP initially at the required amplitude it should be exactly equal to RP it stabilizes at that amplitude of PP this effectively means okay net admittance of the entire network this is inductor is the capacitor so we have omega n equal to 1 over root LC what does it mean omega n L is equal to 1 by omega n C that means the effective susceptance of the whole system is 0 and effective real part is also 0 so the total admittance net admittance of this at omega equal to omega n is equal to 0 this is what makes a sine wave oscillator so please again remember in terms of a network design it is total admittance should be going to 0 at omega equal to omega n precisely that means both the real part and the imaginary part should go to 0 at the same frequency omega equal to omega that is then a candidate for sine wave oscillator negative resistance can be obtained by using tunnel layout for example it is a negative resistance divide which is called n type of negative resistance okay this is voltage that is current say this we had discussed earlier and we discussed devices BJTs and FETs or op amps we have seen how it can be done with the help of an op amp tunnel layout BJTs and FETs are mainly used in the RF and microwave ranges okay op amps are used at lower frequencies non-linear negative resistance how to bring about this a grounded negative resistance can be simulated by using a non-inverting amplifier of gain greater than 1 so if you have let us say a non-inverting amplifier let us see non-inverting amplifier so any gain greater than 1 so this gain is 1 plus R2 over R1 so it is always greater than 1 and longer there is some value of R2 and some value of R1 so it can always simulate therefore a negative resistance it is not necessary that R2 should be equal to R1 so that it is equal to 2 in that case this resistance itself gets reflected as negative otherwise the resistance will be different okay by that extent so what is the negative resistance that case simulated it is going to be just this R divided by 1 – G which is – R2 by R1 so this is known as negative impedance inverter or converter right because depending upon which resistance is being used as a terminating resistance it can be called an inverter or converter so this is an important block so a frequency of oscillation is 1 over root TLC now the amplitude of oscillation if it is not limited by anything all the way can go up to the saturation limit of the device in this case so if you consider that these are the supply voltages VS and – VS after it reaches saturation plus VS or – VS there is no such negative resistance that gets simulated here so the gain is 0 the loop gain goes to 0 and therefore this does not work as a negative resistance any longer so what happens if this is the limit plus VS – VS is the limit up to which it works as a negative resistance then that divided by 2 is the limit up to which the resistance at the input can work as a negative resistance so that is the main contention of this that is what happens to this negative resistance it will be just let us say the negative resistance okay as far as this input is concerned right if this is V and that is I so then ultimately it gets limited to VS by 2 and VS by 2 on either side okay with slope which is determined by – 1 over RS here is 1 over RS here this is I okay so this is positive resistance of 1 over RS in fact okay that is the slope and this is the negative resistance which is – 1 over RS so delta I by delta V is the slope so it is going to be useful only in this range okay that is what is plotted here because this is output is limited to plus VS and – VS because of its saturation limits this negative resistance effect is seen only until this voltage reaches plus VS – VS between this voltage is half of this so up to that value of V it is negative thereafter only the positive resistance because this is constant and this voltage – this constant voltage by RP dash is the resistance it says which is giving you the slope of 1 by RP so this is the point that we have to see here it is V okay – VS by RP then RP dash these are all dash okay so that is the limit here it is going to be equation is V plus VS by RP dash so only here it is negative resistance okay where it is V – 2 V by RP dash let us see what happens due to this the RP dash used is 600 ohms RP is 1 kilo ohms so it is much less than this effective resistance of this parallel combination is negative same C and L are used C is made 1 micro far R is equal to 10 kilo ohm op amp used this LM7 for 1 to simulate this so one can see that is growth of oscillation taking place and then it gets limited to the supply voltage which is nearly plus – 15 volts for this so it gets limited to that you can see the almost okay gets chopped off here and here so we do not want this to depend upon supply voltage and all that so we would like to limit it to voltage lower than this okay so that can be done one way is by using a non-linear resistance okay so what is done is same negative resistance effect is brought about RP L and C gain of 2 stage RP dash – RP dash comes across this except that now there is a positive resistance which is brought only when the voltage across RP reaches the Zener voltage that is in this case chosen to be 1 volt Zener that means it can limit the or it comes into picture or it switches on a shunt resistance across RP okay and lowering the positive resistance or making that become dominant only when the amplitude of voltage is reaching plus – 1 volt this is a very crude mechanism of limiting the amplitude to whatever value you want independent of the power supply so with this kind of limitation one can see that this output in the non-linearity of this Zener brings about this linear resistance in series with it okay and this whole effect okay is coming to picture and limiting the amplitude to something like 3.6 volts in this case okay so very close to let us say a lower value than the supply voltage so and it looks almost like the sine wave however please remember that this amount of current flowing in this which is bringing about the effect of this resistance in shunt with RP is the one that is bringing about non-linearity in this waveform so bringing about amplitude stabilization using this kind of non-linearity or the non-linearity due to the op-amp is causing distortion in the sine wave okay so it is at the expense of distortion that the amplitude is getting limited okay however the frequency is okay almost independent of the amplitude if these are not frequency dependent if these are frequency dependent okay this non-linearity is frequency dependent then that frequency dependent will shift the frequency of the system this will later on see in the next class right. However you can see that I can get a fairly good amount of sort of distortion reduced due to clipping etc by introducing a smooth non-linearity okay with device okay now precision amplitude stabilization this precision amplitude stabilization is brought about by again in the same circuit by using an amplitude stabilization technique using a control systems this is what is called automatic gain control the gain of this loop should be controlled in such a manner that in this case for example okay this whole system is bringing about this RZ effect in shunt with RP by means of a control here so I am sensing the output amplitude how I am sensing it using a multiplier so if you have this this output let us say is to be fixed at certain amplitude let us say VP sine omega t so what is the amplitude here it is VP squared sine squared omega t by 10 so sine squared omega t is 1 – cos 2 omega t by 2 so it is VP squared by 20 which is the DC voltage which gives a measure of the amplitude of this sine wave. So if a low pass filter or an integrator is put that gets rid of this component okay so the DC component corresponds to VP squared by 20 that is compared with this a positive voltage because the square that is compared with the negative voltage okay which is V reference right so that is when the integrator stops integrating and keeps a fixed value of voltage such that the amplitude stabilizes at that point so this is the comparator same control system that we have used earlier in PLL and all that so this voltage is the reference voltage that voltage is 0 that means this current is same as this current so that is under this condition that means VP gets fixed at square root of 20 times V reference so precisely we can adjust the amplitude to remain at this constant value of 20 times V reference. So irrespective of the frequency of oscillation decided by this so this is the gain determining loop here okay such that this compensation occurs precisely okay when this control voltage which is remaining unaltered okay is applied to this and that control voltage gets applied when this condition gets satisfied in this control loop so this voltage should be equal to this that is 0 so that these two currents are exactly one and the same in the steady state DC currents so you can see now this control system working it is there is growth of oscillation taking place initially okay and that is necessary that means this voltage is such that effective resistance is kept negative so that is task building up then as the build up goes on this voltage keeps increasing and therefore it is applied to inverse so this voltage keeps decreasing until it reaches a point it remains constant that is when this voltage is 0 so output amplitude remains constant at this now for this actually V reference has been chosen to be say 0.2 volts so that 0.2 into 20 is 4 square root of 4 is 2 so this stabilizes at a amplitude of nearly equal to 2 volts or here it is taken as 0.45 as V reference in which case it is 9 here and it is square root of 9 which is 3 so that is exactly what it is it is getting stabilized at 3 so I use 2 values 1 is 0.2 and 0.45 so that it is precisely 2 or 3 so this is the stabilization V reference in this case is 0.45 so this is the technique of amplitude stabilization and we can see that VP is equal to precisely equal to 3 volts 3.04 okay this I have sample small portion of this in the steady state so that is indicating exactly 3.04 as simulated amplitude for V reference equal to 0.45 so this is precise amplitude stabilization so what is this system this is what is called automatic gain control system or AGC or EVC which is one of the most popular blocks system block analog system block used in communication receivers cell phones or television receivers or radio receivers from time immemorial so this is path and parcel of a communication system as the front end RF or IF okay all these amplifiers are controlled such that output of these stages remain fairly constant irrespective of the received input at the antenna which keeps fading often okay so that is simulated using this AGC system which is part and parcel of all sine wave oscillators if they have to have precise stabilization of amplitude and part of all communication receivers so VP sine omega t is fed to that let us consider this as a voltage controlled amplifier nothing but a multiplier VX VY by 10 so VY is VC so this amplifier gain is VC by 10 that into VP sine omega t is the output so this is the output that it is here depending upon VC and that is squared just like we did in the amplitude stabilization loop for sensing the output amplitude in terms of DC so we get here VP squared by 20 that is VP O let us call it this is equal to VP O so this VP is different from this VP O it is multiplied by VC by 10 okay so this is VP O squared by 20 is the DC that is produced that okay has to be equal to the negative reference that is put there so what happens here is that if input is 0 for example and V reference is still that this is 0 so this negative voltage will make this go to positive saturation so it starts with positive saturation right that means it starts with the highest gain possible for VC which is let us say 10 volts so the maximum gain of this is 1 okay so this output is going to be input is going to be just reflected at the output okay if it is 0 nothing comes there so whatever is applied comes with gain of 1 as long as this is in saturation as the amplitude is increased at a certain point when this voltage is reached okay then this starts increasing above this then this is positive so this goes towards negative saturation or it becomes less positive as it becomes less positive the gain of this reduces to a value less than 1 so output is going to be sort of decreasing such that this voltage ultimately gets adjusted to be equal to square root of 20 times V reference that is the negative feedback that is working for you as AGC okay so for input voltage of 3 volts and V reference of 0.2 we have simulated the same circuit here using multipliers okay and you can see the control voltage from maximum saturation right it is coming down to the state value of 6.9 volts required to maintain an output voltage of 2 volts because expected output voltage is 2 volts so it is coming out as 2.054 volts right so that is with a reference of 0.2 okay 0.2 into 20 is 4 square root of 4 is 2. So now let us say we change this input voltage now from 3 volts to some other value let us say 10 volts so again this control voltage is adjusting automatically such that now it comes to 2 volts from the original 6.9 volts so as to maintain the output at nearly 2 volts 1.946 volts so this is a precise AGC control system so if the input is greater than 2 volts the AGC takes over if it is less than 2 volts since our multiplier can only work up to 10 volts let us say. So then it is going to be stuck at the saturation value of the op amp or that of the multiplier input which corresponds to 10 volts that is maximum gain it can give is 1 that means up to input voltage of 2 volts the AGC does not work it is going to have a gain of 1 throughout maximum gain possible so the AGC dynamic range that V naught VP naught is going to be in the case of 2 volts remains up to an input voltage VP of 2 volts it follows the input voltage with the slope of 1 and then it remains constant at 2 volts so this is how the AGC functions then beyond 10 volts the multiplier again stops functioning so we have that sort of remaining constant gain remaining constant at that value corresponding to there so there is no point in using this so this is the AGC range lock range. Lock range is determined by the range of operation of the multiplier output can be limited input can be limited to 10 volts let us say saturation range of the op amp whichever is lower loop gain evaluation. So we have here various blocks this is one block so as far as the loop gain is concerned it is delta VP O delta VP O change in output voltage variation for a delta VC that is that of the VCA this being constant okay so it is now going to be equal to VP by 10 because it is directly proportional to VC so it is this is the KVCA of this block VP by 10 it depends upon VP so if VP is more it is going to be more then as far as this is concerned this is going to produce an average delta V average divided by delta VP that is the transfer parameter of this or sensitivity of this so that is going to be nothing but to VP O by 20 that is that of the amplitude conversion AC to DC converter AC to DC converter so that is KACDC this is KVCA and then the integrator okay transfer function is 1 by SCR so minus so the loop gain is nothing but KVCA KACDC into minus 1 by SCR so the change in what is that amplitude for a change in reference this change with respect to this change is what is 1 by 1 plus 1 over loop gain which is going to be K let us say VCA KACDC into 1 over SCR so that is the transfer function the bandwidth of that is controlled by this this is the transfer function for delta the ACDC converter to delta V reference so let us say unity gain right minus this is the transfer function of this loop so in conclusion we have discussed here a very interesting loop which is called AGCAVC and we also discussed the aspect of sine wave oscillators LC oscillator or the harmonic oscillator the fundamental oscillator circuit in electronics and how the amplitude stabilization can occur approximate control of the amplitude using non-linear devices then precise control of the amplitude using gain control loops thank you very much.