 Today we will be having phase and frequency followers discussed in our 36th lecture. Let us consider what we did in the last lecture. We had sine wave generators and function generators discussed in the last lecture. Sine wave generators were blocks which were simulating second order differential equation with the first order coefficient being 0 or in fact it has to be negative in order that it is self starting. So that the sine wave can start from a minimal noise pickup and grow into the ah sine wave of the required frequency at a given amplitude at which point of time the coefficient of the first order should go to 0 at the exactly the given amplitude of oscillation that is required by us. So this requires amplitude stabilization scheme ah AGC ah so that the gain of the amplifier can be appropriately adjusted. So as to make the amplitude take on the required value function generator on the other hand is a simpler solution to obtaining ah stable amplitude of oscillation square wave and triangular wave it can be converted to a sine wave. Again the voltage control oscillators made out of sine wave generators require an amplitude stabilization scheme which is a control circuit for precision amplitude stabilization with the least amount of distortion coming into picture in the oscillating wave form. Whereas ah function generator needs just one multiplier as against the two multipliers required for ah sine wave ah voltage control oscillator. And therefore most of the time it is the function generator circuit that is normally used as voltage controlled oscillator. FM generator and FSK generator can be nothing but DVCOs where FM is equal to the carrier frequency plus delta omega d sine omega mt. So that ah DC it generates the carrier and the frequency deviation delta omega d sine omega mt is the modulating frequency at the input of the VCO in order in addition to the DC. So that if you apply DC with modulating frequency in series with it at the input of the VCO one generates FM. The DC corresponds to the VCO omega c and the modulating frequency amplitude causes the frequency deviation delta omega d. The ah proportionality constant ok which converts voltage to frequency is TK VCO that is the change of output frequency as ah control voltage is changed. It is expressed in terms of radiance per second per volt or in terms of hertz per volt. And ah FSK it is nothing but ah digital ah word input done with the DC over it. So that the frequency of shift is changing from omega c plus delta omega d to omega c minus delta omega d as once and zero streams keep coming and modulating the carrier. So inverse function generation using VCO in a frequency locked loop feedback loop VCO comes in and input is a frequency and the frequency detector is put at the front end that forms what is called frequency locked loop. Obviously a VCO converts voltage to frequency whereas VCO put in a feedback loop converts frequency to voltage inverse function. So we are discussing therefore later on the FLL ok in that form with VCO in a feedback loop. So in the today's lecture we will be starting with PLL the phase locked loop. So phase and frequency followers we call that just like voltage and current followers. We have discussed this already in ah the ah analog systems of usefulness in current day signal processing. So input is an AC signal VP sin omega t output is to be VP dash sin omega t plus phi where phi is the phase shift between input and output this is fixed by the designer it could be 90 degrees 60 degrees 45 degrees 120 degrees anything that we desire ok. So this is fixed how to build a circuit with this kind of phase shift at any frequency is our task. Now this was what was proposed earlier for this phase follower phi i is the input phi naught is the output and this phi i minus phi naught is what is fed in the error detector. So error detector output is going to convert the phase difference to voltage. So this is K1 this will call it as K PED phase detector. So later on we see that this phase 2 voltage conversion is done by a phase detector whose input is a phase difference output is a DC V average. So this is nothing but the sensitivity of the phase 2 voltage converter which is delta VC by delta phi and voltage to phase converter so that this point we have generated an output phase from a voltage. So what this block is is something that we will again discuss. So K2 is nothing but KVCP voltage to phase converter VC P voltage control phase generator we will call it voltage control phase generator. So again KVCP here this is nothing but delta phi naught by delta VC therefore is nothing but KPD into KVCP that is the DC loop gain. Where are the applications of this so called phase lock loop these are used in fine tuning of filters wherein we had already seen this topic wherein what we had done was we had taken a filter applied the input to the filter. So we have for example if you just have let us say a low pass filter of second order we will say we have put a buffer here. So if this is VI and this is R and this is C R and C this is a buffer we get here VI divided by 1 plus SCR square. So it generates a second order low pass output and this output as a phase lag with respect to input so output phase therefore is going to be minus 2 tan inverse omega CR. S is replaced by j omega so we have this as minus 2 tan inverse omega CR. So if you plot this it will start with zero phase and keeps on going up to pi minus pi going through minus pi by 2 at omega equal to 1 by RC each of the network gives a phase shift of 45 degree and if now this R and C or C can be varied with respect to control voltage then it becomes a voltage controlled phase generator. So you can vary this R or C okay or both in order to change the phase. So this is a simple voltage controlled phase generator. So it can be therefore used for tuning filters that means we have used this information to tune sort of universal active filter block exactly to the incoming frequency at which the phase shift at the low pass filter output was adjusted to be pi by 2 exactly. And we have shown that at omega equal to omega naught equal to 1 over RC the phase shift between input and output of the low pass filter is exactly pi by 2 that is the phase locking. So creating quadrature signals needed for communication applications so this quadrature pi by 2 phase shift these are needed normally in communication applications generation of multi phase signals independent of frequency that is from single phase to for example 3 phase conversion. So this can be easily done by shifting the phase independent of frequency to 120 degrees and minus 120 degrees. So phase modulation direct phase modulation this is a circuit we can directly phase modulate frequency remaining the same distortion and spectrum analyzer we have described how it can be used for tuning on to fundamental and the harmonics thereby being able to be used as distortion analyzer that means given a setup whose distortion is to be measured you give the input a pure sine wave and take the output and feed it to a self tune filter which will give the fundamental at is band pass output and the rest of the harmonic at the notch output. So the notch output RMS value directly gives you the % distortion knowing the fundamental amplitude PLL functioning input to the phase lock loop is Phi I which is the reference phase which is taken to be 0 output of concern is Phi naught. So that is fixed let us say at pi by 2 or something like that 60 degree anything that you decide Delta Phi naught by Delta Phi I of such system is going to be K1 K2 by 1 plus K1 K2 this is GL by 1 plus GL okay it is also equal to 1 by 1 plus 1 over the loop gain 1 by 1 plus 1 over loop gain this is the standard ahh feedback ahh system design that we have already discussed time and again where GL is the negative feedback loop gain it is negative feedback that way it becomes 1 plus 1 over loop gain Delta Phi naught by Delta Phi I should be equal to 1 if the loop gain GL is much greater than 1 okay we know that it is going to be phase locked system okay. Now what are these 2 blocks that we have used to convert phase 2 voltage a phase detector is used this again we had discussed while discussing filter tuning so VP sin omega T is 1 input VP dash sin omega T plus Phi is the other input omega is the frequency at which input is coming 2 pi f so this is the low pass filter so VP VP dash by 10 sin omega T sin omega T plus Phi is the output at this point so we get this as VP VP dash by 20 cos Phi cos A minus B minus cos A plus B 2 omega T plus Phi otherwise this is got rid of by the low pass filter that means omega is much greater than 1 by RC. So if it is designed such that this is the case that means RC has to be chosen to be much greater than 1 by omega so if that is done then we get this DC output VP VP dash by 20 cos Phi and this if you want the phase shift to be 90 degree for example this factor V average should be going to 0 that means cos Phi should be 0 so Phi is pi by 2 and K1 is KPD by 1 plus SCR the low pass filter okay and KPD is Delta V average by Delta Phi which is minus VP VP dash by 20 sin Phi. So that is why this product is going to result in a negative feedback okay as far the system is concerned KPD is maximum at Phi is equal to Phi by 2 that is minus VP VP dash by 20 and is 0 at Phi equal to 0 and Phi this is to be noted particularly because it is gradually going to 0 from the maximum the sensitivity. So zero sensitivity means the loop if at all this phase detector is put in a loop the loop gets broken okay so loop gain equal to 0 can be controlled by this by these two limits that means that is the limit over which right loop gain much greater than 1 can be achieved only within this limit right because it is going to be 0 both at Phi equal to 0 and Phi that is in between okay hopefully GL is going to be much greater than 1. So that is the characteristic of the V average cos Phi and the sensitivity is going to be a sin Phi characteristic okay minus sin Phi so it will be going to maximum okay at this point and then going back to zero so that is the sin Phi characteristic minus sin Phi VP dash by 20. In order to detect phase a phase detector is effective only maybe somewhere within this region where this sensitivity into the other sensitivity make the loop gain much greater than 1 that means it is constrained to detect phase difference around Phi by 2 by only a limited magnitude around this range that is what is called the lock range of the phase lock loop okay. Now it is also true that if this is resulting in negative feedback of the entire system okay the slope here being negative okay this one will result in positive feedback that means a shift of phase by Phi from this plus Phi by 2 to minus Phi by 2 changes the slope from negative to positive okay. So what it means is Delta V average by Delta Phi in this case is negative whereas it is positive okay that is around Phi by 2 phase shift cohescent whereas Delta V average by Delta Phi around minus Phi by 2 is positive please remember this because it can if this is resulting in negative feedback the other one the operating range results in positive feedback that is shifting ahh thing from let us say that this is the minus Phi by 2 maybe a lag okay so this may be the resulting in let us say negative feedback then plus Phi by 2 operating region will result in positive feedback and vice versa. Now it can make the sensitivity high maximum and remain same as the maximum throughout by a simple technique of linearizing the phase detector that can be done by putting ahh 2 limiters or these are called zero crossing detectors is convert design wave at the input to a square wave with amplitude let us say VP here and shifted wave square wave okay with let us say shift of Phi with VP dash then what happens is multiplication results in this becoming VP VP dash by 10 and changing from ahh let us say when both are positive negative this is positive or both are positive it is positive okay. So it changes from negative to positive in half to period of the original signal okay and then again changes from negative to positive in the other half the 2 omega component is there apart from that average component corresponds to this which is nothing but minus VP VP dash by 10 that area into Phi that is this area negative area plus the positive area VP VP dash by 10 into Pi minus Phi this is Pi so this is Pi minus Phi. So the average becomes VP VP dash by 10 into 1 minus 2 Phi by Pi and it just becomes a linear phase detector and KPD is constant throughout the operating range and from 0 to Pi which is minus VP VP dash by 10 into 2 by Pi differentiate this this is nothing but Delta V average by Delta Phi which is what we have called as KPD it is negative and remains constant. So it is better to operate this around Phi by 2 so that the dynamic range over which the locking functions phase locking occurs or phase following occurs is maximum of Pi by 2 from Pi by 2 going all the way up to 0 and again Pi by 2 from Pi by 2 going all the way up to Pi. So that is the range over which locking can take place and beyond this it goes if this is negative feedback this will be positive again if you shift it by Pi again it becomes positive feedback. So if this is resulting in positive feedback shift it by Pi and it will become negative feedback shift the input by Pi output by Pi and VD to the phase detector. So this is the technique of negative feedback in this phase lock loop so this is going to be drawn here this way it keeps on repeating this is at Pi by 2 this is the maximum VP VP dash by 10 okay and this should be the operating point or this should be operating point okay. So this is the phase detector which is linearized now this is the voltage control phase generator. So what is now happening is it is nothing but a low pass filter as demonstrated earlier so let us say the low pass filter Omega naught which is 1 over RC okay is controlled by a voltage then DeFi by DVC or the voltage control phase generator which is the second block K2 okay whose KVCP is what we are trying to now define is going to be Delta Phi by Delta VC this is the phase generator output this is the input okay which is coming to the phase lock loop the phase detector also so it is coming to this also so that is going to result in Delta Phi by Delta Omega naught this is characterized by the low pass filter let us say or high pass filter so characteristic and Delta Omega naught by Delta VC is characterized by how we are changing the element R and C by means of a voltage so this is a voltage control resistor or voltage control capacitor or we will see later that multiplier okay feeding its output to a resistance or capacitance can act as a voltage control resistor or capacitor respectively such an arrangement is shown here so here the polarity minus plus okay so this is nothing but an inverting amplifier with this converting voltage to current VI by R that is flowing through this Z okay and results in Z into VI by R so that will result in a transfer function V naught by VI for one block as one because R R so it is one divided by one plus yes C dash C dash is C plus effect of C by 2 so C dash into R so C dash is nothing but C shunted by C by 2 modified by VC divided by 10 so this is VC so this output is VC if this is V naught 1 VC V naught 1 by 10 so the current in this is VC V naught 1 by 10 into SC by 2 this into SC divided by 2 so this results in the capacitor appearing as okay C VC by 20 so that is the C dash so omega naught is equal to 1 by C dash R for this low pass filter so the other one also is exactly identical to this so the transfer function is the square which results in a phase shift from here to here by an amount which is changing from 0 to pi minus pi with that going to minus pi by 2 at omega naught equal to or omega naught by omega I by omega naught equal to 1 omega naught is equal to 1 over CR so at omega equal to 1 over CR the phase shift is pi by 2 so this is normalized frequency omega I by omega naught omega naught equal to 1 over CR so now actually omega I the incoming frequency is fixed what is changed in the voltage control phase generator is the omega naught as decided by this that is voltage control so that is why this is a voltage control phase generator so delta phi by delta VC is delta phi by delta omega naught which can be evaluated from this the phase shift is changing this way 0 to minus pi going through minus pi by 2 at omega equal to omega naught what we are interested in as the variable here is not omega I please remember it is omega naught which is variable that means if this is omega I by omega naught is X omega I by omega naught is X with omega I as variable now omega naught as variable it becomes 1 over X okay that is the plot is going to change the slope when you have omega naught as the variable as against omega I as the variable omega I being fixed omega naught is changed by the control voltage so this characteristic slope is going to change to let us say if this is negative that will become positive this way so this has to be one in mind so we have here these 2 blocks which are put and specific value of R equal to 1 K C equal to 1 micro farad has been used okay this is 0.5 micro farad C by 2 and now the omega naught can be varied over a certain limit okay based on the saturation state of the op amp and the incoming input saturation state here so let us assume that the multiplier is working only up to plus minus 10 volts it is having VXVI by 10 as its output so all these precision multipliers have been designed to work over the dynamic range of plus minus 10 volts. So in which case the capacitance C varies from C minus C by 2 to C dash this is the range C plus C by 2 that means effectively C by 2 to 3 C by 2 that means omega naught changes over a range of 2 by RC to let us say 2 thirds RC so that is the lock range is the highest frequency this is the lowest frequency so that is called the lock range of this phase lock loop we are designed it such that this becomes the lock range let us say for the example chosen. Now you will find that input is not directly fed here this introduces in the loop a negative sign the phase detector itself right along with the phase generator which is corresponding to delta this amplifier is introducing a design the rest of the block therefore has to introduce a positive sign so that it is overall negative feedback that means it is converting the delta phi by delta VC is nothing but delta phi by this is the phase generator here delta omega naught into delta omega naught by delta VC so this has to be positive okay delta omega naught by delta VC here is omega naught is equal to 1 by RC into 1 plus VC by 10 so we have delta omega naught by delta VC negative negative okay and therefore this has to be positive that means delta phi by delta omega naught is delta omega naught by delta VC is negative so this has to be positive okay this whole thing has to be positive that means delta phi by delta omega naught has to be negative so that is facilitated by changing the phase shift of this from 0 to phi here that means this inverter simply changes the phase shift to phi so that the whole contribution due to the voltage to phase converter okay is going to be let us say because the phase detector okay is going to give you a phase of KPD of negative value for increasing let us say frequency the phase shift is negative KPD is negative then this whole thing has to be giving another negative value okay and that is done by this so overall the phase detector okay is going to give you a positive sign for delta phi by delta VC so this is if this is negative there can be this negative and this negative this positive and this positive so these are the combinations if this is positive this has to be positive if this is negative this also has to be negative so that KVCF and the KPD overall should be always positive so that this one negative sign is taken care of okay in the negative feedback topology. So VP sin omega reference T is fed both to the phase detector as well as voltage control phase generator so this is not VCF it is VCPE phase variation of the low pass filter as a function of omega omega means omega I that is what is plotted by us it is this inverse of this that is this characteristic which is going like this with positive slope that is of interest because omega I is fixed and omega naught is variable so if you now plot this as omega I by omega naught what is actually required is in omega naught by omega I plot it will look like this so delta phi naught by delta phi I is going to be equal to 1 by 1 plus GL 1 by 1 plus what is GL here 1 by GL so mistake here 1 by 1 plus 1 over GL as pointed out earlier in the earlier slides and what is GL? GL is nothing but this is negative KVCP into KPD okay KVCP KPD into 1 over okay SC dash or dash of the integrator. So this 1 over GL now becomes 1 by 1 plus SC dash or dash divided by KVCP KPD so which is 1 by 1 plus S by bandwidth of the phase log it is a first order system this way if you now include the gain bandwidth product of the op amp which is acting as an integrator we know that the loop gain as to change from this to okay that 1 over SC dash or dash which is the ideal transfer function of the integrator gets modified this we have seen earlier also by additional phase error caused by the finite gain bandwidth product. So this 1 plus S by GB comes into picture as a product 1 plus S by GB which will make this 1 by 1 plus S by bandwidth plus S square by gain bandwidth product into bandwidth of the phase log loop. So this results in this system becoming a second order and we can optimize this if you make the bandwidth same as the gain bandwidth product of the op amp then Q is equal to 1 so that means there will be just in the transient response of the phase log loop there will be just 1 peak and it comes to steady state as phase is changed by a step suddenly from right the phase jumps from say 5 1 to 5 2 then at the input then the output characteristic of the phase will be just having just 1 peak just the optimum highest speed system rate of rise is maximum and simultaneously the number of ripples are minimized now that loop has been made with these values of resistors and capacitors so you can see that when the control voltage is 0 now contribution of capacitor occurs due to C by 2 so omega naught is 1 by RC which is R is 1K and C is 1 micro field so that frequency is 159 hertz okay and with the limiting values of VC to plus minus 10 volts the range has been obtained for locking of the system at 90 degree phase shift. So this is one frequency close to the positive saturation value so at which point it is 100 hertz so since omega naught is inversely proportional VC this is the minimum frequency on one side okay and when VC is 0 it is 159 hertz and when VC is going negative it is going to 200 hertz here as the input okay frequency is 200 hertz now you can see the control voltage is going negative this is positive near saturation this is negative but much below saturation that means it can go still higher 160 hertz you can see the VC is close to 0 159 most calculated value and again the phase shift is exactly 90 degrees and amplitude because it is phase locked it is magnitude locked also the amplitude if the input remains the same output amplitude also remains same for all frequencies because omega CR is a constant okay so a phase lock loop is also a magnitude lock loop okay range of input frequency for with the specified phase locking occurs is now going to change from 2 by RC to 2 by 3 RC that is 318 hertz 206 hertz we have seen it locking nicely from 200 hertz to 100 nearly about 100 hertz right yes it has gone over much about saturation above 10 volts okay that is why it has slightly changed okay so that is the reason VC has gone above 10 volts right that is why it has gone all the way up to 100 hertz without any problem of locking okay the phase modulation this has been discussed in self tuned filter this was the filter that has been built for self tuning action and we have seen that whatever be the input frequency the filter gets locked on to the incoming frequency so if it is a square wave you get the fundamental at the band pass output okay here and at the notch output you do not get anything so that has been highlighted in this for a specific input incoming frequency which is 2000 hertz we see that it is very nearly control voltage is very nearly settling at zero okay for the given values of RNC and it settling down to zero and this is the band pass output gradually growing and getting maximum this is the notch output going to zero so it has been self tuned tuned to 2000 then what happens let us see it is locked this is the low pass output compared to the input and you can see that it is having 90 degree phase shift low pass output so again another frequency that is 1000 hertz the volt on to negative from zero some value so it has got tuned to that incoming frequency of 1000 and you can see the notch output going to zero once again this is the effect of the second order system that has resulted in the control low okay becoming second order so you can see just one peak here so it is having a Q equal to very nearly equal to one okay so this is the 90 degree phase shift for that frequency of 1000 hertz this is the phase locking occurring at 90 degrees for the low pass filter output so now we can go over to frequency followers so what is a frequency follower please remember that in this particular case right if you change over from low pass to high pass it will become positive feedback and it will not function right one sign change occurs right so in this case of the circuit please remember that R is varied that means delta what is that omega not by delta VC is going to be positive because it is R shunted by 0.8 R okay into 10 by VC and you put a multiplier okay sorry it is this way this is 0.8 R shunted by R into 10 by VC so the resistance that is varied is inversely proportional to VC that means omega naught is going to be directly proportional to VC is directly proportional to VC so delta omega naught by delta VC is positive unlike the previous case of phase lock loop where we had delta omega naught by delta VC becoming negative because C was C VC by 10 so that is reference that means actually this being positive we have to have this delta phi by delta omega naught okay into this being positive in this particular case we have to select it as positive so that overall loop gain is negative so we have directly taken the output from the low pass filter and fed it to the of our universal active filter block okay and fed it to the integrator it gives a negative sign so this conversion factor has to be positive for negative feedback okay okay so we have seen this that we have now frequency lock loop to discuss and what is it this is frequency to voltage converter which is the same as phase detector now what I have to mention is when we feed sin theta 1 sin theta 2 to phase detector it just takes theta 1 minus theta 2 okay of cos and cos theta 1 plus theta 2 this is the overall phase which is made up of omega t frequency plus the actual phase difference phi so if frequencies are different it can be omega 1 t plus phi and this can be omega 2 t so then this component becomes omega 1 minus omega 2 into t okay so that plus phi and this component becomes actually phi becomes meaningless because the change of phase okay is incorporated in change frequency difference in frequency so we have this becoming omega 1 plus omega 2 into t so if this is removed by the low pass filter the low pass filter will correspond to this so the same arrangement of phase detector with low pass filter is going to be sufficient for frequency detection at the input so you do not know the difference as far as K2 is concerned so this K1 remains same as KPD but K2 on the other hand now becomes voltage to frequency converter which is nothing but a VCO so it is an independent oscillator controlled by a voltage so instead of KVCP we have to use KVCO as the sensitivity factor so where are these frequency locked loops used these are popularly known as PLL and used in signal conditioning in repeater stations okay so what is done here is that the signal gets corrupted by the transmission line and gets distorted the it can be restored to higher power level by using a frequency follower so PLL as a frequency follower is used in repeater stations in microwave one microwave station to another microwave station okay several repeater stations are there for signal conditioning that means improve the signal to noise ratio right so that is what is done simply by using a PLL and taking the output from the VCO which is going to be frequency selective VC and therefore it gets rid of the noise and power level can be improved signal power level wanted signal power level can be improved generating stable carrier for and clock signals up to microwave frequencies one of the most important uses of the PLL FSK detection in modem I told you that VCO put in a feedback loop results in inverse function so FSK detection FSK generator is VCO and FSK detection is done in the PLL and they are part of modems modulator means VCO the modulator means PLL FM detection okay and AM detection can also be done okay using this just as FSK detection is that AM detection it can be removing the amplitude modulation by making it go through a limiter and that can recover the carrier and carrier that recovered carrier okay shifted in phase by angle of pi by 2 is again multiplied by the incoming AM to give AM detection we will see this application later clock recovery obviously right so it can synchronize with the incoming clock if it is tuned properly precision speed control of motors the time of voltage reference is over and time reference and frequency references are coming to be used more and more often than voltage or current as reference now phase detector as frequency detector so VP sin 2 omega 2 pi of t VP sin 2 pi of t plus 5 now is replaced by VP sin omega 1 t VP sin omega 2 t as pointed out earlier so this will hopefully give an output VP VP dash by 20 cos omega 1 minus omega 2 t the some component being eliminated by this low pass filter that is omega 1 plus omega 2 should be much greater than 1 by RC so frequency by definition this is very important fundamentally omega is equal to delta 5 by delta t rate of change of phase is radiant frequency output you have to multiply being this sin omega i t sin omega naught t plus 5 you get this as cos omega i minus omega naught t omega i plus omega naught t okay and delta phi naught by delta phi i is going to be 1 by 1 plus okay 1 by GL okay when GL is much greater than 1 then delta phi naught by delta phi i is equal to 1 this we have already shown in PLL in which case delta phi naught by delta t divided by delta phi i by delta t also should be equal to 1 and therefore what is delta phi naught by delta t is output frequency absolute value and delta phi i by delta t is absolute value of input frequency output frequency is exactly equal to incoming frequency so this is what is assumed while analyzing the PLL later on we will see right that output frequency should be always equal to input frequency if it is phase lock so that is why the frequency lock loop got its name as phase lock loop frequency locking phase locking means frequency locking the two frequencies are different these are the basic building blocks let us see incoming frequency is applied to the phase detector one input the other input to the phase detector is one that gets the incoming frequency as the input and converts it into a suitable phase from the incoming frequency so the frequency remains the same as the incoming frequency but phase is now controlled by VC so that is called a voltage control phase generator and that gives the output here and the phase locking takes place we reference if it is 0 and K need not be there if this sensitivities are high enough the loop can be made high so you can just connect the output of the low pass filter directly to this to form a basic phase lock loop or frequency lock loop now in this particular case that phase locking occurs for pi by 2 90 degrees your reference is 0 same thing happens here reference is 0 phase locking occurs at pi by 2 but output frequency is going to be same as incoming frequency so this is replaced by VCO with KVCO as its sensitivity factor so these are the other sensitivity factors here so that is FLL so this link from this to this is broken and it is VCP is replaced by VCO that is the major difference and what does this difference do to FLL so we will see this in the next class on FLL so that is the difference between the true PLL and the FLL in conclusion we have seen that we have to first understand the PLL okay as the true phase lock loop which is primarily used for tuning filters okay or generating multi phase signal from a single phase of any frequency so phase can be maintained absolutely constant and also used for phase modulation directly okay circuit doing phase modulation this is a simple circuit the next one discussed is a variant of the phase lock loop which is popularly called as PLL but it is truly frequency lock loop so this frequency lock loop contains as a basic building block the phase detector and followed by the loop pass filter and an amplifier the blue stop the loop gain if necessary otherwise it may not be necessary if the sensitive factors themselves are pretty high so this is fed to the VCO now which is an independent oscillator controlled by a DC voltage at the input the frequency of each is controlled by a DC voltage it is this phase lock loop which is called as FLL that we are going to discuss in detail in the next class thank you very much.